loo/0000755000176200001440000000000014411522703011042 5ustar liggesusersloo/NAMESPACE0000644000176200001440000001007514411130104012252 0ustar liggesusers# Generated by roxygen2: do not edit by hand S3method("$",loo) S3method("[",loo) S3method("[[",loo) S3method(E_loo,default) S3method(E_loo,matrix) S3method(ap_psis,array) S3method(ap_psis,default) S3method(ap_psis,matrix) S3method(crps,matrix) S3method(crps,numeric) S3method(dim,importance_sampling) S3method(dim,kfold) S3method(dim,loo) S3method(dim,psis_loo) S3method(dim,waic) S3method(elpd,array) S3method(elpd,matrix) S3method(importance_sampling,array) S3method(importance_sampling,default) S3method(importance_sampling,matrix) S3method(loo,"function") S3method(loo,array) S3method(loo,matrix) S3method(loo_approximate_posterior,"function") S3method(loo_approximate_posterior,array) S3method(loo_approximate_posterior,matrix) S3method(loo_compare,default) S3method(loo_crps,matrix) S3method(loo_model_weights,default) S3method(loo_moment_match,default) S3method(loo_predictive_metric,matrix) S3method(loo_scrps,matrix) S3method(loo_subsample,"function") S3method(nobs,psis_loo_ss) S3method(plot,loo) S3method(plot,psis) S3method(plot,psis_loo) S3method(print,compare.loo) S3method(print,compare.loo_ss) S3method(print,importance_sampling) S3method(print,importance_sampling_loo) S3method(print,loo) S3method(print,pareto_k_table) S3method(print,pseudobma_bb_weights) S3method(print,pseudobma_weights) S3method(print,psis) S3method(print,psis_loo) S3method(print,psis_loo_ap) S3method(print,stacking_weights) S3method(print,waic) S3method(print_dims,importance_sampling) S3method(print_dims,importance_sampling_loo) S3method(print_dims,kfold) S3method(print_dims,psis_loo) S3method(print_dims,psis_loo_ss) S3method(print_dims,waic) S3method(psis,array) S3method(psis,default) S3method(psis,matrix) S3method(relative_eff,"function") S3method(relative_eff,array) S3method(relative_eff,default) S3method(relative_eff,importance_sampling) S3method(relative_eff,matrix) S3method(scrps,matrix) S3method(scrps,numeric) S3method(sis,array) S3method(sis,default) S3method(sis,matrix) S3method(tis,array) S3method(tis,default) S3method(tis,matrix) S3method(update,psis_loo_ss) S3method(waic,"function") S3method(waic,array) S3method(waic,matrix) S3method(weights,importance_sampling) export(.compute_point_estimate) export(.ndraws) export(.thin_draws) export(E_loo) export(compare) export(crps) export(elpd) export(example_loglik_array) export(example_loglik_matrix) export(extract_log_lik) export(find_model_names) export(gpdfit) export(is.kfold) export(is.loo) export(is.psis) export(is.psis_loo) export(is.sis) export(is.tis) export(is.waic) export(kfold) export(kfold_split_grouped) export(kfold_split_random) export(kfold_split_stratified) export(loo) export(loo.array) export(loo.function) export(loo.matrix) export(loo_approximate_posterior) export(loo_approximate_posterior.array) export(loo_approximate_posterior.function) export(loo_approximate_posterior.matrix) export(loo_compare) export(loo_crps) export(loo_i) export(loo_model_weights) export(loo_model_weights.default) export(loo_moment_match) export(loo_moment_match.default) export(loo_predictive_metric) export(loo_scrps) export(loo_subsample) export(loo_subsample.function) export(mcse_loo) export(nlist) export(obs_idx) export(pareto_k_ids) export(pareto_k_influence_values) export(pareto_k_table) export(pareto_k_values) export(print_dims) export(pseudobma_weights) export(psis) export(psis_n_eff_values) export(psislw) export(relative_eff) export(scrps) export(sis) export(stacking_weights) export(tis) export(waic) export(waic.array) export(waic.function) export(waic.matrix) export(weights.importance_sampling) importFrom(matrixStats,colLogSumExps) importFrom(matrixStats,colMaxs) importFrom(matrixStats,colSums2) importFrom(matrixStats,colVars) importFrom(matrixStats,logSumExp) importFrom(parallel,makePSOCKcluster) importFrom(parallel,mclapply) importFrom(parallel,parLapply) importFrom(parallel,stopCluster) importFrom(stats,constrOptim) importFrom(stats,nobs) importFrom(stats,qnorm) importFrom(stats,quantile) importFrom(stats,rgamma) importFrom(stats,rnorm) importFrom(stats,sd) importFrom(stats,setNames) importFrom(stats,update) importFrom(stats,var) importFrom(stats,weights) loo/data/0000755000176200001440000000000014411130104011741 5ustar liggesusersloo/data/milk.rda0000644000176200001440000000260213267637066013420 0ustar liggesusersBZh91AY&SYB)?E$C:Vq^he$I MM шdd24y#5 @4 2 "I@2LFM2h42bi0 b #hi!M4(FTlj=Cڧz# h=G4iQzbSLFhM e5AfUhOt8MMKn6)c?V2ډy UQn{ ,AkO4#-Xlcm7 Ryxukae E|CꠓKŸ@;jQ$ %mm&: T=8$3^W G$ߌ-(W4M/|Fp \ k\8=BrnLxj6ğ51 ^B9xCQY[i$UPs0AA-sU l5PÑgLásU+܁J2V nQ/JK"(Hv!Floo/data/Kline.rda0000644000176200001440000000066513267637066013535 0ustar liggesusersBZh91AY&SYA^0_TIMdߐ@0F@AZ mLSLMFL4GCRi2ibF44ESzjd @@J*Ij#xIB9J 'aMTuC19F 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See below for more on interpreting \code{p_loo} when there are warnings about high Pareto k diagnostic values. } \section{Pareto k estimates}{ The Pareto \code{k} estimate is a diagnostic for Pareto smoothed importance sampling (PSIS), which is used to compute components of \code{elpd_loo}. In importance-sampling LOO (the full posterior distribution is used as the proposal distribution). The Pareto k diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance sampling is not able to give reliable estimate. If \code{k<0.5}, then the corresponding component of \code{elpd_loo} is estimated with high accuracy. If \verb{0.50.7}, then importance sampling is not able to provide useful estimate for that component/observation. Pareto k is also useful as a measure of influence of an observation. Highly influential observations have high k values. Very high k values often indicate model misspecification, outliers or mistakes in data processing. See Section 6 of Gabry et al. (2019) for an example. \subsection{Interpreting \code{p_loo} when Pareto \code{k} is large}{ If \code{k > 0.7} then we can also look at the \code{p_loo} estimate for some additional information about the problem: \itemize{ \item If \verb{p_loo << p} (the total number of parameters in the model), then the model is likely to be misspecified. Posterior predictive checks (PPCs) are then likely to also detect the problem. Try using an overdispersed model, or add more structural information (nonlinearity, mixture model, etc.). \item If \code{p_loo < p} and the number of parameters \code{p} is relatively large compared to the number of observations (e.g., \code{p>N/5}), it is likely that the model is so flexible or the population prior so weak that it’s difficult to predict the left out observation (even for the true model). This happens, for example, in the simulated 8 schools (in VGG2017), random effect models with a few observations per random effect, and Gaussian processes and spatial models with short correlation lengths. \item If \code{p_loo > p}, then the model is likely to be badly misspecified. If the number of parameters \verb{p<N/5} (more accurately we should count number of observations influencing each parameter as in hierarchical models some groups may have few observations and other groups many), it is possible that PPCs won't detect the problem. } } } \section{elpd_diff}{ \code{elpd_diff} is the difference in \code{elpd_loo} for two models. If more than two models are compared, the difference is computed relative to the model with highest \code{elpd_loo}. } \section{se_diff}{ The standard error of component-wise differences of elpd_loo (Eq 24 in VGG2017) between two models. This SE is \emph{smaller} than the SE for individual models due to correlation (i.e., if some observations are easier and some more difficult to predict for all models). } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. \emph{J. R. Stat. Soc. A}, 182: 389-402. doi:10.1111/rssa.12378 (\href{https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssa.12378}{journal version}, \href{https://arxiv.org/abs/1709.01449}{preprint arXiv:1709.01449}, \href{https://github.com/jgabry/bayes-vis-paper}{code on GitHub}) } loo/man/loo_moment_match_split.Rd0000644000176200001440000000720214214417101016640 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/split_moment_matching.R \name{loo_moment_match_split} \alias{loo_moment_match_split} \title{Split moment matching for efficient approximate leave-one-out cross-validation (LOO)} \usage{ loo_moment_match_split( x, upars, cov, total_shift, total_scaling, total_mapping, i, log_prob_upars, log_lik_i_upars, r_eff_i, cores, is_method, ... ) } \arguments{ \item{x}{A fitted model object.} \item{upars}{A matrix containing the model parameters in unconstrained space where they can have any real value.} \item{cov}{Logical; Indicate whether to match the covariance matrix of the samples or not. If \code{FALSE}, only the mean and marginal variances are matched.} \item{total_shift}{A vector representing the total shift made by the moment matching algorithm.} \item{total_scaling}{A vector representing the total scaling of marginal variance made by the moment matching algorithm.} \item{total_mapping}{A vector representing the total covariance transformation made by the moment matching algorithm.} \item{i}{Observation index.} \item{log_prob_upars}{A function that takes arguments \code{x} and \code{upars} and returns a matrix of log-posterior density values of the unconstrained posterior draws passed via \code{upars}.} \item{log_lik_i_upars}{A function that takes arguments \code{x}, \code{upars}, and \code{i} and returns a vector of log-likeliood draws of the \code{i}th observation based on the unconstrained posterior draws passed via \code{upars}.} \item{r_eff_i}{MCMC relative effective sample size of the \code{i}'th log likelihood draws.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{is_method}{The importance sampling method to use. The following methods are implemented: \itemize{ \item \code{\link[=psis]{"psis"}}: Pareto-Smoothed Importance Sampling (PSIS). Default method. \item \code{\link[=tis]{"tis"}}: Truncated Importance Sampling (TIS) with truncation at \code{sqrt(S)}, where \code{S} is the number of posterior draws. \item \code{\link[=sis]{"sis"}}: Standard Importance Sampling (SIS). }} \item{...}{Further arguments passed to the custom functions documented above.} } \value{ A list containing the updated log-importance weights and log-likelihood values. Also returns the updated MCMC effective sample size and the integrand-specific log-importance weights. } \description{ A function that computes the split moment matching importance sampling loo. Takes in the moment matching total transformation, transforms only half of the draws, and computes a single elpd using multiple importance sampling. } \references{ Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2021). Implicitly adaptive importance sampling. \emph{Statistics and Computing}, 31, 16. doi:10.1007/s11222-020-09982-2. arXiv preprint arXiv:1906.08850. } \seealso{ \code{\link[=loo]{loo()}}, \code{\link[=loo_moment_match]{loo_moment_match()}} } loo/man/nobs.psis_loo_ss.Rd0000644000176200001440000000065013575772017015420 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{nobs.psis_loo_ss} \alias{nobs.psis_loo_ss} \title{The number of observations in a \code{psis_loo_ss} object.} \usage{ \method{nobs}{psis_loo_ss}(object, ...) } \arguments{ \item{object}{a \code{psis_loo_ss} object.} \item{...}{Currently unused.} } \description{ The number of observations in a \code{psis_loo_ss} object. } loo/man/pareto-k-diagnostic.Rd0000644000176200001440000001730414407123455015763 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/diagnostics.R \name{pareto-k-diagnostic} \alias{pareto-k-diagnostic} \alias{pareto_k_table} \alias{pareto_k_ids} \alias{pareto_k_values} \alias{pareto_k_influence_values} \alias{psis_n_eff_values} \alias{mcse_loo} \alias{plot.psis_loo} \alias{plot.loo} \alias{plot.psis} \title{Diagnostics for Pareto smoothed importance sampling (PSIS)} \usage{ pareto_k_table(x) pareto_k_ids(x, threshold = 0.5) pareto_k_values(x) pareto_k_influence_values(x) psis_n_eff_values(x) mcse_loo(x, threshold = 0.7) \method{plot}{psis_loo}( x, diagnostic = c("k", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot" ) \method{plot}{psis}( x, diagnostic = c("k", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot" ) } \arguments{ \item{x}{An object created by \code{\link[=loo]{loo()}} or \code{\link[=psis]{psis()}}.} \item{threshold}{For \code{pareto_k_ids()}, \code{threshold} is the minimum \eqn{k} value to flag (default is \code{0.5}). For \code{mcse_loo()}, if any \eqn{k} estimates are greater than \code{threshold} the MCSE estimate is returned as \code{NA} (default is \code{0.7}). See \strong{Details} for the motivation behind these defaults.} \item{diagnostic}{For the \code{plot} method, which diagnostic should be plotted? The options are \code{"k"} for Pareto \eqn{k} estimates (the default) or \code{"n_eff"} for PSIS effective sample size estimates.} \item{label_points, ...}{For the \code{plot()} method, if \code{label_points} is \code{TRUE} the observation numbers corresponding to any values of \eqn{k} greater than 0.5 will be displayed in the plot. Any arguments specified in \code{...} will be passed to \code{\link[graphics:text]{graphics::text()}} and can be used to control the appearance of the labels.} \item{main}{For the \code{plot()} method, a title for the plot.} } \value{ \code{pareto_k_table()} returns an object of class \code{"pareto_k_table"}, which is a matrix with columns \code{"Count"}, \code{"Proportion"}, and \code{"Min. n_eff"}, and has its own print method. \code{pareto_k_ids()} returns an integer vector indicating which observations have Pareto \eqn{k} estimates above \code{threshold}. \code{pareto_k_values()} returns a vector of the estimated Pareto \eqn{k} parameters. These represent the reliability of sampling. \code{pareto_k_influence_values()} returns a vector of the estimated Pareto \eqn{k} parameters. These represent influence of the observations on the model posterior distribution. \code{psis_n_eff_values()} returns a vector of the estimated PSIS effective sample sizes. \code{mcse_loo()} returns the Monte Carlo standard error (MCSE) estimate for PSIS-LOO. MCSE will be NA if any Pareto \eqn{k} values are above \code{threshold}. The \code{plot()} method is called for its side effect and does not return anything. If \code{x} is the result of a call to \code{\link[=loo]{loo()}} or \code{\link[=psis]{psis()}} then \code{plot(x, diagnostic)} produces a plot of the estimates of the Pareto shape parameters (\code{diagnostic = "k"}) or estimates of the PSIS effective sample sizes (\code{diagnostic = "n_eff"}). } \description{ Print a diagnostic table summarizing the estimated Pareto shape parameters and PSIS effective sample sizes, find the indexes of observations for which the estimated Pareto shape parameter \eqn{k} is larger than some \code{threshold} value, or plot observation indexes vs. diagnostic estimates. The \strong{Details} section below provides a brief overview of the diagnostics, but we recommend consulting Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for full details. } \details{ The reliability and approximate convergence rate of the PSIS-based estimates can be assessed using the estimates for the shape parameter \eqn{k} of the generalized Pareto distribution: \itemize{ \item If \eqn{k < 0.5} then the distribution of raw importance ratios has finite variance and the central limit theorem holds. However, as \eqn{k} approaches \eqn{0.5} the RMSE of plain importance sampling (IS) increases significantly while PSIS has lower RMSE. \item If \eqn{0.5 \leq k < 1}{0.5 <= k < 1} then the variance of the raw importance ratios is infinite, but the mean exists. TIS and PSIS estimates have finite variance by accepting some bias. The convergence of the estimate is slower with increasing \eqn{k}. If \eqn{k} is between 0.5 and approximately 0.7 then we observe practically useful convergence rates and Monte Carlo error estimates with PSIS (the bias of TIS increases faster than the bias of PSIS). If \eqn{k > 0.7} we observe impractical convergence rates and unreliable Monte Carlo error estimates. \item If \eqn{k \geq 1}{k >= 1} then neither the variance nor the mean of the raw importance ratios exists. The convergence rate is close to zero and bias can be large with practical sample sizes. } \subsection{What if the estimated tail shape parameter \eqn{k} exceeds \eqn{0.5}}{ Importance sampling is likely to work less well if the marginal posterior \eqn{p(\theta^s | y)} and LOO posterior \eqn{p(\theta^s | y_{-i})} are very different, which is more likely to happen with a non-robust model and highly influential observations. If the estimated tail shape parameter \eqn{k} exceeds \eqn{0.5}, the user should be warned. (Note: If \eqn{k} is greater than \eqn{0.5} then WAIC is also likely to fail, but WAIC lacks its own diagnostic.) In practice, we have observed good performance for values of \eqn{k} up to 0.7. When using PSIS in the context of approximate LOO-CV, we recommend one of the following actions when \eqn{k > 0.7}: \itemize{ \item With some additional computations, it is possible to transform the MCMC draws from the posterior distribution to obtain more reliable importance sampling estimates. This results in a smaller shape parameter \eqn{k}. See \code{\link[=loo_moment_match]{loo_moment_match()}} for an example of this. \item Sampling directly from \eqn{p(\theta^s | y_{-i})} for the problematic observations \eqn{i}, or using \eqn{k}-fold cross-validation will generally be more stable. \item Using a model that is more robust to anomalous observations will generally make approximate LOO-CV more stable. } } \subsection{Observation influence statistics}{ The estimated shape parameter \eqn{k} for each observation can be used as a measure of the observation's influence on posterior distribution of the model. These can be obtained with \code{pareto_k_influence_values()}. } \subsection{Effective sample size and error estimates}{ In the case that we obtain the samples from the proposal distribution via MCMC the \strong{loo} package also computes estimates for the Monte Carlo error and the effective sample size for importance sampling, which are more accurate for PSIS than for IS and TIS (see Vehtari et al (2019) for details). However, the PSIS effective sample size estimate will be \strong{over-optimistic when the estimate of \eqn{k} is greater than 0.7}. } } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \itemize{ \item \code{\link[=psis]{psis()}} for the implementation of the PSIS algorithm. \item The \href{https://mc-stan.org/loo/articles/online-only/faq.html}{FAQ page} on the \strong{loo} website for answers to frequently asked questions. } } loo/man/loo_subsample.Rd0000644000176200001440000002166614407123455014771 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{loo_subsample} \alias{loo_subsample} \alias{loo_subsample.function} \title{Efficient approximate leave-one-out cross-validation (LOO) using subsampling} \usage{ loo_subsample(x, ...) \method{loo_subsample}{`function`}( x, ..., data = NULL, draws = NULL, observations = 400, log_p = NULL, log_g = NULL, r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), loo_approximation = "plpd", loo_approximation_draws = NULL, estimator = "diff_srs", llgrad = NULL, llhess = NULL ) } \arguments{ \item{x}{A function. The \strong{Methods (by class)} section, below, has detailed descriptions of how to specify the inputs.} \item{data, draws, ...}{For \code{loo_subsample.function()}, these are the data, posterior draws, and other arguments to pass to the log-likelihood function. Note that for some \code{loo_approximation}s, the draws will be replaced by the posteriors summary statistics to compute loo approximations. See argument \code{loo_approximation} for details.} \item{observations}{The subsample observations to use. The argument can take four (4) types of arguments: \itemize{ \item \code{NULL} to use all observations. The algorithm then just uses standard \code{loo()} or \code{loo_approximate_posterior()}. \item A single integer to specify the number of observations to be subsampled. \item A vector of integers to provide the indices used to subset the data. \emph{These observations need to be subsampled with the same scheme as given by the \code{estimator} argument}. \item A \code{psis_loo_ss} object to use the same observations that were used in a previous call to \code{loo_subsample()}. }} \item{log_p, log_g}{Should be supplied only if approximate posterior draws are used. The default (\code{NULL}) indicates draws are from "true" posterior (i.e. using MCMC). If not \code{NULL} then they should be specified as described in \code{\link[=loo_approximate_posterior]{loo_approximate_posterior()}}.} \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo_subsample()} be saved in the returned object? See \code{\link[=loo]{loo()}} for details.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{loo_approximation}{What type of approximation of the loo_i's should be used? The default is \code{"plpd"} (the log predictive density using the posterior expectation). There are six different methods implemented to approximate loo_i's (see the references for more details): \itemize{ \item \code{"plpd"}: uses the lpd based on point estimates (i.e., \eqn{p(y_i|\hat{\theta})}). \item \code{"lpd"}: uses the lpds (i,e., \eqn{p(y_i|y)}). \item \code{"tis"}: uses truncated importance sampling to approximate PSIS-LOO. \item \code{"waic"}: uses waic (i.e., \eqn{p(y_i|y) - p_{waic}}). \item \code{"waic_grad_marginal"}: uses waic approximation using first order delta method and posterior marginal variances to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad_marginal). Requires gradient of likelihood function. \item \code{"waic_grad"}: uses waic approximation using first order delta method and posterior covariance to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient of likelihood function. \item \code{"waic_hess"}: uses waic approximation using second order delta method and posterior covariance to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient and Hessian of likelihood function. } As point estimates of \eqn{\hat{\theta}}, the posterior expectations of the parameters are used.} \item{loo_approximation_draws}{The number of posterior draws used when integrating over the posterior. This is used if \code{loo_approximation} is set to \code{"lpd"}, \code{"waic"}, or \code{"tis"}.} \item{estimator}{How should \code{elpd_loo}, \code{p_loo} and \code{looic} be estimated? The default is \code{"diff_srs"}. \itemize{ \item \code{"diff_srs"}: uses the difference estimator with simple random sampling (srs). \code{p_loo} is estimated using standard srs. \item \code{"hh"}: uses the Hansen-Hurwitz estimator with sampling proportional to size, where \code{abs} of loo_approximation is used as size. \item \code{"srs"}: uses simple random sampling and ordinary estimation. }} \item{llgrad}{The gradient of the log-likelihood. This is only used when \code{loo_approximation} is \code{"waic_grad"}, \code{"waic_grad_marginal"}, or \code{"waic_hess"}. The default is \code{NULL}.} \item{llhess}{The hessian of the log-likelihood. This is only used with \code{loo_approximation = "waic_hess"}. The default is \code{NULL}.} } \value{ \code{loo_subsample()} returns a named list with class \code{c("psis_loo_ss", "psis_loo", "loo")}. This has the same structure as objects returned by \code{\link[=loo]{loo()}} but with the additional slot: \itemize{ \item \code{loo_subsampling}: A list with two vectors, \code{log_p} and \code{log_g}, of the same length containing the posterior density and the approximation density for the individual draws. } } \description{ Efficient approximate leave-one-out cross-validation (LOO) using subsampling } \details{ The \code{loo_subsample()} function is an S3 generic and a methods is currently provided for log-likelihood functions. The implementation works for both MCMC and for posterior approximations where it is possible to compute the log density for the approximation. } \section{Methods (by class)}{ \itemize{ \item \code{loo_subsample(`function`)}: A function \code{f()} that takes arguments \code{data_i} and \code{draws} and returns a vector containing the log-likelihood for a single observation \code{i} evaluated at each posterior draw. The function should be written such that, for each observation \code{i} in \code{1:N}, evaluating \if{html}{\out{

}}\preformatted{f(data_i = data[i,, drop=FALSE], draws = draws) }\if{html}{\out{
}} results in a vector of length \code{S} (size of posterior sample). The log-likelihood function can also have additional arguments but \code{data_i} and \code{draws} are required. If using the function method then the arguments \code{data} and \code{draws} must also be specified in the call to \code{loo()}: \itemize{ \item \code{data}: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation \code{i}, the \code{i}th row of \code{data} will be passed to the \code{data_i} argument of the log-likelihood function. \item \code{draws}: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike \code{data}, which is indexed by observation, for each observation the entire object \code{draws} will be passed to the \code{draws} argument of the log-likelihood function. \item The \code{...} can be used if your log-likelihood function takes additional arguments. These arguments are used like the \code{draws} argument in that they are recycled for each observation. } }} \references{ Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2019). Leave-One-Out Cross-Validation for Large Data. In \emph{International Conference on Machine Learning} Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. In \emph{International Conference on Artificial Intelligence and Statistics (AISTATS)} } \seealso{ \code{\link[=loo]{loo()}}, \code{\link[=psis]{psis()}}, \code{\link[=loo_compare]{loo_compare()}} } loo/man/print_dims.Rd0000644000176200001440000000162013575772017014272 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/print.R \name{print_dims} \alias{print_dims} \alias{print_dims.importance_sampling} \alias{print_dims.psis_loo} \alias{print_dims.importance_sampling_loo} \alias{print_dims.waic} \alias{print_dims.kfold} \alias{print_dims.psis_loo_ss} \title{Print dimensions of log-likelihood or log-weights matrix} \usage{ print_dims(x, ...) \method{print_dims}{importance_sampling}(x, ...) \method{print_dims}{psis_loo}(x, ...) \method{print_dims}{importance_sampling_loo}(x, ...) \method{print_dims}{waic}(x, ...) \method{print_dims}{kfold}(x, ...) \method{print_dims}{psis_loo_ss}(x, ...) } \arguments{ \item{x}{The object returned by \code{\link[=psis]{psis()}}, \code{\link[=loo]{loo()}}, or \code{\link[=waic]{waic()}}.} \item{...}{Ignored.} } \description{ Print dimensions of log-likelihood or log-weights matrix } \keyword{internal} loo/man/E_loo.Rd0000644000176200001440000000763714407123455013164 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/E_loo.R \name{E_loo} \alias{E_loo} \alias{E_loo.default} \alias{E_loo.matrix} \title{Compute weighted expectations} \usage{ E_loo(x, psis_object, ...) \method{E_loo}{default}( x, psis_object, ..., type = c("mean", "variance", "quantile"), probs = NULL, log_ratios = NULL ) \method{E_loo}{matrix}( x, psis_object, ..., type = c("mean", "variance", "quantile"), probs = NULL, log_ratios = NULL ) } \arguments{ \item{x}{A numeric vector or matrix.} \item{psis_object}{An object returned by \code{\link[=psis]{psis()}}.} \item{...}{Arguments passed to individual methods.} \item{type}{The type of expectation to compute. The options are \code{"mean"}, \code{"variance"}, and \code{"quantile"}.} \item{probs}{For computing quantiles, a vector of probabilities.} \item{log_ratios}{Optionally, a vector or matrix (the same dimensions as \code{x}) of raw (not smoothed) log ratios. If working with log-likelihood values, the log ratios are the \strong{negative} of those values. If \code{log_ratios} is specified we are able to compute \link[=pareto-k-diagnostic]{Pareto k} diagnostics specific to \code{E_loo()}.} } \value{ A named list with the following components: \describe{ \item{\code{value}}{ The result of the computation. For the matrix method, \code{value} is a vector with \code{ncol(x)} elements, with one exception: when \code{type="quantile"} and multiple values are specified in \code{probs} the \code{value} component of the returned object is a \code{length(probs)} by \code{ncol(x)} matrix. For the default/vector method the \code{value} component is scalar, with one exception: when \code{type} is \code{"quantile"} and multiple values are specified in \code{probs} the \code{value} component is a vector with \code{length(probs)} elements. } \item{\code{pareto_k}}{ Function-specific diagnostic. If \code{log_ratios} is not specified when calling \code{E_loo()}, \code{pareto_k} will be \code{NULL}. Otherwise, for the matrix method it will be a vector of length \code{ncol(x)} containing estimates of the shape parameter \eqn{k} of the generalized Pareto distribution. For the default/vector method, the estimate is a scalar. } } } \description{ The \code{E_loo()} function computes weighted expectations (means, variances, quantiles) using the importance weights obtained from the \link[=psis]{PSIS} smoothing procedure. The expectations estimated by the \code{E_loo()} function assume that the PSIS approximation is working well. \strong{A small \link[=pareto-k-diagnostic]{Pareto k} estimate is necessary, but not sufficient, for \code{E_loo()} to give reliable estimates.} Additional diagnostic checks for gauging the reliability of the estimates are in development and will be added in a future release. } \examples{ \donttest{ if (requireNamespace("rstanarm", quietly = TRUE)) { # Use rstanarm package to quickly fit a model and get both a log-likelihood # matrix and draws from the posterior predictive distribution library("rstanarm") # data from help("lm") ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14) trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69) d <- data.frame( weight = c(ctl, trt), group = gl(2, 10, 20, labels = c("Ctl","Trt")) ) fit <- stan_glm(weight ~ group, data = d, refresh = 0) yrep <- posterior_predict(fit) dim(yrep) log_ratios <- -1 * log_lik(fit) dim(log_ratios) r_eff <- relative_eff(exp(-log_ratios), chain_id = rep(1:4, each = 1000)) psis_object <- psis(log_ratios, r_eff = r_eff, cores = 2) E_loo(yrep, psis_object, type = "mean") E_loo(yrep, psis_object, type = "var") E_loo(yrep, psis_object, type = "quantile", probs = 0.5) # median E_loo(yrep, psis_object, type = "quantile", probs = c(0.1, 0.9)) # To get Pareto k diagnostic with E_loo we also need to provide the negative # log-likelihood values using the log_ratios argument. E_loo(yrep, psis_object, type = "mean", log_ratios = log_ratios) } } } loo/man/example_loglik_array.Rd0000644000176200001440000000203013575772017016310 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/example_log_lik_array.R \name{example_loglik_array} \alias{example_loglik_array} \alias{example_loglik_matrix} \title{Objects to use in examples and tests} \usage{ example_loglik_array() example_loglik_matrix() } \value{ \code{example_loglik_array()} returns a 500 (draws) x 2 (chains) x 32 (observations) pointwise log-likelihood array. \code{example_loglik_matrix()} returns the same pointwise log-likelihood values as \code{example_loglik_array()} but reshaped into a 1000 (draws*chains) x 32 (observations) matrix. } \description{ Example pointwise log-likelihood objects to use in demonstrations and tests. See the \strong{Value} and \strong{Examples} sections below. } \examples{ LLarr <- example_loglik_array() (dim_arr <- dim(LLarr)) LLmat <- example_loglik_matrix() (dim_mat <- dim(LLmat)) all.equal(dim_mat[1], dim_arr[1] * dim_arr[2]) all.equal(dim_mat[2], dim_arr[3]) all.equal(LLarr[, 1, ], LLmat[1:500, ]) all.equal(LLarr[, 2, ], LLmat[501:1000, ]) } loo/man/loo-package.Rd0000644000176200001440000001217714407123455014304 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo-package.R \docType{package} \name{loo-package} \alias{loo-package} \title{Efficient LOO-CV and WAIC for Bayesian models} \description{ \if{html}{ \figure{stanlogo.png}{options: width="50" alt="mc-stan.org"} } \emph{Stan Development Team} This package implements the methods described in Vehtari, Gelman, and Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2019), and Yao et al. (2018). To get started see the \strong{loo} package \href{https://mc-stan.org/loo/articles/index.html}{vignettes}, the \code{\link[=loo]{loo()}} function for efficient approximate leave-one-out cross-validation (LOO-CV), the \code{\link[=psis]{psis()}} function for the Pareto smoothed importance sampling (PSIS) algorithm, or \code{\link[=loo_model_weights]{loo_model_weights()}} for an implementation of Bayesian stacking of predictive distributions from multiple models. } \details{ Leave-one-out cross-validation (LOO-CV) and the widely applicable information criterion (WAIC) are methods for estimating pointwise out-of-sample prediction accuracy from a fitted Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameter values. LOO-CV and WAIC have various advantages over simpler estimates of predictive error such as AIC and DIC but are less used in practice because they involve additional computational steps. This package implements the fast and stable computations for approximate LOO-CV laid out in Vehtari, Gelman, and Gabry (2017). From existing posterior simulation draws, we compute LOO-CV using Pareto smoothed importance sampling (PSIS; Vehtari, Simpson, Gelman, Yao, and Gabry, 2019), a new procedure for stabilizing and diagnosing importance weights. As a byproduct of our calculations, we also obtain approximate standard errors for estimated predictive errors and for comparing of predictive errors between two models. We recommend PSIS-LOO-CV instead of WAIC, because PSIS provides useful diagnostics and effective sample size and Monte Carlo standard error estimates. } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. \emph{Bayesian Analysis}, advance publication, doi:10.1214/17-BA1091. (\href{https://projecteuclid.org/euclid.ba/1516093227}{online}). Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2019). Leave-One-Out Cross-Validation for Large Data. In \emph{International Conference on Machine Learning} Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. In \emph{International Conference on Artificial Intelligence and Statistics (AISTATS)} Epifani, I., MacEachern, S. N., and Peruggia, M. (2008). Case-deletion importance sampling estimators: Central limit theorems and related results. \emph{Electronic Journal of Statistics} \strong{2}, 774-806. Gelfand, A. E. (1996). Model determination using sampling-based methods. In \emph{Markov Chain Monte Carlo in Practice}, ed. W. R. Gilks, S. Richardson, D. J. Spiegelhalter, 145-162. London: Chapman and Hall. Gelfand, A. E., Dey, D. K., and Chang, H. (1992). Model determination using predictive distributions with implementation via sampling-based methods. In \emph{Bayesian Statistics 4}, ed. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, 147-167. Oxford University Press. Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. \emph{Statistics and Computing} \strong{24}, 997-1016. Ionides, E. L. (2008). Truncated importance sampling. \emph{Journal of Computational and Graphical Statistics} \strong{17}, 295-311. Koopman, S. J., Shephard, N., and Creal, D. (2009). Testing the assumptions behind importance sampling. \emph{Journal of Econometrics} \strong{149}, 2-11. Peruggia, M. (1997). On the variability of case-deletion importance sampling weights in the Bayesian linear model. \emph{Journal of the American Statistical Association} \strong{92}, 199-207. Stan Development Team (2017). The Stan C++ Library, Version 2.17.0. \url{https://mc-stan.org}. Stan Development Team (2018). RStan: the R interface to Stan, Version 2.17.3. \url{https://mc-stan.org}. Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. \emph{Journal of Machine Learning Research} \strong{11}, 3571-3594. Zhang, J., and Stephens, M. A. (2009). A new and efficient estimation method for the generalized Pareto distribution. \emph{Technometrics} \strong{51}, 316-325. } loo/man/weights.importance_sampling.Rd0000644000176200001440000000177013701164066017622 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/importance_sampling.R \name{weights.importance_sampling} \alias{weights.importance_sampling} \title{Extract importance sampling weights} \usage{ \method{weights}{importance_sampling}(object, ..., log = TRUE, normalize = TRUE) } \arguments{ \item{object}{An object returned by \code{\link[=psis]{psis()}}, \code{\link[=tis]{tis()}}, or \code{\link[=sis]{sis()}}.} \item{...}{Ignored.} \item{log}{Should the weights be returned on the log scale? Defaults to \code{TRUE}.} \item{normalize}{Should the weights be normalized? Defaults to \code{TRUE}.} } \value{ The \code{weights()} method returns an object with the same dimensions as the \code{log_weights} component of \code{object}. The \code{normalize} and \code{log} arguments control whether the returned weights are normalized and whether or not to return them on the log scale. } \description{ Extract importance sampling weights } \examples{ # See the examples at help("psis") } loo/man/waic.Rd0000644000176200001440000001227014407123455013037 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/waic.R \name{waic} \alias{waic} \alias{waic.array} \alias{waic.matrix} \alias{waic.function} \alias{is.waic} \title{Widely applicable information criterion (WAIC)} \usage{ waic(x, ...) \method{waic}{array}(x, ...) \method{waic}{matrix}(x, ...) \method{waic}{`function`}(x, ..., data = NULL, draws = NULL) is.waic(x) } \arguments{ \item{x}{A log-likelihood array, matrix, or function. The \strong{Methods (by class)} section, below, has detailed descriptions of how to specify the inputs for each method.} \item{draws, data, ...}{For the function method only. See the \strong{Methods (by class)} section below for details on these arguments.} } \value{ A named list (of class \code{c("waic", "loo")}) with components: \describe{ \item{\code{estimates}}{ A matrix with two columns (\code{"Estimate"}, \code{"SE"}) and three rows (\code{"elpd_waic"}, \code{"p_waic"}, \code{"waic"}). This contains point estimates and standard errors of the expected log pointwise predictive density (\code{elpd_waic}), the effective number of parameters (\code{p_waic}) and the information criterion \code{waic} (which is just \code{-2 * elpd_waic}, i.e., converted to deviance scale). } \item{\code{pointwise}}{ A matrix with three columns (and number of rows equal to the number of observations) containing the pointwise contributions of each of the above measures (\code{elpd_waic}, \code{p_waic}, \code{waic}). } } } \description{ The \code{waic()} methods can be used to compute WAIC from the pointwise log-likelihood. However, we recommend LOO-CV using PSIS (as implemented by the \code{\link[=loo]{loo()}} function) because PSIS provides useful diagnostics as well as effective sample size and Monte Carlo estimates. } \section{Methods (by class)}{ \itemize{ \item \code{waic(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{waic(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{waic(`function`)}: A function \code{f()} that takes arguments \code{data_i} and \code{draws} and returns a vector containing the log-likelihood for a single observation \code{i} evaluated at each posterior draw. The function should be written such that, for each observation \code{i} in \code{1:N}, evaluating \if{html}{\out{
}}\preformatted{f(data_i = data[i,, drop=FALSE], draws = draws) }\if{html}{\out{
}} results in a vector of length \code{S} (size of posterior sample). The log-likelihood function can also have additional arguments but \code{data_i} and \code{draws} are required. If using the function method then the arguments \code{data} and \code{draws} must also be specified in the call to \code{loo()}: \itemize{ \item \code{data}: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation \code{i}, the \code{i}th row of \code{data} will be passed to the \code{data_i} argument of the log-likelihood function. \item \code{draws}: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike \code{data}, which is indexed by observation, for each observation the entire object \code{draws} will be passed to the \code{draws} argument of the log-likelihood function. \item The \code{...} can be used if your log-likelihood function takes additional arguments. These arguments are used like the \code{draws} argument in that they are recycled for each observation. } }} \examples{ ### Array and matrix methods LLarr <- example_loglik_array() dim(LLarr) LLmat <- example_loglik_matrix() dim(LLmat) waic_arr <- waic(LLarr) waic_mat <- waic(LLmat) identical(waic_arr, waic_mat) \dontrun{ log_lik1 <- extract_log_lik(stanfit1) log_lik2 <- extract_log_lik(stanfit2) (waic1 <- waic(log_lik1)) (waic2 <- waic(log_lik2)) print(compare(waic1, waic2), digits = 2) } } \references{ Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. \emph{Journal of Machine Learning Research} \strong{11}, 3571-3594. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \itemize{ \item The \strong{loo} package \href{https://mc-stan.org/loo/articles/}{vignettes} and Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for more details on why we prefer \code{loo()} to \code{waic()}. \item \code{\link[=loo_compare]{loo_compare()}} for comparing models on approximate LOO-CV or WAIC. } } loo/man/tis.Rd0000644000176200001440000001151614407123455012715 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/tis.R \name{tis} \alias{tis} \alias{tis.array} \alias{tis.matrix} \alias{tis.default} \title{Truncated importance sampling (TIS)} \usage{ tis(log_ratios, ...) \method{tis}{array}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) \method{tis}{matrix}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) \method{tis}{default}(log_ratios, ..., r_eff = NULL) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log scale (for Importance sampling LOO, these are \emph{negative} log-likelihood values). See the \strong{Methods (by class)} section below for a detailed description of how to specify the inputs for each method.} \item{...}{Arguments passed on to the various methods.} \item{r_eff}{Vector of relative effective sample size estimates containing one element per observation. The values provided should be the relative effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. See the \code{\link[=relative_eff]{relative_eff()}} helper function for computing \code{r_eff}. If using \code{psis} with draws of the \code{log_ratios} not obtained from MCMC then the warning message thrown when not specifying \code{r_eff} can be disabled by setting \code{r_eff} to \code{NA}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} } \value{ The \code{tis()} methods return an object of class \code{"tis"}, which is a named list with the following components: \describe{ \item{\code{log_weights}}{ Vector or matrix of smoothed (and truncated) but \emph{unnormalized} log weights. To get normalized weights use the \code{\link[=weights.importance_sampling]{weights()}} method provided for objects of class \code{tis}. } \item{\code{diagnostics}}{ A named list containing one vector: \itemize{ \item \code{pareto_k}: Not used in \code{tis}, all set to 0. \item \code{n_eff}: Effective sample size estimates. } } } Objects of class \code{"tis"} also have the following \link[=attributes]{attributes}: \describe{ \item{\code{norm_const_log}}{ Vector of precomputed values of \code{colLogSumExps(log_weights)} that are used internally by the \code{\link[=weights]{weights()}}method to normalize the log weights. } \item{\code{r_eff}}{ If specified, the user's \code{r_eff} argument. } \item{\code{tail_len}}{ Not used for \code{tis}. } \item{\code{dims}}{ Integer vector of length 2 containing \code{S} (posterior sample size) and \code{N} (number of observations). } \item{\code{method}}{ Method used for importance sampling, here \code{tis}. } } } \description{ Implementation of truncated (self-normalized) importance sampling (TIS), truncated at S^(1/2) as recommended by Ionides (2008). } \section{Methods (by class)}{ \itemize{ \item \code{tis(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{tis(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{tis(default)}: A vector of length \eqn{S} (posterior sample size). }} \examples{ log_ratios <- -1 * example_loglik_array() r_eff <- relative_eff(exp(-log_ratios)) tis_result <- tis(log_ratios, r_eff = r_eff) str(tis_result) # extract smoothed weights lw <- weights(tis_result) # default args are log=TRUE, normalize=TRUE ulw <- weights(tis_result, normalize=FALSE) # unnormalized log-weights w <- weights(tis_result, log=FALSE) # normalized weights (not log-weights) uw <- weights(tis_result, log=FALSE, normalize = FALSE) # unnormalized weights } \references{ Ionides, Edward L. (2008). Truncated importance sampling. \emph{Journal of Computational and Graphical Statistics} 17(2): 295--311. } \seealso{ \itemize{ \item \code{\link[=psis]{psis()}} for approximate LOO-CV using PSIS. \item \code{\link[=loo]{loo()}} for approximate LOO-CV. \item \link{pareto-k-diagnostic} for PSIS diagnostics. } } loo/man/obs_idx.Rd0000644000176200001440000000123013575772017013546 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{obs_idx} \alias{obs_idx} \title{Get observation indices used in subsampling} \usage{ obs_idx(x, rep = TRUE) } \arguments{ \item{x}{A \code{psis_loo_ss} object.} \item{rep}{If sampling with replacement is used, an observation can have multiple samples and these are then repeated in the returned object if \code{rep=TRUE} (e.g., a vector \code{c(1,1,2)} indicates that observation 1 has been subampled two times). If \code{rep=FALSE} only the unique indices are returned.} } \value{ An integer vector. } \description{ Get observation indices used in subsampling } loo/man/dot-compute_point_estimate.Rd0000644000176200001440000000130013575772017017461 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{.compute_point_estimate} \alias{.compute_point_estimate} \title{Compute a point estimate from a draws object} \usage{ .compute_point_estimate(draws) } \arguments{ \item{draws}{A draws object with draws from the posterior.} } \value{ A 1 by P matrix with point estimates from a draws object. } \description{ Compute a point estimate from a draws object } \details{ This is a generic function to thin draws from arbitrary draws objects. The function is internal and should only be used by developers to enable \code{\link[=loo_subsample]{loo_subsample()}} for arbitrary draws objects. } \keyword{internal} loo/man/importance_sampling.matrix.Rd0000644000176200001440000000527013575772017017465 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/importance_sampling.R \name{importance_sampling.matrix} \alias{importance_sampling.matrix} \title{Importance sampling of matrices} \usage{ \method{importance_sampling}{matrix}( log_ratios, method, ..., r_eff = NULL, cores = getOption("mc.cores", 1) ) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log scale (for PSIS-LOO these are \emph{negative} log-likelihood values). See the \strong{Methods (by class)} section below for a detailed description of how to specify the inputs for each method.} \item{...}{Arguments passed on to the various methods.} \item{r_eff}{Vector of relative effective sample size estimates containing one element per observation. The values provided should be the relative effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. See the \code{\link[=relative_eff]{relative_eff()}} helper function for computing \code{r_eff}. If using \code{psis} with draws of the \code{log_ratios} not obtained from MCMC then the warning message thrown when not specifying \code{r_eff} can be disabled by setting \code{r_eff} to \code{NA}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{is_method}{The importance sampling method to use. The following methods are implemented: \itemize{ \item \code{\link[=psis]{"psis"}}: Pareto-Smoothed Importance Sampling (PSIS). Default method. \item \code{\link[=tis]{"tis"}}: Truncated Importance Sampling (TIS) with truncation at \code{sqrt(S)}, where \code{S} is the number of posterior draws. \item \code{\link[=sis]{"sis"}}: Standard Importance Sampling (SIS). }} } \description{ Importance sampling of matrices } \keyword{internal} loo/man/loo_compare.Rd0000644000176200001440000001072314407123455014414 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_compare.R, R/loo_compare.psis_loo_ss_list.R \name{loo_compare} \alias{loo_compare} \alias{loo_compare.default} \alias{print.compare.loo} \alias{print.compare.loo_ss} \title{Model comparison} \usage{ loo_compare(x, ...) \method{loo_compare}{default}(x, ...) \method{print}{compare.loo}(x, ..., digits = 1, simplify = TRUE) \method{print}{compare.loo_ss}(x, ..., digits = 1, simplify = TRUE) } \arguments{ \item{x}{An object of class \code{"loo"} or a list of such objects. If a list is used then the list names will be used as the model names in the output. See \strong{Examples}.} \item{...}{Additional objects of class \code{"loo"}, if not passed in as a single list.} \item{digits}{For the print method only, the number of digits to use when printing.} \item{simplify}{For the print method only, should only the essential columns of the summary matrix be printed? The entire matrix is always returned, but by default only the most important columns are printed.} } \value{ A matrix with class \code{"compare.loo"} that has its own print method. See the \strong{Details} section. } \description{ Compare fitted models based on \link[=loo-glossary]{ELPD}. By default the print method shows only the most important information. Use \code{print(..., simplify=FALSE)} to print a more detailed summary. } \details{ When comparing two fitted models, we can estimate the difference in their expected predictive accuracy by the difference in \code{\link[=loo-glossary]{elpd_loo}} or \code{elpd_waic} (or multiplied by \eqn{-2}, if desired, to be on the deviance scale). When using \code{loo_compare()}, the returned matrix will have one row per model and several columns of estimates. The values in the \code{\link[=loo-glossary]{elpd_diff}} and \code{\link[=loo-glossary]{se_diff}} columns of the returned matrix are computed by making pairwise comparisons between each model and the model with the largest ELPD (the model in the first row). For this reason the \code{elpd_diff} column will always have the value \code{0} in the first row (i.e., the difference between the preferred model and itself) and negative values in subsequent rows for the remaining models. To compute the standard error of the difference in \link[=loo-glossary]{ELPD} --- which should not be expected to equal the difference of the standard errors --- we use a paired estimate to take advantage of the fact that the same set of \eqn{N} data points was used to fit both models. These calculations should be most useful when \eqn{N} is large, because then non-normality of the distribution is not such an issue when estimating the uncertainty in these sums. These standard errors, for all their flaws, should give a better sense of uncertainty than what is obtained using the current standard approach of comparing differences of deviances to a Chi-squared distribution, a practice derived for Gaussian linear models or asymptotically, and which only applies to nested models in any case. } \examples{ # very artificial example, just for demonstration! LL <- example_loglik_array() loo1 <- loo(LL, r_eff = NA) # should be worst model when compared loo2 <- loo(LL + 1, r_eff = NA) # should be second best model when compared loo3 <- loo(LL + 2, r_eff = NA) # should be best model when compared comp <- loo_compare(loo1, loo2, loo3) print(comp, digits = 2) # show more details with simplify=FALSE # (will be the same for all models in this artificial example) print(comp, simplify = FALSE, digits = 3) # can use a list of objects with custom names # will use apple, banana, and cherry, as the names in the output loo_compare(list("apple" = loo1, "banana" = loo2, "cherry" = loo3)) \dontrun{ # works for waic (and kfold) too loo_compare(waic(LL), waic(LL - 10)) } } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \itemize{ \item The \href{https://mc-stan.org/loo/articles/online-only/faq.html}{FAQ page} on the \strong{loo} website for answers to frequently asked questions. } } loo/man/print.loo.Rd0000644000176200001440000000263013701164066014036 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/print.R \name{print.loo} \alias{print.loo} \alias{print.waic} \alias{print.psis_loo} \alias{print.importance_sampling_loo} \alias{print.psis_loo_ap} \alias{print.psis} \alias{print.importance_sampling} \title{Print methods} \usage{ \method{print}{loo}(x, digits = 1, ...) \method{print}{waic}(x, digits = 1, ...) \method{print}{psis_loo}(x, digits = 1, plot_k = FALSE, ...) \method{print}{importance_sampling_loo}(x, digits = 1, plot_k = FALSE, ...) \method{print}{psis_loo_ap}(x, digits = 1, plot_k = FALSE, ...) \method{print}{psis}(x, digits = 1, plot_k = FALSE, ...) \method{print}{importance_sampling}(x, digits = 1, plot_k = FALSE, ...) } \arguments{ \item{x}{An object returned by \code{\link[=loo]{loo()}}, \code{\link[=psis]{psis()}}, or \code{\link[=waic]{waic()}}.} \item{digits}{An integer passed to \code{\link[base:Round]{base::round()}}.} \item{...}{Arguments passed to \code{\link[=plot.psis_loo]{plot.psis_loo()}} if \code{plot_k} is \code{TRUE}.} \item{plot_k}{Logical. If \code{TRUE} the estimates of the Pareto shape parameter \eqn{k} are plotted. Ignored if \code{x} was generated by \code{\link[=waic]{waic()}}. To just plot \eqn{k} without printing use the \link[=pareto-k-diagnostic]{plot()} method for 'loo' objects.} } \value{ \code{x}, invisibly. } \description{ Print methods } \seealso{ \link{pareto-k-diagnostic} } loo/man/loo.Rd0000644000176200001440000003321214407123455012704 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo.R \name{loo} \alias{loo} \alias{loo.array} \alias{loo.matrix} \alias{loo.function} \alias{loo_i} \alias{is.loo} \alias{is.psis_loo} \title{Efficient approximate leave-one-out cross-validation (LOO)} \usage{ loo(x, ...) \method{loo}{array}( x, ..., r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") ) \method{loo}{matrix}( x, ..., r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") ) \method{loo}{`function`}( x, ..., data = NULL, draws = NULL, r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") ) loo_i( i, llfun, ..., data = NULL, draws = NULL, r_eff = NULL, is_method = "psis" ) is.loo(x) is.psis_loo(x) } \arguments{ \item{x}{A log-likelihood array, matrix, or function. The \strong{Methods (by class)} section, below, has detailed descriptions of how to specify the inputs for each method.} \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be saved in the returned object? The \code{loo()} function calls \code{\link[=psis]{psis()}} internally but by default discards the (potentially large) \code{"psis"} object after using it to compute the LOO-CV summaries. Setting \code{save_psis=TRUE} will add a \code{psis_object} component to the list returned by \code{loo}. Currently this is only needed if you plan to use the \code{\link[=E_loo]{E_loo()}} function to compute weighted expectations after running \code{loo}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{is_method}{The importance sampling method to use. The following methods are implemented: \itemize{ \item \code{\link[=psis]{"psis"}}: Pareto-Smoothed Importance Sampling (PSIS). Default method. \item \code{\link[=tis]{"tis"}}: Truncated Importance Sampling (TIS) with truncation at \code{sqrt(S)}, where \code{S} is the number of posterior draws. \item \code{\link[=sis]{"sis"}}: Standard Importance Sampling (SIS). }} \item{data, draws, ...}{For the \code{loo.function()} method and the \code{loo_i()} function, these are the data, posterior draws, and other arguments to pass to the log-likelihood function. See the \strong{Methods (by class)} section below for details on how to specify these arguments.} \item{i}{For \code{loo_i()}, an integer in \code{1:N}.} \item{llfun}{For \code{loo_i()}, the same as \code{x} for the \code{loo.function()} method. A log-likelihood function as described in the \strong{Methods (by class)} section.} } \value{ The \code{loo()} methods return a named list with class \code{c("psis_loo", "loo")} and components: \describe{ \item{\code{estimates}}{ A matrix with two columns (\code{Estimate}, \code{SE}) and three rows (\code{elpd_loo}, \code{p_loo}, \code{looic}). This contains point estimates and standard errors of the expected log pointwise predictive density (\code{\link[=loo-glossary]{elpd_loo}}), the effective number of parameters (\code{\link[=loo-glossary]{p_loo}}) and the LOO information criterion \code{looic} (which is just \code{-2 * elpd_loo}, i.e., converted to deviance scale). } \item{\code{pointwise}}{ A matrix with five columns (and number of rows equal to the number of observations) containing the pointwise contributions of the measures (\code{elpd_loo}, \code{mcse_elpd_loo}, \code{p_loo}, \code{looic}, \code{influence_pareto_k}). in addition to the three measures in \code{estimates}, we also report pointwise values of the Monte Carlo standard error of \code{\link[=loo-glossary]{elpd_loo}} (\code{\link[=loo-glossary]{mcse_elpd_loo}}), and statistics describing the influence of each observation on the posterior distribution (\code{influence_pareto_k}). These are the estimates of the shape parameter \eqn{k} of the generalized Pareto fit to the importance ratios for each leave-one-out distribution (see the \link{pareto-k-diagnostic} page for details). } \item{\code{diagnostics}}{ A named list containing two vectors: \itemize{ \item \code{pareto_k}: Importance sampling reliability diagnostics. By default, these are equal to the \code{influence_pareto_k} in \code{pointwise}. Some algorithms can improve importance sampling reliability and modify these diagnostics. See the \link{pareto-k-diagnostic} page for details. \item \code{n_eff}: PSIS effective sample size estimates. } } \item{\code{psis_object}}{ This component will be \code{NULL} unless the \code{save_psis} argument is set to \code{TRUE} when calling \code{loo()}. In that case \code{psis_object} will be the object of class \code{"psis"} that is created when the \code{loo()} function calls \code{\link[=psis]{psis()}} internally to do the PSIS procedure. } } The \code{loo_i()} function returns a named list with components \code{pointwise} and \code{diagnostics}. These components have the same structure as the \code{pointwise} and \code{diagnostics} components of the object returned by \code{loo()} except they contain results for only a single observation. } \description{ The \code{loo()} methods for arrays, matrices, and functions compute PSIS-LOO CV, efficient approximate leave-one-out (LOO) cross-validation for Bayesian models using Pareto smoothed importance sampling (\link[=psis]{PSIS}). This is an implementation of the methods described in Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019). The \code{loo_i()} function enables testing log-likelihood functions for use with the \code{loo.function()} method. } \details{ The \code{loo()} function is an S3 generic and methods are provided for 3-D pointwise log-likelihood arrays, pointwise log-likelihood matrices, and log-likelihood functions. The array and matrix methods are the most convenient, but for models fit to very large datasets the \code{loo.function()} method is more memory efficient and may be preferable. } \section{Methods (by class)}{ \itemize{ \item \code{loo(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{loo(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{loo(`function`)}: A function \code{f()} that takes arguments \code{data_i} and \code{draws} and returns a vector containing the log-likelihood for a single observation \code{i} evaluated at each posterior draw. The function should be written such that, for each observation \code{i} in \code{1:N}, evaluating \if{html}{\out{
}}\preformatted{f(data_i = data[i,, drop=FALSE], draws = draws) }\if{html}{\out{
}} results in a vector of length \code{S} (size of posterior sample). The log-likelihood function can also have additional arguments but \code{data_i} and \code{draws} are required. If using the function method then the arguments \code{data} and \code{draws} must also be specified in the call to \code{loo()}: \itemize{ \item \code{data}: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation \code{i}, the \code{i}th row of \code{data} will be passed to the \code{data_i} argument of the log-likelihood function. \item \code{draws}: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike \code{data}, which is indexed by observation, for each observation the entire object \code{draws} will be passed to the \code{draws} argument of the log-likelihood function. \item The \code{...} can be used if your log-likelihood function takes additional arguments. These arguments are used like the \code{draws} argument in that they are recycled for each observation. } }} \section{Defining \code{loo()} methods in a package}{ Package developers can define \code{loo()} methods for fitted models objects. See the example \code{loo.stanfit()} method in the \strong{Examples} section below for an example of defining a method that calls \code{loo.array()}. The \code{loo.stanreg()} method in the \strong{rstanarm} package is an example of defining a method that calls \code{loo.function()}. } \examples{ ### Array and matrix methods (using example objects included with loo package) # Array method LLarr <- example_loglik_array() rel_n_eff <- relative_eff(exp(LLarr)) loo(LLarr, r_eff = rel_n_eff, cores = 2) # Matrix method LLmat <- example_loglik_matrix() rel_n_eff <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = 500)) loo(LLmat, r_eff = rel_n_eff, cores = 2) ### Using log-likelihood function instead of array or matrix set.seed(124) # Simulate data and draw from posterior N <- 50; K <- 10; S <- 100; a0 <- 3; b0 <- 2 p <- rbeta(1, a0, b0) y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) dim(fake_posterior) # S x 1 fake_data <- data.frame(y,K) dim(fake_data) # N x 2 llfun <- function(data_i, draws) { # each time called internally within loo the arguments will be equal to: # data_i: ith row of fake_data (fake_data[i,, drop=FALSE]) # draws: entire fake_posterior matrix dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } # Use the loo_i function to check that llfun works on a single observation # before running on all obs. For example, using the 3rd obs in the data: loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) print(loo_3$pointwise[, "elpd_loo"]) # Use loo.function method (setting r_eff=NA since this posterior not obtained via MCMC) loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data, r_eff = NA) # If we look at the elpd_loo contribution from the 3rd obs it should be the # same as what we got above with the loo_i function and i=3: print(loo_with_fn$pointwise[3, "elpd_loo"]) print(loo_3$pointwise[, "elpd_loo"]) # Check that the loo.matrix method gives same answer as loo.function method log_lik_matrix <- sapply(1:N, function(i) { llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior) }) loo_with_mat <- loo(log_lik_matrix, r_eff = NA) all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE! \dontrun{ ### For package developers: defining loo methods # An example of a possible loo method for 'stanfit' objects (rstan package). # A similar method is included in the rstan package. # In order for users to be able to call loo(stanfit) instead of # loo.stanfit(stanfit) the NAMESPACE needs to be handled appropriately # (roxygen2 and devtools packages are good for that). # loo.stanfit <- function(x, pars = "log_lik", ..., save_psis = FALSE, cores = getOption("mc.cores", 1)) { stopifnot(length(pars) == 1L) LLarray <- loo::extract_log_lik(stanfit = x, parameter_name = pars, merge_chains = FALSE) r_eff <- loo::relative_eff(x = exp(LLarray), cores = cores) loo::loo.array(LLarray, r_eff = r_eff, cores = cores, save_psis = save_psis) } } } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \itemize{ \item The \strong{loo} package \href{https://mc-stan.org/loo/articles/index.html}{vignettes} for demonstrations. \item The \href{https://mc-stan.org/loo/articles/online-only/faq.html}{FAQ page} on the \strong{loo} website for answers to frequently asked questions. \item \code{\link[=psis]{psis()}} for the underlying Pareto Smoothed Importance Sampling (PSIS) procedure used in the LOO-CV approximation. \item \link{pareto-k-diagnostic} for convenience functions for looking at diagnostics. \item \code{\link[=loo_compare]{loo_compare()}} for model comparison. } } loo/man/importance_sampling.Rd0000644000176200001440000000215713575772017016163 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/importance_sampling.R \name{importance_sampling} \alias{importance_sampling} \title{A parent class for different importance sampling methods.} \usage{ importance_sampling(log_ratios, method, ...) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log scale (for PSIS-LOO these are \emph{negative} log-likelihood values). See the \strong{Methods (by class)} section below for a detailed description of how to specify the inputs for each method.} \item{method}{The importance sampling method to use. The following methods are implemented: \itemize{ \item \code{\link[=psis]{"psis"}}: Pareto-Smoothed Importance Sampling (PSIS). Default method. \item \code{\link[=tis]{"tis"}}: Truncated Importance Sampling (TIS) with truncation at \code{sqrt(S)}, where \code{S} is the number of posterior draws. \item \code{\link[=sis]{"sis"}}: Standard Importance Sampling (SIS). }} \item{...}{Arguments passed on to the various methods.} } \description{ A parent class for different importance sampling methods. } \keyword{internal} loo/man/figures/0000755000176200001440000000000014407123455013267 5ustar liggesusersloo/man/figures/stanlogo.png0000644000176200001440000003745413267637066015653 0ustar liggesusersPNG  IHDRwx+sBIT|d pHYs&:4tEXtSoftwarewww.inkscape.org< IDATxw|ՙsf$w+$` ؘbQCB ے'7B6R6uq6e7M ل$ ؒm Л$W43+WY{nyޯ/lIw >̙31 B2J #7¡# #a@H P U] ]tuW~V-nUt+ˊ@l sy#/f TILxC&f~I&`= PX]&b.{gʘɋE 邞0t hrh=OuO\Κ;gnnՓ zt2L¾xYIqAsb?l_3bw끳1s+WAŮmZ􇕻OA_LӀsxH`9pO_Can5 j.͠gڪ' UX;i\}2Ə A|g2xEnE٬Z;),V%sNLbALpCfX3j8w5O+~WϪg}1~X%L]PSyL1|/cʽ atC=؟{9ROLl;-!/aKH> `<` 4u-7%ʽNiܻ ;)x+֑|1c^"Qs.ȇ} hLOSq#cʽ-p+5o P;)7Ŵ0o܋|F dS |1J7(`-Nczʽ,a؈#~ܔgw3VE`ܗyBwS o0{V,sQ?|}1K"{/Y.+q5Jy9NZx "j9UX\oӶwa^2[xmoG!F@ǘ,٤׆2O2X{Lã)A¿6ҲwrdgK?%F#c]JF>;H9rϓJ?#ti;/evyʁ{4Qs%AFb_ .YB*2wc K ^;Kri*oC}1@J;-ߙ 0=Q=S8NRJܳZRGӠ_.[|s~5wS JBja킾 ˘ʎ7՞6rfjߣOASdb1E 8y)PF҄we߁ʑ{-aї1hnY@ʽG1a8wc Jл,Exq@f_VsaLy%p",CþYTFnwc r =U[(H_,N?+LpleK鲑;ɕt\/;}g1&V.NpTi/}2W徘 z+YɒdFɒzd˽ ^ r,C<{hyt$CʶR}&{)R{)-`ʲQ} VĘUnw*{9+EԾW¦mDe_鑲*&j&` J5Kgw Sʦܛ -=;r\Ȕ(&j?s^KY;)M%_¿aʚMޕ )|wSvv 9A;[Y;)?%9r^#ࣾ@,]NߙLy+ro?@;)i1"ДܴLfnrdP%Xs(%5roS5sJE2^n1Ţdʽ+\O 7oV.܂9/E)*%Q~ <5}" 9~wc˽F)&̞*"ܔ.%AQd mĉ_1( $W69I1ٗ۩{AP!Ug[S2r_ȸG,003.;1&rA5 Z[hL}15O89}4 5U|'1GNK}؍)/9-SypZ.a0B"Fq xN\t뮐z˘ÑY{sO;hnLƏ Vӕ|bYﾓc= [܊"&p 4a/71 b؍)5xSLSXHwyܛȋ@U6kЖ&u-y4,er$S''L&o+Hl;#wV7؍g9;U0i S#'!> |*[6rof\<?dϫevA)TiB2k$~*?+/z1ٷqnT 2 =϶vO][V-e쨈j`@6gzF3 9zk|g1X)d"]8&B8ɄY=&(Vy ؍f L z2ȉ'v]qx pe9Xhg7f?ׁ͋la\v(o#y/Vy'r{)w9$`*N2S+'r4/2zYҟ+H35O`F5&sSZjXOG+i2_#b3e­L<"E<]$EO9`-x]Eh]L^o ˜@j'(LKΡSA|BKĻ1sKЇzg;}12ۀ'+:ݡ')LVuG`j=ˋɻ$-T pCNQS]T[?^}'(7&(-3K"_ie&?1G"'55'_{DVA+)\Z]U]yBJ ~ړq rES>lV|*=NHftɚ*7.zX=ZD8Vlk> Cꩂ:8;) NՇ U4p{a۽~\n68Ȕ%Nu]#1\;y͵ԥb*SaG~ȅq{&kuϋLo88d3ɂW}M\Vy߳bYc# z:l8͘Cp!ɥa]!kLOll}(UK86">8]f88"3zOMqpn˽R&IoRHe5h!JR&W=[3߹ɂeF6m;t;rBA$sh 9pTzZÖωHW 6hJKcC}ae%V55ss 3dގw)u̦.%YOd?P-vxΑmph66*7H߬ Z|qȟ&jM4y"i;wBڣ8`&pv1HN͚nF\2Jpe|TQ{<{X87A'eLq.Kn޴ V!e9(3Ag>ksw$g*6Le2|#m#W.5g8$ru9c~;D?WթzUvB$:̍?}ș 5Cs@,!zKqtu;ZE.>v#azblbɳCh=<#?QB + \1>}`;U]oorF&VˠԞ -L˸JTo9vO >ҮɻSW\8G!` \'.慃س}ޠQc;Bڈ.$S#!=g&`TOjuW>fX|a_uLz7:.ΊǎQ#ԖBm)}LxE< zZ>s~c撷"8o8* 83d1Y9z6}0SP"~c_N.ʘqQ`+yc"!5sorדRږ_]~q+UAѱcZ2Jޘ~C'*3 88jB{*zx`.ΑmR2HLX'u{~ V<+EGhn%(/c␩Q@13ݞLF>zn7=η3zsnv\]ld+$!p|r|ƌ) `o7E\½ 4(dBߚ81.wE&o)cB-scae~*m[Dy0ˆcc 6ecʀ1j2Vd]wRuOJݛ\2WrK'Ȉq(({p)aw 2&ϡe{j{ڵo5 q k*:88ҾMQdr,%Ikw?t^n+a vQ ##jqH͛1]a(5M+ { >w s8`R؍v;Miџ*;9m]<FK<-ܘ;YȸG'dҗ]bJcj0(և<~d^ҒyKAĈD8&maSƅ [=/Vw]Edt8P+wpVg :*Xś%_R, eN r/E\Qh?5%V0Q|RÉMQゎ .`x,XVfˎ"i)PF694/\|YivT8W]sA !S P Δ=ӵ<qVgtӣЖ^[nHžw sU*483p^N{8HIKvb'^q / !τ123  A6;W|1sd831]Ɍ(dXsx;*:Xk-EB`&;*T8. &R67ފNBlRxw]Bf)qzgA Agĺx$"R P`j”b^ bۍ-f?/u}_#e{9%dsed 'D!6Ɓ-4X'>;͡WrJ=mSGM.aetM" wWtp~Gm%MQ, Ч|1:y]5U8!VE'ɲ jS3f(mABKK솗25@:*f_TbD`<@Eiq Ͷ /;hnQLAT8Mԧ+9% 9v,§"kX ]7)V\QACΐV%o(:mGҢ E 6r76H)qȔ^vѝ0IDAT .Rgv|G1r+Y0~RLҢuYԊb&{jTbRccbeKX^%0fdaqJne2WgD&Hhu1ĊH)ʃN.IWlrU>ØBLNx9Hhski+1Q cN陂tM,Ua^S(뜲&iu VE!+` zZ=K!Qg2JJu,]ɬt%ӣБV޼:{K$gO7Q 0<ɒK,c=|$OБb\bw5zHUgy]3!P;jbv&$]2g9.x޿٧yr7%Axqe%%촢Ϫ];dL 0:qNu+Niu-Ab ^kSiyv4N0"IqJ]f+M _ ҜzD^3S v42nEct>Bb}P˘0L9J.(d-{wn*\PצnBNBwc|쒁*^nh6Ӣ.aUb6gJQc)/Joc{ bd#&2,,FnØči¿Jճvd2x^GFgeLQ;岢 O.TOeŖ?xL61#j‚8Si Zu..7$2T;v8 A61+E湱oHѐ(Ű[q3Dݦ;*~ R=htܝ"ywci 'D'D] fK(7!$M+_QJ.B7D9cLu.fe&(i)xaz-)TN8g[zWS`|03EcΐE2ms4$Ċ!]lݘz%J.8,إ'U~^y>;C.9Id9]*U\@J=Az[N"`D9C}>ہrsHU 'DJ蘒&?͵ w}KNyc{cU>{97d71$1cgBbuM Wo|mlcYy `rE c鋁*{.cyCƷw}c7L~h7ЃuM> z\"cay*;wHJwmof@%h M_h]7ݘ4> :*CdItŞ+P[ \ϒhNA_AM85Q&jZKk]a+g olߴP_LjGi0onɴ 4e-1&%uTpNg\x Ck+vE?O;LCoN[~2q800"sT"EaCǺ!ȦdAOG從Q< `S)*!prrtH*]RqM?=@9џ]\Ý@0!*Q1:ݛb,2 Y =|u/'&E>Nk]#-}Cw;w!Yzp=MAy8O1S4'Oy8)L߭urG]flnLaFZ>| =-v` 1&T`A@2bM[0Y 7/CAz:cLN$@Sۥ=6S}"bh7/c1~%W+ <+I "z~cX k1/z^ neq+3^(6Tŋ޼3_cCPޓbpDƊ`}|zy0ч17t~-ٖC E_^rn}}1[OhW='3ۚyBo19w_ 4rPGZ=?>Rws֭Դ>z_ OY+D3V8t`}}u:kzc \=`bm_YQ{AgJ_D2vTDq)#i&%[RWǸ )Y ;#YM X<Ê|2"_OB*v +B"ءs{"G3ۂڲHc*l<;̡dls=AۋYK{cJD$$pˌ@Յ\ U؞C[/#0_cSYp pͪuMԾ'KlHS6*̇y_¦T Ɓ2rv4oBЛҶ<|3ɍ}g1;V5S}"1)iл/f߁jW`,WO뙽yRw6rHo ,1$ p.Re߁Vj @;9l;dyǰ.ެ7r@5QȧJy!@yEy\cr(^'pk#k}ʆLJrC?pnVG%dqLYwlzԷZqs@coLYwl[L }حԋm Łs2r̕_\Ø hSi!Sx;Dq_s:mL!ɃaZ)yw ڜ.߁r)˄K}?h=P;ɪ6\ >nstqk^Ϩ?^ÝO!$ﳟ$R@Yyrk?>?Քd K}?S4_ne[@;-FZs~g.4QspIg*;,ͮq@0p* odA@8SΖB>NpwֳY߁e)+#җ]^A६DDNfjyN㝂> !s=XHC\;CFZ)_kgvzmGeB5^S.s@13tu;9M959.p @ܗd` \͡\\Oy>gپ3%)NEt$ֳv@LxE%3L(H{}޼]P[@>od2S-LqoF \Z:@-M):?.7Qn|Vpp#>3=AV*:T%(:[tY@l_g2-GĖy!w[PM30Uq 0))&d[@Q |+w2g"] ^;!H) \( b7י,R}gDR;sdLѻqnK4!""A.dJZ;i":R80w".Kp*S9 Id&91QXƔ0J|,AR@O(,sݻH[.oo#zp>0^2@}o; y{H];Or)d΂|AO@މr@[<~20SLY v(ٴ\;Au[9@c:oȧhSD/{.p"h=ɩn8Aꀩsiw(=яU) /)Kۮ|?0iBBtVW}yNeܱBw{6v5f AuJR'TcKam+QlȻOf98vM|g+~ZNƟ )rt{j1EqWLӺcKymS^Νͪud{7v/?k9rcJIK@xlV$ܻ۬dZq)r}VN{e]pϡe$PN}m+Ӏscf% Xw\)r_DI7KɌk}g2xd@Rc_Qs N#B|lZ7kw|hdwŸ|g1xs_;Y[e2-R&WFo.W^_9ޭvoP63ƔE!o+]av&Y-3crGJG[VʲwkKW)ƔLė/E^!诀c]};Oe_ >! 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Vf>?+}piܱ++G0YS 4^~әnD r{gC8F05M`U~k'yZ&>00<)Ky#bڴkKup\!S9 Hq;<,VfV14wC8y +Vp{.fzUU ޠ*X|X5j9W/ׯyAEy qK鱽oA )9 3"2p_}x}qOq$sCuO+o}LJNDÖ+w'F CdFg98ɶ)fAy6K|Yk++G 7*=`GPy7/z\lerj5*0MHLf|gˣx^yOͦe 9pQEeˍ#N ԝae4`F,\|Y[jd (0O1.`+dUׯW%z<{5wzΗs6lS]os$7^%LI4!@&&1F(:BHō t3f 3c}f]yJX9Q4Gn@~NERCI;D8'WݴsByK)j0~D%P! d޹EQpm,Fi*RڂX U`$j2G1l?Πɡ"^@p_zNƙa$FF:u5ȏ2IENDB`loo/man/figures/logo.svg0000644000176200001440000002316214407123455014754 0ustar liggesusers loo/man/dot-thin_draws.Rd0000644000176200001440000000126613575772017015056 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{.thin_draws} \alias{.thin_draws} \title{Thin a draws object} \usage{ .thin_draws(draws, loo_approximation_draws) } \arguments{ \item{draws}{A draws object with posterior draws.} \item{loo_approximation_draws}{The number of posterior draws to return (ie after thinning).} } \value{ A thinned draws object. } \description{ Thin a draws object } \details{ This is a generic function to thin draws from arbitrary draws objects. The function is internal and should only be used by developers to enable \code{\link[=loo_subsample]{loo_subsample()}} for arbitrary draws objects. } \keyword{internal} loo/man/update.psis_loo_ss.Rd0000644000176200001440000001207414407123455015733 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{update.psis_loo_ss} \alias{update.psis_loo_ss} \title{Update \code{psis_loo_ss} objects} \usage{ \method{update}{psis_loo_ss}( object, ..., data = NULL, draws = NULL, observations = NULL, r_eff = NULL, cores = getOption("mc.cores", 1), loo_approximation = NULL, loo_approximation_draws = NULL, llgrad = NULL, llhess = NULL ) } \arguments{ \item{object}{A \code{psis_loo_ss} object to update.} \item{...}{Currently not used.} \item{data, draws}{See \code{\link[=loo_subsample.function]{loo_subsample.function()}}.} \item{observations}{The subsample observations to use. The argument can take four (4) types of arguments: \itemize{ \item \code{NULL} to use all observations. The algorithm then just uses standard \code{loo()} or \code{loo_approximate_posterior()}. \item A single integer to specify the number of observations to be subsampled. \item A vector of integers to provide the indices used to subset the data. \emph{These observations need to be subsampled with the same scheme as given by the \code{estimator} argument}. \item A \code{psis_loo_ss} object to use the same observations that were used in a previous call to \code{loo_subsample()}. }} \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{loo_approximation}{What type of approximation of the loo_i's should be used? The default is \code{"plpd"} (the log predictive density using the posterior expectation). There are six different methods implemented to approximate loo_i's (see the references for more details): \itemize{ \item \code{"plpd"}: uses the lpd based on point estimates (i.e., \eqn{p(y_i|\hat{\theta})}). \item \code{"lpd"}: uses the lpds (i,e., \eqn{p(y_i|y)}). \item \code{"tis"}: uses truncated importance sampling to approximate PSIS-LOO. \item \code{"waic"}: uses waic (i.e., \eqn{p(y_i|y) - p_{waic}}). \item \code{"waic_grad_marginal"}: uses waic approximation using first order delta method and posterior marginal variances to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad_marginal). Requires gradient of likelihood function. \item \code{"waic_grad"}: uses waic approximation using first order delta method and posterior covariance to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient of likelihood function. \item \code{"waic_hess"}: uses waic approximation using second order delta method and posterior covariance to approximate \eqn{p_{waic}} (ie. \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient and Hessian of likelihood function. } As point estimates of \eqn{\hat{\theta}}, the posterior expectations of the parameters are used.} \item{loo_approximation_draws}{The number of posterior draws used when integrating over the posterior. This is used if \code{loo_approximation} is set to \code{"lpd"}, \code{"waic"}, or \code{"tis"}.} \item{llgrad}{The gradient of the log-likelihood. This is only used when \code{loo_approximation} is \code{"waic_grad"}, \code{"waic_grad_marginal"}, or \code{"waic_hess"}. The default is \code{NULL}.} \item{llhess}{The hessian of the log-likelihood. This is only used with \code{loo_approximation = "waic_hess"}. The default is \code{NULL}.} } \value{ A \code{psis_loo_ss} object. } \description{ Update \code{psis_loo_ss} objects } \details{ If \code{observations} is updated then if a vector of indices or a \code{psis_loo_ss} object is supplied the updated object will have exactly the observations indicated by the vector or \code{psis_loo_ss} object. If a single integer is supplied, new observations will be sampled to reach the supplied sample size. } loo/man/psislw.Rd0000644000176200001440000000470514407123455013441 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/psislw.R \name{psislw} \alias{psislw} \title{Pareto smoothed importance sampling (deprecated, old version)} \usage{ psislw( lw, wcp = 0.2, wtrunc = 3/4, cores = getOption("mc.cores", 1), llfun = NULL, llargs = NULL, ... ) } \arguments{ \item{lw}{A matrix or vector of log weights. For computing LOO, \code{lw = -log_lik}, the \emph{negative} of an \eqn{S} (simulations) by \eqn{N} (data points) pointwise log-likelihood matrix.} \item{wcp}{The proportion of importance weights to use for the generalized Pareto fit. The \code{100*wcp}\\% largest weights are used as the sample from which to estimate the parameters of the generalized Pareto distribution.} \item{wtrunc}{For truncating very large weights to \eqn{S}^\code{wtrunc}. Set to zero for no truncation.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}, the old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until it is removed. \strong{As of version 2.0.0, the default is now 1 core if \code{mc.cores} is not set, but we recommend using as many (or close to as many) cores as possible.}} \item{llfun, llargs}{See \code{\link[=loo.function]{loo.function()}}.} \item{...}{Ignored when \code{psislw()} is called directly. The \code{...} is only used internally when \code{psislw()} is called by the \code{\link[=loo]{loo()}} function.} } \value{ A named list with components \code{lw_smooth} (modified log weights) and \code{pareto_k} (estimated generalized Pareto shape parameter(s) k). } \description{ As of version \verb{2.0.0} this function is \strong{deprecated}. Please use the \code{\link[=psis]{psis()}} function for the new PSIS algorithm. } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \link{pareto-k-diagnostic} for PSIS diagnostics. } loo/man/kfold-helpers.Rd0000644000176200001440000000365613575772017014674 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/kfold-helpers.R \name{kfold-helpers} \alias{kfold-helpers} \alias{kfold_split_random} \alias{kfold_split_stratified} \alias{kfold_split_grouped} \title{Helper functions for K-fold cross-validation} \usage{ kfold_split_random(K = 10, N = NULL) kfold_split_stratified(K = 10, x = NULL) kfold_split_grouped(K = 10, x = NULL) } \arguments{ \item{K}{The number of folds to use.} \item{N}{The number of observations in the data.} \item{x}{A discrete variable of length \code{N} with at least \code{K} levels (unique values). Will be coerced to a \link[=factor]{factor}.} } \value{ An integer vector of length \code{N} where each element is an index in \code{1:K}. } \description{ These functions can be used to generate indexes for use with K-fold cross-validation. See the \strong{Details} section for explanations. } \details{ \code{kfold_split_random()} splits the data into \code{K} groups of equal size (or roughly equal size). For a categorical variable \code{x} \code{kfold_split_stratified()} splits the observations into \code{K} groups ensuring that relative category frequencies are approximately preserved. For a grouping variable \code{x}, \code{kfold_split_grouped()} places all observations in \code{x} from the same group/level together in the same fold. The selection of which groups/levels go into which fold (relevant when when there are more groups than folds) is randomized. } \examples{ ids <- kfold_split_random(K = 5, N = 20) print(ids) table(ids) x <- sample(c(0, 1), size = 200, replace = TRUE, prob = c(0.05, 0.95)) table(x) ids <- kfold_split_stratified(K = 5, x = x) print(ids) table(ids, x) grp <- gl(n = 50, k = 15, labels = state.name) length(grp) head(table(grp)) ids_10 <- kfold_split_grouped(K = 10, x = grp) (tab_10 <- table(grp, ids_10)) colSums(tab_10) ids_9 <- kfold_split_grouped(K = 9, x = grp) (tab_9 <- table(grp, ids_9)) colSums(tab_9) } loo/man/crps.Rd0000644000176200001440000000745114411130104013050 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/crps.R \name{crps} \alias{crps} \alias{scrps} \alias{loo_crps} \alias{loo_scrps} \alias{crps.matrix} \alias{crps.numeric} \alias{loo_crps.matrix} \alias{scrps.matrix} \alias{scrps.numeric} \alias{loo_scrps.matrix} \title{Continuously ranked probability score} \usage{ crps(x, ...) scrps(x, ...) loo_crps(x, ...) loo_scrps(x, ...) \method{crps}{matrix}(x, x2, y, ..., permutations = 1) \method{crps}{numeric}(x, x2, y, ..., permutations = 1) \method{loo_crps}{matrix}( x, x2, y, log_lik, ..., permutations = 1, r_eff = NULL, cores = getOption("mc.cores", 1) ) \method{scrps}{matrix}(x, x2, y, ..., permutations = 1) \method{scrps}{numeric}(x, x2, y, ..., permutations = 1) \method{loo_scrps}{matrix}( x, x2, y, log_lik, ..., permutations = 1, r_eff = NULL, cores = getOption("mc.cores", 1) ) } \arguments{ \item{x}{A \code{S} by \code{N} matrix (draws by observations), or a vector of length \code{S} when only single observation is provided in \code{y}.} \item{...}{Passed on to \code{\link[=E_loo]{E_loo()}} in the \verb{loo_*()} version of these functions.} \item{x2}{Independent draws from the same distribution as draws in \code{x}. Should be of the identical dimension.} \item{y}{A vector of observations or a single value.} \item{permutations}{An integer, with default value of 1, specifying how many times the expected value of |X - X'| (\verb{|x - x2|}) is computed. The row order of \code{x2} is shuffled as elements \code{x} and \code{x2} are typically drawn given the same values of parameters. This happens, e.g., when one calls \code{posterior_predict()} twice for a fitted \pkg{rstanarm} or \pkg{brms} model. Generating more permutations is expected to decrease the variance of the computed expected value.} \item{log_lik}{A log-likelihood matrix the same size as \code{x}.} \item{r_eff}{An optional vector of relative effective sample size estimates containing one element per observation. See \code{\link[=psis]{psis()}} for details.} \item{cores}{The number of cores to use for parallelization of \verb{[psis()]}. See \code{\link[=psis]{psis()}} for details.} } \value{ A list containing two elements: \code{estimates} and \code{pointwise}. The former reports estimator and standard error and latter the pointwise values. } \description{ The \code{crps()} and \code{scrps()} functions and their \verb{loo_*()} counterparts can be used to compute the continuously ranked probability score (CRPS) and scaled CRPS (SCRPS) (see Bolin and Wallin, 2022). CRPS is a proper scoring rule, and strictly proper when the first moment of the predictive distribution is finite. Both can be expressed in terms of samples form the predictive distribution. See e.g. Gneiting and Raftery (2007) for a comprehensive discussion on CRPS. } \details{ To compute (S)CRPS, the user needs to provide two sets of draws, \code{x} and \code{x2}, from the predictive distribution. This is due to the fact that formulas used to compute CRPS involve an expectation of the absolute difference of \code{x} and \code{x2}, both having the same distribution. See the \code{permutations} argument, as well as Gneiting and Raftery (2007) for details. } \examples{ \dontrun{ # An example using rstanarm library(rstanarm) data("kidiq") fit <- stan_glm(kid_score ~ mom_hs + mom_iq, data = kidiq) ypred1 <- posterior_predict(fit) ypred2 <- posterior_predict(fit) crps(ypred1, ypred2, y = fit$y) loo_crps(ypred1, ypred2, y = fit$y, log_lik = log_lik(fit)) } } \references{ Bolin, D., & Wallin, J. (2022). Local scale invariance and robustness of proper scoring rules. arXiv. \doi{10.48550/arXiv.1912.05642} Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), 359–378. } loo/man/importance_sampling.default.Rd0000644000176200001440000000356013575772017017605 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/importance_sampling.R \name{importance_sampling.default} \alias{importance_sampling.default} \title{Importance sampling (default)} \usage{ \method{importance_sampling}{default}(log_ratios, method, ..., r_eff = NULL) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log scale (for PSIS-LOO these are \emph{negative} log-likelihood values). See the \strong{Methods (by class)} section below for a detailed description of how to specify the inputs for each method.} \item{...}{Arguments passed on to the various methods.} \item{r_eff}{Vector of relative effective sample size estimates containing one element per observation. The values provided should be the relative effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. See the \code{\link[=relative_eff]{relative_eff()}} helper function for computing \code{r_eff}. If using \code{psis} with draws of the \code{log_ratios} not obtained from MCMC then the warning message thrown when not specifying \code{r_eff} can be disabled by setting \code{r_eff} to \code{NA}.} \item{is_method}{The importance sampling method to use. The following methods are implemented: \itemize{ \item \code{\link[=psis]{"psis"}}: Pareto-Smoothed Importance Sampling (PSIS). Default method. \item \code{\link[=tis]{"tis"}}: Truncated Importance Sampling (TIS) with truncation at \code{sqrt(S)}, where \code{S} is the number of posterior draws. \item \code{\link[=sis]{"sis"}}: Standard Importance Sampling (SIS). }} } \description{ Importance sampling (default) } \keyword{internal} loo/man/find_model_names.Rd0000644000176200001440000000070713575772017015412 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_compare.R \name{find_model_names} \alias{find_model_names} \title{Find the model names associated with \code{"loo"} objects} \usage{ find_model_names(x) } \arguments{ \item{x}{List of \code{"loo"} objects.} } \value{ Character vector of model names the same length as \code{x.} } \description{ Find the model names associated with \code{"loo"} objects } \keyword{internal} loo/man/loo-datasets.Rd0000644000176200001440000000346614407123455014522 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/datasets.R \name{loo-datasets} \alias{loo-datasets} \alias{Kline} \alias{milk} \alias{voice} \alias{voice_loo} \title{Datasets for loo examples and vignettes} \description{ Small datasets for use in \strong{loo} examples and vignettes. The \code{Kline} and \code{milk} datasets are also included in the \strong{rethinking} package (McElreath, 2016a), but we include them here as \strong{rethinking} is not on CRAN. } \details{ Currently the data sets included are: \itemize{ \item \code{Kline}: Small dataset from Kline and Boyd (2010) on tool complexity and demography in Oceanic islands societies. This data is discussed in detail in McElreath (2016a,2016b). \href{https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/Kline}{(Link to variable descriptions)} \item \code{milk}: Small dataset from Hinde and Milligan (2011) on primate milk composition.This data is discussed in detail in McElreath (2016a,2016b). \href{https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/milk}{(Link to variable descriptions)} \item \code{voice}: Voice rehabilitation data from Tsanas et al. (2014). } } \examples{ str(Kline) str(milk) } \references{ Hinde and Milligan. 2011. \emph{Evolutionary Anthropology} 20:9-23. Kline, M.A. and R. Boyd. 2010. \emph{Proc R Soc B} 277:2559-2564. McElreath, R. (2016a). rethinking: Statistical Rethinking book package. R package version 1.59. McElreath, R. (2016b). \emph{Statistical rethinking: A Bayesian course with examples in R and Stan}. Chapman & Hall/CRC. A. Tsanas, M.A. Little, C. Fox, L.O. Ramig: Objective automatic assessment of rehabilitative speech treatment in Parkinson's disease, IEEE Transactions on Neural Systems and Rehabilitation Engineering, Vol. 22, pp. 181-190, January 2014 } loo/man/kfold-generic.Rd0000644000176200001440000000317513575772017014642 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/kfold-generic.R \name{kfold-generic} \alias{kfold-generic} \alias{kfold} \alias{is.kfold} \title{Generic function for K-fold cross-validation for developers} \usage{ kfold(x, ...) is.kfold(x) } \arguments{ \item{x}{A fitted model object.} \item{...}{Arguments to pass to specific methods.} } \value{ For developers defining a \code{kfold()} method for a class \code{"foo"}, the \code{kfold.foo()} function should return a list with class \code{c("kfold", "loo")} with at least the following named elements: \itemize{ \item \code{"estimates"}: A \verb{1x2} matrix containing the ELPD estimate and its standard error. The matrix must have row name "\code{elpd_kfold}" and column names \code{"Estimate"} and \code{"SE"}. \item \code{"pointwise"}: A \code{Nx1} matrix with column name \code{"elpd_kfold"} containing the pointwise contributions for each data point. } It is important for the object to have at least these classes and components so that it is compatible with other functions like \code{\link[=loo_compare]{loo_compare()}} and \code{print()} methods. } \description{ For developers of Bayesian modeling packages, \strong{loo} includes a generic function \code{kfold()} so that methods may be defined for K-fold CV without name conflicts between packages. See, for example, the \code{kfold()} methods in the \strong{rstanarm} and \strong{brms} packages. The \strong{Value} section below describes the objects that \code{kfold()} methods should return in order to be compatible with \code{\link[=loo_compare]{loo_compare()}} and the \strong{loo} package print methods. } loo/man/relative_eff.Rd0000644000176200001440000001073514407123455014553 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/effective_sample_sizes.R \name{relative_eff} \alias{relative_eff} \alias{relative_eff.default} \alias{relative_eff.matrix} \alias{relative_eff.array} \alias{relative_eff.function} \alias{relative_eff.importance_sampling} \title{Convenience function for computing relative efficiencies} \usage{ relative_eff(x, ...) \method{relative_eff}{default}(x, chain_id, ...) \method{relative_eff}{matrix}(x, chain_id, ..., cores = getOption("mc.cores", 1)) \method{relative_eff}{array}(x, ..., cores = getOption("mc.cores", 1)) \method{relative_eff}{`function`}( x, chain_id, ..., cores = getOption("mc.cores", 1), data = NULL, draws = NULL ) \method{relative_eff}{importance_sampling}(x, ...) } \arguments{ \item{x}{A vector, matrix, 3-D array, or function. See the \strong{Methods (by class)} section below for details on specifying \code{x}, but where "log-likelihood" is mentioned replace it with one of the following depending on the use case: \itemize{ \item For use with the \code{\link[=loo]{loo()}} function, the values in \code{x} (or generated by \code{x}, if a function) should be \strong{likelihood} values (i.e., \code{exp(log_lik)}), not on the log scale. \item For generic use with \code{\link[=psis]{psis()}}, the values in \code{x} should be the reciprocal of the importance ratios (i.e., \code{exp(-log_ratios)}). }} \item{chain_id}{A vector of length \code{NROW(x)} containing MCMC chain indexes for each each row of \code{x} (if a matrix) or each value in \code{x} (if a vector). No \code{chain_id} is needed if \code{x} is a 3-D array. If there are \code{C} chains then valid chain indexes are values in \code{1:C}.} \item{cores}{The number of cores to use for parallelization.} \item{data, draws, ...}{Same as for the \code{\link[=loo]{loo()}} function method.} } \value{ A vector of relative effective sample sizes. } \description{ \code{relative_eff()} computes the the MCMC effective sample size divided by the total sample size. } \section{Methods (by class)}{ \itemize{ \item \code{relative_eff(default)}: A vector of length \eqn{S} (posterior sample size). \item \code{relative_eff(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{relative_eff(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{relative_eff(`function`)}: A function \code{f()} that takes arguments \code{data_i} and \code{draws} and returns a vector containing the log-likelihood for a single observation \code{i} evaluated at each posterior draw. The function should be written such that, for each observation \code{i} in \code{1:N}, evaluating \if{html}{\out{

}}\preformatted{f(data_i = data[i,, drop=FALSE], draws = draws) }\if{html}{\out{
}} results in a vector of length \code{S} (size of posterior sample). The log-likelihood function can also have additional arguments but \code{data_i} and \code{draws} are required. If using the function method then the arguments \code{data} and \code{draws} must also be specified in the call to \code{loo()}: \itemize{ \item \code{data}: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation \code{i}, the \code{i}th row of \code{data} will be passed to the \code{data_i} argument of the log-likelihood function. \item \code{draws}: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike \code{data}, which is indexed by observation, for each observation the entire object \code{draws} will be passed to the \code{draws} argument of the log-likelihood function. \item The \code{...} can be used if your log-likelihood function takes additional arguments. These arguments are used like the \code{draws} argument in that they are recycled for each observation. } \item \code{relative_eff(importance_sampling)}: If \code{x} is an object of class \code{"psis"}, \code{relative_eff()} simply returns the \code{r_eff} attribute of \code{x}. }} \examples{ LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() dim(LLarr) dim(LLmat) rel_n_eff_1 <- relative_eff(exp(LLarr)) rel_n_eff_2 <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = 500)) all.equal(rel_n_eff_1, rel_n_eff_2) } loo/man/loo_approximate_posterior.Rd0000644000176200001440000001366714407123455017437 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_approximate_posterior.R \name{loo_approximate_posterior} \alias{loo_approximate_posterior} \alias{loo_approximate_posterior.array} \alias{loo_approximate_posterior.matrix} \alias{loo_approximate_posterior.function} \title{Efficient approximate leave-one-out cross-validation (LOO) for posterior approximations} \usage{ loo_approximate_posterior(x, log_p, log_g, ...) \method{loo_approximate_posterior}{array}( x, log_p, log_g, ..., save_psis = FALSE, cores = getOption("mc.cores", 1) ) \method{loo_approximate_posterior}{matrix}( x, log_p, log_g, ..., save_psis = FALSE, cores = getOption("mc.cores", 1) ) \method{loo_approximate_posterior}{`function`}( x, ..., data = NULL, draws = NULL, log_p = NULL, log_g = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1) ) } \arguments{ \item{x}{A log-likelihood array, matrix, or function. The \strong{Methods (by class)} section, below, has detailed descriptions of how to specify the inputs for each method.} \item{log_p}{The log-posterior (target) evaluated at S samples from the proposal distribution (g). A vector of length S.} \item{log_g}{The log-density (proposal) evaluated at S samples from the proposal distribution (g). A vector of length S.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo_approximate_posterior()} be saved in the returned object? See \code{\link[=loo]{loo()}} for details.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{data, draws, ...}{For the \code{loo_approximate_posterior.function()} method, these are the data, posterior draws, and other arguments to pass to the log-likelihood function. See the \strong{Methods (by class)} section below for details on how to specify these arguments.} } \value{ The \code{loo_approximate_posterior()} methods return a named list with class \code{c("psis_loo_ap", "psis_loo", "loo")}. It has the same structure as the objects returned by \code{\link[=loo]{loo()}} but with the additional slot: \describe{ \item{\code{posterior_approximation}}{ A list with two vectors, \code{log_p} and \code{log_g} of the same length containing the posterior density and the approximation density for the individual draws. } } } \description{ Efficient approximate leave-one-out cross-validation (LOO) for posterior approximations } \details{ The \code{loo_approximate_posterior()} function is an S3 generic and methods are provided for 3-D pointwise log-likelihood arrays, pointwise log-likelihood matrices, and log-likelihood functions. The implementation works for posterior approximations where it is possible to compute the log density for the posterior approximation. } \section{Methods (by class)}{ \itemize{ \item \code{loo_approximate_posterior(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{loo_approximate_posterior(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{loo_approximate_posterior(`function`)}: A function \code{f()} that takes arguments \code{data_i} and \code{draws} and returns a vector containing the log-likelihood for a single observation \code{i} evaluated at each posterior draw. The function should be written such that, for each observation \code{i} in \code{1:N}, evaluating \if{html}{\out{
}}\preformatted{f(data_i = data[i,, drop=FALSE], draws = draws) }\if{html}{\out{
}} results in a vector of length \code{S} (size of posterior sample). The log-likelihood function can also have additional arguments but \code{data_i} and \code{draws} are required. If using the function method then the arguments \code{data} and \code{draws} must also be specified in the call to \code{loo()}: \itemize{ \item \code{data}: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation \code{i}, the \code{i}th row of \code{data} will be passed to the \code{data_i} argument of the log-likelihood function. \item \code{draws}: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike \code{data}, which is indexed by observation, for each observation the entire object \code{draws} will be passed to the \code{draws} argument of the log-likelihood function. \item The \code{...} can be used if your log-likelihood function takes additional arguments. These arguments are used like the \code{draws} argument in that they are recycled for each observation. } }} \references{ Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2019). Leave-One-Out Cross-Validation for Large Data. In \emph{International Conference on Machine Learning} Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. In \emph{International Conference on Artificial Intelligence and Statistics (AISTATS)} } \seealso{ \code{\link[=loo]{loo()}}, \code{\link[=psis]{psis()}}, \code{\link[=loo_compare]{loo_compare()}} } loo/man/dot-ndraws.Rd0000644000176200001440000000121313575772017014202 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_subsample.R \name{.ndraws} \alias{.ndraws} \title{The number of posterior draws in a draws object.} \usage{ .ndraws(x) } \arguments{ \item{x}{A draws object with posterior draws.} } \value{ An integer with the number of draws. } \description{ The number of posterior draws in a draws object. } \details{ This is a generic function to return the total number of draws from an arbitrary draws objects. The function is internal and should only be used by developers to enable \code{\link[=loo_subsample]{loo_subsample()}} for arbitrary draws objects. } \keyword{internal} loo/man/psis.Rd0000644000176200001440000001405214407123455013072 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/psis.R, R/sis.R, R/tis.R \name{psis} \alias{psis} \alias{psis.array} \alias{psis.matrix} \alias{psis.default} \alias{is.psis} \alias{is.sis} \alias{is.tis} \title{Pareto smoothed importance sampling (PSIS)} \usage{ psis(log_ratios, ...) \method{psis}{array}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) \method{psis}{matrix}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) \method{psis}{default}(log_ratios, ..., r_eff = NULL) is.psis(x) is.sis(x) is.tis(x) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log scale (for PSIS-LOO these are \emph{negative} log-likelihood values). See the \strong{Methods (by class)} section below for a detailed description of how to specify the inputs for each method.} \item{...}{Arguments passed on to the various methods.} \item{r_eff}{Vector of relative effective sample size estimates containing one element per observation. The values provided should be the relative effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. See the \code{\link[=relative_eff]{relative_eff()}} helper function for computing \code{r_eff}. If using \code{psis} with draws of the \code{log_ratios} not obtained from MCMC then the warning message thrown when not specifying \code{r_eff} can be disabled by setting \code{r_eff} to \code{NA}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{x}{For \code{is.psis()}, an object to check.} } \value{ The \code{psis()} methods return an object of class \code{"psis"}, which is a named list with the following components: \describe{ \item{\code{log_weights}}{ Vector or matrix of smoothed (and truncated) but \emph{unnormalized} log weights. To get normalized weights use the \code{\link[=weights.importance_sampling]{weights()}} method provided for objects of class \code{"psis"}. } \item{\code{diagnostics}}{ A named list containing two vectors: \itemize{ \item \code{pareto_k}: Estimates of the shape parameter \eqn{k} of the generalized Pareto distribution. See the \link{pareto-k-diagnostic} page for details. \item \code{n_eff}: PSIS effective sample size estimates. } } } Objects of class \code{"psis"} also have the following \link[=attributes]{attributes}: \describe{ \item{\code{norm_const_log}}{ Vector of precomputed values of \code{colLogSumExps(log_weights)} that are used internally by the \code{weights} method to normalize the log weights. } \item{\code{tail_len}}{ Vector of tail lengths used for fitting the generalized Pareto distribution. } \item{\code{r_eff}}{ If specified, the user's \code{r_eff} argument. } \item{\code{dims}}{ Integer vector of length 2 containing \code{S} (posterior sample size) and \code{N} (number of observations). } \item{\code{method}}{ Method used for importance sampling, here \code{psis}. } } } \description{ Implementation of Pareto smoothed importance sampling (PSIS), a method for stabilizing importance ratios. The version of PSIS implemented here corresponds to the algorithm presented in Vehtari, Simpson, Gelman, Yao, and Gabry (2019). For PSIS diagnostics see the \link{pareto-k-diagnostic} page. } \section{Methods (by class)}{ \itemize{ \item \code{psis(array)}: An \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I} is the number of MCMC iterations per chain, \eqn{C} is the number of chains, and \eqn{N} is the number of data points. \item \code{psis(matrix)}: An \eqn{S} by \eqn{N} matrix, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. \item \code{psis(default)}: A vector of length \eqn{S} (posterior sample size). }} \examples{ log_ratios <- -1 * example_loglik_array() r_eff <- relative_eff(exp(-log_ratios)) psis_result <- psis(log_ratios, r_eff = r_eff) str(psis_result) plot(psis_result) # extract smoothed weights lw <- weights(psis_result) # default args are log=TRUE, normalize=TRUE ulw <- weights(psis_result, normalize=FALSE) # unnormalized log-weights w <- weights(psis_result, log=FALSE) # normalized weights (not log-weights) uw <- weights(psis_result, log=FALSE, normalize = FALSE) # unnormalized weights } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \itemize{ \item \code{\link[=loo]{loo()}} for approximate LOO-CV using PSIS. \item \link{pareto-k-diagnostic} for PSIS diagnostics. \item The \strong{loo} package \href{https://mc-stan.org/loo/articles/index.html}{vignettes} for demonstrations. \item The \href{https://mc-stan.org/loo/articles/online-only/faq.html}{FAQ page} on the \strong{loo} website for answers to frequently asked questions. } } loo/man/extract_log_lik.Rd0000644000176200001440000000461313762013700015262 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/extract_log_lik.R \name{extract_log_lik} \alias{extract_log_lik} \title{Extract pointwise log-likelihood from a Stan model} \usage{ extract_log_lik(stanfit, parameter_name = "log_lik", merge_chains = TRUE) } \arguments{ \item{stanfit}{A \code{stanfit} object (\pkg{rstan} package).} \item{parameter_name}{A character string naming the parameter (or generated quantity) in the Stan model corresponding to the log-likelihood.} \item{merge_chains}{If \code{TRUE} (the default), all Markov chains are merged together (i.e., stacked) and a matrix is returned. If \code{FALSE} they are kept separate and an array is returned.} } \value{ If \code{merge_chains=TRUE}, an \eqn{S} by \eqn{N} matrix of (post-warmup) extracted draws, where \eqn{S} is the size of the posterior sample and \eqn{N} is the number of data points. If \code{merge_chains=FALSE}, an \eqn{I} by \eqn{C} by \eqn{N} array, where \eqn{I \times C = S}{I * C = S}. } \description{ Convenience function for extracting the pointwise log-likelihood matrix or array from a \code{stanfit} object from the \pkg{rstan} package. Note: recent versions of \pkg{rstan} now include a \code{loo()} method for \code{stanfit} objects that handles this internally. } \details{ Stan does not automatically compute and store the log-likelihood. It is up to the user to incorporate it into the Stan program if it is to be extracted after fitting the model. In a Stan model, the pointwise log likelihood can be coded as a vector in the transformed parameters block (and then summed up in the model block) or it can be coded entirely in the generated quantities block. We recommend using the generated quantities block so that the computations are carried out only once per iteration rather than once per HMC leapfrog step. For example, the following is the \verb{generated quantities} block for computing and saving the log-likelihood for a linear regression model with \code{N} data points, outcome \code{y}, predictor matrix \code{X}, coefficients \code{beta}, and standard deviation \code{sigma}: \verb{vector[N] log_lik;} \code{for (n in 1:N) log_lik[n] = normal_lpdf(y[n] | X[n, ] * beta, sigma);} } \references{ Stan Development Team (2017). The Stan C++ Library, Version 2.16.0. \url{https://mc-stan.org/} Stan Development Team (2017). RStan: the R interface to Stan, Version 2.16.1. \url{https://mc-stan.org/} } loo/man/loo_model_weights.Rd0000644000176200001440000002311514410420140015600 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_model_weights.R \name{loo_model_weights} \alias{loo_model_weights} \alias{loo_model_weights.default} \alias{stacking_weights} \alias{pseudobma_weights} \title{Model averaging/weighting via stacking or pseudo-BMA weighting} \usage{ loo_model_weights(x, ...) \method{loo_model_weights}{default}( x, ..., method = c("stacking", "pseudobma"), optim_method = "BFGS", optim_control = list(), BB = TRUE, BB_n = 1000, alpha = 1, r_eff_list = NULL, cores = getOption("mc.cores", 1) ) stacking_weights(lpd_point, optim_method = "BFGS", optim_control = list()) pseudobma_weights(lpd_point, BB = TRUE, BB_n = 1000, alpha = 1) } \arguments{ \item{x}{A list of \code{"psis_loo"} objects (objects returned by \code{\link[=loo]{loo()}}) or pointwise log-likelihood matrices or , one for each model. If the list elements are named the names will be used to label the models in the results. Each matrix/object should have dimensions \eqn{S} by \eqn{N}, where \eqn{S} is the size of the posterior sample (with all chains merged) and \eqn{N} is the number of data points. If \code{x} is a list of log-likelihood matrices then \code{\link[=loo]{loo()}} is called internally on each matrix. Currently the \code{loo_model_weights()} function is not implemented to be used with results from K-fold CV, but you can still obtain weights using K-fold CV results by calling the \code{stacking_weights()} function directly.} \item{...}{Unused, except for the generic to pass arguments to individual methods.} \item{method}{Either \code{"stacking"} (the default) or \code{"pseudobma"}, indicating which method to use for obtaining the weights. \code{"stacking"} refers to stacking of predictive distributions and \code{"pseudobma"} refers to pseudo-BMA+ weighting (or plain pseudo-BMA weighting if argument \code{BB} is \code{FALSE}).} \item{optim_method}{If \code{method="stacking"}, a string passed to the \code{method} argument of \code{\link[stats:constrOptim]{stats::constrOptim()}} to specify the optimization algorithm. The default is \code{optim_method="BFGS"}, but other options are available (see \code{\link[stats:optim]{stats::optim()}}).} \item{optim_control}{If \code{method="stacking"}, a list of control parameters for optimization passed to the \code{control} argument of \code{\link[stats:constrOptim]{stats::constrOptim()}}.} \item{BB}{Logical used when \code{"method"}=\code{"pseudobma"}. If \code{TRUE} (the default), the Bayesian bootstrap will be used to adjust the pseudo-BMA weighting, which is called pseudo-BMA+ weighting. It helps regularize the weight away from 0 and 1, so as to reduce the variance.} \item{BB_n}{For pseudo-BMA+ weighting only, the number of samples to use for the Bayesian bootstrap. The default is \code{BB_n=1000}.} \item{alpha}{Positive scalar shape parameter in the Dirichlet distribution used for the Bayesian bootstrap. The default is \code{alpha=1}, which corresponds to a uniform distribution on the simplex space.} \item{r_eff_list}{Optionally, a list of relative effective sample size estimates for the likelihood \code{(exp(log_lik))} of each observation in each model. See \code{\link[=psis]{psis()}} and \code{\link[=relative_eff]{relative_eff()}} helper function for computing \code{r_eff}. If \code{x} is a list of \code{"psis_loo"} objects then \code{r_eff_list} is ignored.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{lpd_point}{If calling \code{stacking_weights()} or \code{pseudobma_weights()} directly, a matrix of pointwise leave-one-out (or K-fold) log likelihoods evaluated for different models. It should be a \eqn{N} by \eqn{K} matrix where \eqn{N} is sample size and \eqn{K} is the number of models. Each column corresponds to one model. These values can be calculated approximately using \code{\link[=loo]{loo()}} or by running exact leave-one-out or K-fold cross-validation.} } \value{ A numeric vector containing one weight for each model. } \description{ Model averaging via stacking of predictive distributions, pseudo-BMA weighting or pseudo-BMA+ weighting with the Bayesian bootstrap. See Yao et al. (2018), Vehtari, Gelman, and Gabry (2017), and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for background. } \details{ \code{loo_model_weights()} is a wrapper around the \code{stacking_weights()} and \code{pseudobma_weights()} functions that implements stacking, pseudo-BMA, and pseudo-BMA+ weighting for combining multiple predictive distributions. We can use approximate or exact leave-one-out cross-validation (LOO-CV) or K-fold CV to estimate the expected log predictive density (ELPD). The stacking method (\code{method="stacking"}), which is the default for \code{loo_model_weights()}, combines all models by maximizing the leave-one-out predictive density of the combination distribution. That is, it finds the optimal linear combining weights for maximizing the leave-one-out log score. The pseudo-BMA method (\code{method="pseudobma"}) finds the relative weights proportional to the ELPD of each model. However, when \code{method="pseudobma"}, the default is to also use the Bayesian bootstrap (\code{BB=TRUE}), which corresponds to the pseudo-BMA+ method. The Bayesian bootstrap takes into account the uncertainty of finite data points and regularizes the weights away from the extremes of 0 and 1. In general, we recommend stacking for averaging predictive distributions, while pseudo-BMA+ can serve as a computationally easier alternative. } \examples{ \dontrun{ ### Demonstrating usage after fitting models with RStan library(rstan) # generate fake data from N(0,1). N <- 100 y <- rnorm(N, 0, 1) # Suppose we have three models: N(-1, sigma), N(0.5, sigma) and N(0.6,sigma). stan_code <- " data { int N; vector[N] y; real mu_fixed; } parameters { real sigma; } model { sigma ~ exponential(1); y ~ normal(mu_fixed, sigma); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = normal_lpdf(y[n]| mu_fixed, sigma); }" mod <- stan_model(model_code = stan_code) fit1 <- sampling(mod, data=list(N=N, y=y, mu_fixed=-1)) fit2 <- sampling(mod, data=list(N=N, y=y, mu_fixed=0.5)) fit3 <- sampling(mod, data=list(N=N, y=y, mu_fixed=0.6)) model_list <- list(fit1, fit2, fit3) log_lik_list <- lapply(model_list, extract_log_lik) # optional but recommended r_eff_list <- lapply(model_list, function(x) { ll_array <- extract_log_lik(x, merge_chains = FALSE) relative_eff(exp(ll_array)) }) # stacking method: wts1 <- loo_model_weights( log_lik_list, method = "stacking", r_eff_list = r_eff_list, optim_control = list(reltol=1e-10) ) print(wts1) # can also pass a list of psis_loo objects to avoid recomputing loo loo_list <- lapply(1:length(log_lik_list), function(j) { loo(log_lik_list[[j]], r_eff = r_eff_list[[j]]) }) wts2 <- loo_model_weights( loo_list, method = "stacking", optim_control = list(reltol=1e-10) ) all.equal(wts1, wts2) # can provide names to be used in the results loo_model_weights(setNames(loo_list, c("A", "B", "C"))) # pseudo-BMA+ method: set.seed(1414) loo_model_weights(loo_list, method = "pseudobma") # pseudo-BMA method (set BB = FALSE): loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) # calling stacking_weights or pseudobma_weights directly lpd1 <- loo(log_lik_list[[1]], r_eff = r_eff_list[[1]])$pointwise[,1] lpd2 <- loo(log_lik_list[[2]], r_eff = r_eff_list[[2]])$pointwise[,1] lpd3 <- loo(log_lik_list[[3]], r_eff = r_eff_list[[3]])$pointwise[,1] stacking_weights(cbind(lpd1, lpd2, lpd3)) pseudobma_weights(cbind(lpd1, lpd2, lpd3)) pseudobma_weights(cbind(lpd1, lpd2, lpd3), BB = FALSE) } } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018) Using stacking to average Bayesian predictive distributions. \emph{Bayesian Analysis}, advance publication, doi:10.1214/17-BA1091. (\href{https://projecteuclid.org/euclid.ba/1516093227}{online}). } \seealso{ \itemize{ \item The \strong{loo} package \href{https://mc-stan.org/loo/articles/}{vignettes}, particularly \href{https://mc-stan.org/loo/articles/loo2-weights.html}{Bayesian Stacking and Pseudo-BMA weights using the \strong{loo} package}. \item \code{\link[=loo]{loo()}} for details on leave-one-out ELPD estimation. \item \code{\link[=constrOptim]{constrOptim()}} for the choice of optimization methods and control-parameters. \item \code{\link[=relative_eff]{relative_eff()}} for computing \code{r_eff}. } } loo/man/compare.Rd0000644000176200001440000000645014407123455013545 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/compare.R \name{compare} \alias{compare} \title{Model comparison (deprecated, old version)} \usage{ compare(..., x = list()) } \arguments{ \item{...}{At least two objects returned by \code{\link[=loo]{loo()}} (or \code{\link[=waic]{waic()}}).} \item{x}{A list of at least two objects returned by \code{\link[=loo]{loo()}} (or \code{\link[=waic]{waic()}}). This argument can be used as an alternative to specifying the objects in \code{...}.} } \value{ A vector or matrix with class \code{'compare.loo'} that has its own print method. If exactly two objects are provided in \code{...} or \code{x}, then the difference in expected predictive accuracy and the standard error of the difference are returned. If more than two objects are provided then a matrix of summary information is returned (see \strong{Details}). } \description{ \strong{This function is deprecated}. Please use the new \code{\link[=loo_compare]{loo_compare()}} function instead. } \details{ When comparing two fitted models, we can estimate the difference in their expected predictive accuracy by the difference in \code{elpd_loo} or \code{elpd_waic} (or multiplied by -2, if desired, to be on the deviance scale). \emph{When that difference, \code{elpd_diff}, is positive then the expected predictive accuracy for the second model is higher. A negative \code{elpd_diff} favors the first model.} When using \code{compare()} with more than two models, the values in the \code{elpd_diff} and \code{se_diff} columns of the returned matrix are computed by making pairwise comparisons between each model and the model with the best ELPD (i.e., the model in the first row). Although the \code{elpd_diff} column is equal to the difference in \code{elpd_loo}, do not expect the \code{se_diff} column to be equal to the the difference in \code{se_elpd_loo}. To compute the standard error of the difference in ELPD we use a paired estimate to take advantage of the fact that the same set of \emph{N} data points was used to fit both models. These calculations should be most useful when \emph{N} is large, because then non-normality of the distribution is not such an issue when estimating the uncertainty in these sums. These standard errors, for all their flaws, should give a better sense of uncertainty than what is obtained using the current standard approach of comparing differences of deviances to a Chi-squared distribution, a practice derived for Gaussian linear models or asymptotically, and which only applies to nested models in any case. } \examples{ \dontrun{ loo1 <- loo(log_lik1) loo2 <- loo(log_lik2) print(compare(loo1, loo2), digits = 3) print(compare(x = list(loo1, loo2))) waic1 <- waic(log_lik1) waic2 <- waic(log_lik2) compare(waic1, waic2) } } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } loo/man/loo_predictive_metric.Rd0000644000176200001440000000660114411130104016447 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/loo_predictive_metric.R \name{loo_predictive_metric} \alias{loo_predictive_metric} \alias{loo_predictive_metric.matrix} \title{Estimate leave-one-out predictive performance..} \usage{ loo_predictive_metric(x, ...) \method{loo_predictive_metric}{matrix}( x, y, log_lik, ..., metric = c("mae", "rmse", "mse", "acc", "balanced_acc"), r_eff = NULL, cores = getOption("mc.cores", 1) ) } \arguments{ \item{x}{A numeric matrix of predictions.} \item{...}{Additional arguments passed on to \code{\link[=E_loo]{E_loo()}}} \item{y}{A numeric vector of observations. Length should be equal to the number of rows in \code{x}.} \item{log_lik}{A matrix of pointwise log-likelihoods. Should be of same dimension as \code{x}.} \item{metric}{The type of predictive metric to be used. Currently supported options are \code{"mae"}, \code{"rmse"} and \code{"mse"} for regression and for binary classification \code{"acc"} and \code{"balanced_acc"}. \describe{ \item{\code{"mae"}}{ Mean absolute error. } \item{\code{"mse"}}{ Mean squared error. } \item{\code{"rmse"}}{ Root mean squared error, given by as the square root of \code{MSE}. } \item{\code{"acc"}}{ The proportion of predictions indicating the correct outcome. } \item{\code{"balanced_acc"}}{ Balanced accuracy is given by the average of true positive and true negative rates. } }} \item{r_eff}{A Vector of relative effective sample size estimates containing one element per observation. See \code{\link[=psis]{psis()}} for more details.} \item{cores}{The number of cores to use for parallelization of \verb{[psis()]}. See \code{\link[=psis]{psis()}} for details.} } \value{ A list with the following components: \describe{ \item{\code{estimate}}{ Estimate of the given metric. } \item{\code{se}}{ Standard error of the estimate. } } } \description{ The \code{loo_predictive_metric()} function computes estimates of leave-one-out predictive metrics given a set of predictions and observations. Currently supported metrics are mean absolute error, mean squared error and root mean squared error for continuous predictions and accuracy and balanced accuracy for binary classification. Predictions are passed on to the \code{\link[=E_loo]{E_loo()}} function, so this function assumes that the PSIS approximation is working well. } \examples{ \donttest{ if (requireNamespace("rstanarm", quietly = TRUE)) { # Use rstanarm package to quickly fit a model and get both a log-likelihood # matrix and draws from the posterior predictive distribution library("rstanarm") # data from help("lm") ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14) trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69) d <- data.frame( weight = c(ctl, trt), group = gl(2, 10, 20, labels = c("Ctl","Trt")) ) fit <- stan_glm(weight ~ group, data = d, refresh = 0) ll <- log_lik(fit) r_eff <- relative_eff(exp(-ll), chain_id = rep(1:4, each = 1000)) mu_pred <- posterior_epred(fit) # Leave-one-out mean absolute error of predictions mae <- loo_predictive_metric(x = mu_pred, y = d$weight, log_lik = ll, pred_error = 'mae', r_eff = r_eff) # Leave-one-out 90\%-quantile of mean absolute error mae_90q <- loo_predictive_metric(x = mu_pred, y = d$weight, log_lik = ll, pred_error = 'mae', r_eff = r_eff, type = 'quantile', probs = 0.9) } } } loo/man/psis_approximate_posterior.Rd0000644000176200001440000000624114407123455017612 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/psis_approximate_posterior.R \name{psis_approximate_posterior} \alias{psis_approximate_posterior} \title{Diagnostics for Laplace and ADVI approximations and Laplace-loo and ADVI-loo} \usage{ psis_approximate_posterior( log_p = NULL, log_g = NULL, log_liks = NULL, cores, save_psis, ..., log_q = NULL ) } \arguments{ \item{log_p}{The log-posterior (target) evaluated at S samples from the proposal distribution (g). A vector of length S.} \item{log_g}{The log-density (proposal) evaluated at S samples from the proposal distribution (g). A vector of length S.} \item{log_liks}{A log-likelihood matrix of size S * N, where N is the number of observations and S is the number of samples from q. See \code{\link[=loo.matrix]{loo.matrix()}} for details. Default is \code{NULL}. Then only the posterior is evaluated using the k_hat diagnostic.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by \code{options(mc.cores = NUMBER)}. The old option \code{loo.cores} is now deprecated but will be given precedence over \code{mc.cores} until \code{loo.cores} is removed in a future release. \strong{As of version 2.0.0 the default is now 1 core if \code{mc.cores} is not set}, but we recommend using as many (or close to as many) cores as possible. \itemize{ \item Note for Windows 10 users: it is \strong{strongly} \href{https://github.com/stan-dev/loo/issues/94}{recommended} to avoid using the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argument or setting \code{mc.cores} interactively or in a script is fine). }} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be saved in the returned object? The \code{loo()} function calls \code{\link[=psis]{psis()}} internally but by default discards the (potentially large) \code{"psis"} object after using it to compute the LOO-CV summaries. Setting \code{save_psis=TRUE} will add a \code{psis_object} component to the list returned by \code{loo}. Currently this is only needed if you plan to use the \code{\link[=E_loo]{E_loo()}} function to compute weighted expectations after running \code{loo}.} \item{log_q}{Deprecated argument name (the same as log_g).} } \value{ If log likelihoods are supplied, the function returns a \code{"loo"} object, otherwise the function returns a \code{"psis"} object. } \description{ Diagnostics for Laplace and ADVI approximations and Laplace-loo and ADVI-loo } \references{ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. \emph{Statistics and Computing}. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } \seealso{ \code{\link[=loo]{loo()}} and \code{\link[=psis]{psis()}} } \keyword{internal} loo/DESCRIPTION0000644000176200001440000000471614411522703012560 0ustar liggesusersPackage: loo Type: Package Title: Efficient Leave-One-Out Cross-Validation and WAIC for Bayesian Models Version: 2.6.0 Date: 2023-03-30 Authors@R: c(person("Aki", "Vehtari", email = "Aki.Vehtari@aalto.fi", role = c("aut")), person("Jonah", "Gabry", email = "jsg2201@columbia.edu", role = c("cre", "aut")), person("Mans", "Magnusson", role = c("aut")), person("Yuling", "Yao", role = c("aut")), person("Paul-Christian", "Bürkner", role = c("aut")), person("Topi", "Paananen", role = c("aut")), person("Andrew", "Gelman", role = c("aut")), person("Ben", "Goodrich", role = c("ctb")), person("Juho", "Piironen", role = c("ctb")), person("Bruno", "Nicenboim", role = c("ctb")), person("Leevi", "Lindgren", role = c("ctb"))) Maintainer: Jonah Gabry URL: https://mc-stan.org/loo/, https://discourse.mc-stan.org BugReports: https://github.com/stan-dev/loo/issues Description: Efficient approximate leave-one-out cross-validation (LOO) for Bayesian models fit using Markov chain Monte Carlo, as described in Vehtari, Gelman, and Gabry (2017) . The approximation uses Pareto smoothed importance sampling (PSIS), a new procedure for regularizing importance weights. As a byproduct of the calculations, we also obtain approximate standard errors for estimated predictive errors and for the comparison of predictive errors between models. The package also provides methods for using stacking and other model weighting techniques to average Bayesian predictive distributions. License: GPL (>= 3) LazyData: TRUE Depends: R (>= 3.1.2) Imports: checkmate, matrixStats (>= 0.52), parallel, stats Suggests: bayesplot (>= 1.7.0), brms (>= 2.10.0), ggplot2, graphics, knitr, posterior, rmarkdown, rstan, rstanarm (>= 2.19.0), rstantools, spdep, testthat (>= 2.1.0) VignetteBuilder: knitr Encoding: UTF-8 SystemRequirements: pandoc (>= 1.12.3), pandoc-citeproc RoxygenNote: 7.2.3 NeedsCompilation: no Packaged: 2023-03-31 05:10:34 UTC; jgabry Author: Aki Vehtari [aut], Jonah Gabry [cre, aut], Mans Magnusson [aut], Yuling Yao [aut], Paul-Christian Bürkner [aut], Topi Paananen [aut], Andrew Gelman [aut], Ben Goodrich [ctb], Juho Piironen [ctb], Bruno Nicenboim [ctb], Leevi Lindgren [ctb] Repository: CRAN Date/Publication: 2023-03-31 09:20:03 UTC loo/build/0000755000176200001440000000000014411465510012143 5ustar liggesusersloo/build/vignette.rds0000644000176200001440000000112314411465510014477 0ustar liggesusersTMs0u@JQn:rqk:LCUccMe#)ĿFXr"աpAvQQ?|ǵqEѐKy/e\вTM ɨra{+قi=lPY0IIMZ01R$9MTt́M ;҆ K|\ۂ9پafѻ:;w[m A"kRwhWQl+r뾢S*j=Truloo/build/partial.rdb0000644000176200001440000000007514411455165014277 0ustar liggesusersb```b`abb`b1 H020piּb C".X7loo/tests/0000755000176200001440000000000014407123455012212 5ustar liggesusersloo/tests/testthat/0000755000176200001440000000000014411522703014044 5ustar liggesusersloo/tests/testthat/test_relative_eff.R0000644000176200001440000000314714411130104017654 0ustar liggesuserslibrary(loo) options(mc.cores = 1) set.seed(123) context("relative_eff methods") LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() test_that("relative_eff results haven't changed", { expect_equal_to_reference(relative_eff(exp(LLarr)), "reference-results/relative_eff.rds") }) test_that("relative_eff is equal to ESS / S", { dims <- dim(LLarr) ess <- r_eff <- rep(NA, dims[3]) for (j in 1:dims[3]) { r_eff[j] <- relative_eff(LLarr[,,1, drop=FALSE]) ess[j] <- ess_rfun(LLarr[,,1]) } S <- prod(dim(LLarr)[1:2]) expect_equal(r_eff, ess / S) }) test_that("relative_eff array and matrix methods return identical output", { r_eff_arr <- relative_eff(exp(LLarr)) r_eff_mat <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = nrow(LLarr))) expect_identical(r_eff_arr, r_eff_mat) }) test_that("relative_eff matrix and function methods return identical output", { source(test_path("data-for-tests/function_method_stuff.R")) chain <- rep(1, nrow(draws)) r_eff_mat <- relative_eff(llmat_from_fn, chain_id = chain) r_eff_fn <- relative_eff(llfun, chain_id = chain, data = data, draws = draws, cores = 1) expect_identical(r_eff_mat, r_eff_fn) }) test_that("relative_eff with multiple cores runs", { skip_on_cran() source(test_path("data-for-tests/function_method_stuff.R")) dim(llmat_from_fn) <- c(nrow(llmat_from_fn), 1, ncol(llmat_from_fn)) r_eff_arr <- relative_eff(llmat_from_fn, cores = 2) r_eff_fn <- relative_eff( llfun, chain_id = rep(1, nrow(draws)), data = data, draws = draws, cores = 2 ) expect_identical(r_eff_arr, r_eff_fn) }) loo/tests/testthat/test_loo_approximate_posterior.R0000644000176200001440000001031113576706411022545 0ustar liggesuserssuppressPackageStartupMessages(library(loo)) context("loo_approximate_posterior") # Create test data # Checked by Mans M and Paul B 24th of June 2019 set.seed(123) N <- 50 K <- 10 S <- 1000 a0 <- 1 b0 <- 1 p <- 0.5 y <- rbinom(N, size = K, prob = p) fake_data <- data.frame(y,K) # The log posterior log_post <- function(p, y, a0, b0, K) { log_lik <- sum(dbinom(x = y, size = K, prob = p, log = TRUE)) # the log likelihood log_post <- log_lik + dbeta(x = p, shape1 = a0, shape2 = b0, log = TRUE) # the log prior log_post } it <- optim(par = 0.5, fn = log_post, control = list(fnscale = -1), hessian = TRUE, y = y, a0 = a0, b0 = b0, K = K, lower = 0.01, upper = 0.99, method = "Brent") lap_params <- c(mu = it$par, sd = sqrt(solve(-it$hessian))) a <- a0 + sum(y) b <- b0 + N * K - sum(y) fake_true_posterior <- as.matrix(rbeta(S, a, b)) fake_laplace_posterior <- as.matrix(rnorm(n = S, mean = lap_params["mu"], sd = lap_params["sd"])) # mean(fake_laplace_posterior); sd(fake_laplace_posterior) p_draws <- as.vector(fake_laplace_posterior) log_p <- numeric(S) for(s in 1:S){ log_p[s] <- log_post(p_draws[s], y = y, a0 = a0, b0 = b0, K = K) } log_g <- as.vector(dnorm(as.vector(fake_laplace_posterior), mean = lap_params["mu"], sd = lap_params["sd"], log = TRUE)) llfun <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } ll <- matrix(0, nrow = S, ncol = N) for(j in 1:N){ ll[, j] <- llfun(data_i = fake_data[j, , drop=FALSE], draws = fake_laplace_posterior) } test_that("loo_approximate_posterior.array works as loo_approximate_posterior.matrix", { # Create array with two "chains" log_p_mat <- matrix(log_p, nrow = (S/2), ncol = 2) log_g_mat <- matrix(log_g, nrow = (S/2), ncol = 2) ll_array <- array(0, dim = c((S/2), 2 , ncol(ll))) ll_array[,1,] <- ll[1:(S/2),] ll_array[,2,] <- ll[(S/2 + 1):S,] # Assert that they are ok expect_equivalent(ll_array[1:2,1,1:2], ll[1:2,1:2]) expect_equivalent(ll_array[1:2,2,1:2], ll[(S/2+1):((S/2)+2),1:2]) # Compute aploo expect_silent(aploo1 <- loo_approximate_posterior.matrix(x = ll, log_p = log_p, log_g = log_g)) expect_silent(aploo2 <- loo_approximate_posterior.array(x = ll_array, log_p = log_p_mat, log_g = log_g_mat)) expect_silent(aploo1b <- loo.matrix(x = ll, r_eff = rep(1, N))) # Check equivalence expect_equal(aploo1$estimates, aploo2$estimates) expect_equal(class(aploo1), class(aploo2)) expect_failure(expect_equal(aploo1b$estimates, aploo2$estimates)) expect_failure(expect_equal(class(aploo1), class(aploo1b))) # Should fail with matrix expect_error(aploo2 <- loo_approximate_posterior.matrix(x = ll, log_p = as.matrix(log_p), log_g = log_g)) expect_error(aploo2 <- loo_approximate_posterior.matrix(x = ll, log_p = as.matrix(log_p), log_g = as.matrix(log_g))) # Expect log_p and log_g be stored in the approximate_posterior in the same way expect_length(aploo1$approximate_posterior$log_p, nrow(ll)) expect_length(aploo1$approximate_posterior$log_g, nrow(ll)) expect_equal(aploo1$approximate_posterior$log_p, aploo2$approximate_posterior$log_p) expect_equal(aploo1$approximate_posterior$log_g, aploo2$approximate_posterior$log_g) }) test_that("loo_approximate_posterior.function works as loo_approximate_posterior.matrix", { # Compute aploo expect_silent(aploo1 <- loo_approximate_posterior.matrix(x = ll, log_p = log_p, log_g = log_g)) expect_silent(aploo1b <- loo.matrix(x = ll, r_eff = rep(1, N))) expect_silent(aploo2 <- loo_approximate_posterior.function(x = llfun, log_p = log_p, log_g = log_g, data = fake_data, draws = fake_laplace_posterior)) # Check equivalence expect_equal(aploo1$estimates, aploo2$estimates) expect_equal(class(aploo1), class(aploo2)) expect_failure(expect_equal(aploo1b$estimates, aploo2$estimates)) # Check equivalence # Expect log_p and log_g be stored in the approximate_posterior in the same way expect_length(aploo2$approximate_posterior$log_p, nrow(fake_laplace_posterior)) expect_length(aploo2$approximate_posterior$log_g, nrow(fake_laplace_posterior)) expect_equal(aploo1$approximate_posterior$log_p, aploo2$approximate_posterior$log_p) expect_equal(aploo1$approximate_posterior$log_g, aploo2$approximate_posterior$log_g) }) loo/tests/testthat/test_kfold_helpers.R0000644000176200001440000000651613575772017020076 0ustar liggesuserslibrary(loo) set.seed(14014) context("kfold helper functions") test_that("kfold_split_random works", { fold_rand <- kfold_split_random(10, 100) expect_length(fold_rand, 100) expect_equal(sort(unique(fold_rand)), 1:10) expect_equal(sum(fold_rand == 2), sum(fold_rand == 9)) }) test_that("kfold_split_stratified works", { y <- rep(c(0, 1), times = c(10, 190)) fold_strat <- kfold_split_stratified(5, y) expect_true(all(table(fold_strat) == 40)) y <- rep(c(1, 2, 3), times = c(15, 33, 42)) fold_strat <- kfold_split_stratified(7, y) expect_equal(range(table(fold_strat)), c(12, 13)) y <- mtcars$cyl fold_strat <- kfold_split_stratified(10, y) expect_equal(range(table(fold_strat)), c(3, 4)) }) test_that("kfold_split_grouped works", { grp <- gl(n = 50, k = 15, labels = state.name) fold_group <- kfold_split_grouped(x = grp) expect_true(all(table(fold_group) == 75)) expect_equal(sum(table(fold_group)), length(grp)) fold_group <- kfold_split_grouped(K = 9, x = grp) expect_false(all(table(fold_group) == 75)) expect_equal(sum(table(fold_group)), length(grp)) grp <- gl(n = 50, k = 4, labels = state.name) grp[grp == "Montana"] <- "Utah" fold_group <- kfold_split_grouped(K = 10, x = grp) expect_equal(sum(table(fold_group)), length(grp) - 4) grp <- rep(c("A","B"), each = 20) fold_group <- kfold_split_grouped(K = 2, x = grp) expect_equal(fold_group, as.integer(as.factor(grp))) }) test_that("kfold helpers throw correct errors", { expect_error(kfold_split_random(10), "!is.null(N) is not TRUE", fixed = TRUE) expect_error(kfold_split_random(10.5, 100), "K == as.integer(K) is not TRUE", fixed = TRUE) expect_error(kfold_split_random(10, 100.5), "N == as.integer(N) is not TRUE", fixed = TRUE) expect_error(kfold_split_random(K = c(1,1), N = 100), "length(K) == 1 is not TRUE", fixed = TRUE) expect_error(kfold_split_random(N = c(100, 100)), "length(N) == 1 is not TRUE", fixed = TRUE) expect_error(kfold_split_random(K = 5, N = 4), "K <= N is not TRUE", fixed = TRUE) expect_error(kfold_split_random(K = 1, N = 4), "K > 1 is not TRUE", fixed = TRUE) y <- sample(c(0, 1), size = 200, replace = TRUE, prob = c(0.05, 0.95)) expect_error(kfold_split_stratified(10), "!is.null(x) is not TRUE", fixed = TRUE) expect_error(kfold_split_stratified(10.5, y), "K == as.integer(K) is not TRUE", fixed = TRUE) expect_error(kfold_split_stratified(K = c(1,1), y), "length(K) == 1 is not TRUE", fixed = TRUE) expect_error(kfold_split_stratified(K = 201, y), "K <= length(x) is not TRUE", fixed = TRUE) expect_error(kfold_split_stratified(K = 1, y), "K > 1 is not TRUE", fixed = TRUE) grp <- gl(n = 50, k = 15) expect_error(kfold_split_grouped(10), "!is.null(x) is not TRUE", fixed = TRUE) expect_error(kfold_split_grouped(3, c(1,1,1)), "'K' must not be bigger than the number of levels/groups in 'x'", fixed = TRUE) expect_error(kfold_split_grouped(10.5, grp), "K == as.integer(K) is not TRUE", fixed = TRUE) expect_error(kfold_split_grouped(K = c(1,1), grp), "length(K) == 1 is not TRUE", fixed = TRUE) expect_error(kfold_split_grouped(K = 1, grp), "K > 1 is not TRUE", fixed = TRUE) }) test_that("print_dims.kfold works", { xx <- structure(list(), K = 17, class = c("kfold", "loo")) expect_output(print_dims(xx), "Based on 17-fold cross-validation") attr(xx, "K") <- NULL expect_silent(print_dims(xx)) }) loo/tests/testthat/test_psislw.R0000644000176200001440000000445513267637066016600 0ustar liggesuserslibrary(loo) SW <- suppressWarnings context("psislw") set.seed(123) x <- matrix(rnorm(5000), 100, 50) expect_deprecated <- function(object) { testthat::expect_warning(object, "deprecated", ignore.case = TRUE) } test_that("psislw throws deprecation warning", { expect_deprecated(psislw(x[, 1])) }) test_that("psislw handles special cases, throws appropriate errors/warnings", { expect_warning( psis <- psislw(x[, 1], wcp = 0.01), regexp = "All tail values are the same. Weights are truncated but not smoothed" ) expect_true(is.infinite(psis$pareto_k)) expect_warning( psislw(x[, 1], wcp = 0.01), regexp = "Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details", fixed = TRUE ) expect_error( expect_deprecated(psislw(wcp = 0.2)), regexp = "'lw' or 'llfun' and 'llargs' must be specified" ) }) test_that("psislw returns expected results", { psis <- SW(psislw(x[, 1])) lw <- psis$lw_smooth expect_equal(length(psis), 2L) expect_equal(nrow(lw), nrow(x)) expect_equal(lw[1], -5.6655489517740527106) expect_equal(lw[50], -5.188442371693668953) expect_equal(range(lw), c(-7.4142421808626526314, -2.6902215137943321643)) expect_equal(psis$pareto_k, 0.17364505906017813075) }) test_that("psislw function and matrix methods return same result", { set.seed(024) # fake data and posterior draws N <- 50; K <- 10; S <- 100; a0 <- 3; b0 <- 2 p <- rbeta(1, a0, b0) y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) draws <- rbeta(S, a, b) data <- data.frame(y,K) llfun <- function(i, data, draws) { dbinom(data$y, size = data$K, prob = draws, log = TRUE) } psislw_with_fn <- SW(psislw(llfun = llfun, llargs = nlist(data, draws, N, S))) # Check that we get same answer if using log-likelihood matrix ll <- sapply(1:N, function(i) llfun(i, data[i,, drop=FALSE], draws)) psislw_with_mat <- SW(psislw(-ll)) expect_equal(psislw_with_fn, psislw_with_mat) }) test_that("psislw_warnings helper works properly", { k <- c(0, 0.1, 0.55, 0.75) expect_silent(psislw_warnings(k[1:2])) expect_warning(psislw_warnings(k[1:3]), "Some Pareto k diagnostic values are slightly high") expect_warning(psislw_warnings(k), "Some Pareto k diagnostic values are too high") }) loo/tests/testthat/test_loo_subsampling.R0000644000176200001440000017305314411130104020422 0ustar liggesuserslibrary(loo) options(mc.cores = 1) context("loo_subsampling") test_that("overall loo_subampling works as expected (compared with loo) for diff_est", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { # each time called internally within loo the arguments will be equal to: # data_i: ith row of fdata (fake_data[i,, drop=FALSE]) # draws: entire fake_posterior matrix dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(true_loo, "psis_loo") expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 500, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss, "psis_loo_ss") # Check consistency expect_equivalent(loo_ss$pointwise[, "elpd_loo_approx"], loo_ss$loo_subsampling$elpd_loo_approx[loo_ss$pointwise[, "idx"]]) # Expect values z <- 2 expect_lte(loo_ss$estimates["elpd_loo", "Estimate"] - z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss$estimates["elpd_loo", "Estimate"] + z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss$estimates["p_loo", "Estimate"] - z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss$estimates["p_loo", "Estimate"] + z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss$estimates["looic", "Estimate"] - z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss$estimates["looic", "Estimate"] + z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss$estimates["looic", "Estimate"], tol = 0.00000001)) # Test that observations works as expected expect_message(loo_ss2 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = obs_idx(loo_ss), loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_equal(loo_ss2$estimates, loo_ss$estimates, tol = 0.00000001) expect_silent(loo_ss2 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = loo_ss, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_equal(loo_ss2$estimates, loo_ss$estimates, tol = 0.00000001) # Test lpd expect_silent(loo_ss_lpd <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 500, loo_approximation = "lpd", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss_lpd, "psis_loo_ss") z <- 2 expect_lte(loo_ss_lpd$estimates["elpd_loo", "Estimate"] - z * loo_ss_lpd$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss_lpd$estimates["elpd_loo", "Estimate"] + z * loo_ss_lpd$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss_lpd$estimates["p_loo", "Estimate"] - z * loo_ss_lpd$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss_lpd$estimates["p_loo", "Estimate"] + z * loo_ss_lpd$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss_lpd$estimates["looic", "Estimate"] - z * loo_ss_lpd$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss_lpd$estimates["looic", "Estimate"] + z * loo_ss_lpd$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss_lpd$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss_lpd$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss_lpd$estimates["looic", "Estimate"], tol = 0.00000001)) expect_silent(loo_ss_lpd10 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 500, loo_approximation = "lpd", loo_approximation_draws = 10, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss_lpd10, "psis_loo_ss") z <- 2 expect_lte(loo_ss_lpd10$estimates["elpd_loo", "Estimate"] - z * loo_ss_lpd10$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss_lpd10$estimates["elpd_loo", "Estimate"] + z * loo_ss_lpd10$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss_lpd10$estimates["p_loo", "Estimate"] - z * loo_ss_lpd10$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss_lpd10$estimates["p_loo", "Estimate"] + z * loo_ss_lpd10$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss_lpd10$estimates["looic", "Estimate"] - z * loo_ss_lpd10$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss_lpd10$estimates["looic", "Estimate"] + z * loo_ss_lpd10$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss_lpd10$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss_lpd10$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss_lpd10$estimates["looic", "Estimate"], tol = 0.00000001)) # Test conversion of objects expect_silent(true_loo_2 <- loo:::as.psis_loo.psis_loo(true_loo)) expect_silent(true_loo_ss <- loo:::as.psis_loo_ss.psis_loo(true_loo)) expect_s3_class(true_loo_ss, "psis_loo_ss") expect_silent(true_loo_conv <- loo:::as.psis_loo.psis_loo_ss(true_loo_ss)) expect_failure(expect_s3_class(true_loo_conv, "psis_loo_ss")) expect_equal(true_loo_conv, true_loo) expect_error(loo:::as.psis_loo.psis_loo_ss(loo_ss)) }) test_that("loo with subsampling of all observations works as ordinary loo.", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(true_loo, "psis_loo") expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1000, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss, "psis_loo_ss") expect_error(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1001, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001) expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss$estimates["p_loo", "Estimate"], tol = 0.00000001) expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss$estimates["looic", "Estimate"], tol = 0.00000001) expect_equal(dim(true_loo),dim(loo_ss)) expect_equal(true_loo$diagnostics, loo_ss$diagnostics) expect_equal(max(loo_ss$pointwise[, "m_i"]), 1) }) test_that("overall loo_subsample works with diff_srs as expected (compared with loo)", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(x = llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 200, loo_approximation = "plpd", estimator = "diff_srs", r_eff = rep(1, nrow(fake_data)))) expect_equal(true_loo$estimates[1,1], loo_ss$estimates[1,1], tol = 0.1) }) test_that("Test the srs estimator with 'none' approximation", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(true_loo, "psis_loo") expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 200, loo_approximation = "none", estimator = "srs", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss, "psis_loo_ss") expect_error(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1100, loo_approximation = "none", estimator = "srs", r_eff = rep(1, nrow(fake_data)))) expect_equal(length(obs_idx(loo_ss)), nobs(loo_ss)) # Check consistency expect_equivalent(loo_ss$pointwise[, "elpd_loo_approx"], loo_ss$loo_subsampling$elpd_loo_approx[loo_ss$pointwise[, "idx"]]) # Expect values z <- 2 expect_lte(loo_ss$estimates["elpd_loo", "Estimate"] - z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss$estimates["elpd_loo", "Estimate"] + z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss$estimates["p_loo", "Estimate"] - z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss$estimates["p_loo", "Estimate"] + z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss$estimates["looic", "Estimate"] - z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss$estimates["looic", "Estimate"] + z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss$estimates["looic", "Estimate"], tol = 0.00000001)) }) test_that("Test the Hansen-Hurwitz estimator", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(true_loo, "psis_loo") expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 300, loo_approximation = "plpd", estimator = "hh_pps", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss, "psis_loo_ss") expect_silent(loo_ss_max <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1100, loo_approximation = "plpd", estimator = "hh_pps", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss_max, "psis_loo_ss") expect_silent(loo_ss_max2 <- update(loo_ss, draws = fake_posterior, data = fake_data, observations = 1100, r_eff = rep(1, nrow(fake_data)))) expect_equal(nobs(loo_ss_max2), 1100) expect_gt(max(loo_ss_max2$pointwise[, "m_i"]), 1) expect_error(loo_ss_max2 <- update(loo_ss_max2, draws = fake_posterior, data = fake_data, observations = 300, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_max3 <- update(loo_ss, draws = fake_posterior, data = fake_data, observations = 1500, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss2 <- update(loo_ss, draws = fake_posterior, data = fake_data, observations = loo_ss, r_eff = rep(1, nrow(fake_data)))) expect_error(loo_ss2 <- update(loo_ss, draws = fake_posterior, data = fake_data, observations = loo_ss, loo_approximation = "lpd", r_eff = rep(1, nrow(fake_data)))) expect_equal(loo_ss$estimates, loo_ss2$estimates) expect_equal(length(obs_idx(loo_ss_max)), length(obs_idx(loo_ss_max2))) expect_equal(length(obs_idx(loo_ss_max)), nobs(loo_ss_max)) # Check consistency expect_equivalent(loo_ss$pointwise[, "elpd_loo_approx"], loo_ss$loo_subsampling$elpd_loo_approx[loo_ss$pointwise[, "idx"]]) # Check consistency expect_equivalent(loo_ss_max$pointwise[, "elpd_loo_approx"], loo_ss_max$loo_subsampling$elpd_loo_approx[loo_ss_max$pointwise[, "idx"]]) # Expect values z <- 2 expect_lte(loo_ss$estimates["elpd_loo", "Estimate"] - z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss$estimates["elpd_loo", "Estimate"] + z * loo_ss$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss$estimates["p_loo", "Estimate"] - z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss$estimates["p_loo", "Estimate"] + z * loo_ss$estimates["p_loo", "subsampling SE"], true_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss$estimates["looic", "Estimate"] - z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss$estimates["looic", "Estimate"] + z * loo_ss$estimates["looic", "subsampling SE"], true_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(true_loo$estimates["elpd_loo", "Estimate"], loo_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["p_loo", "Estimate"], loo_ss$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(true_loo$estimates["looic", "Estimate"], loo_ss$estimates["looic", "Estimate"], tol = 0.00000001)) expect_lte(loo_ss_max$estimates["elpd_loo", "Estimate"] - z * loo_ss_max$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss_max$estimates["elpd_loo", "Estimate"] + z * loo_ss_max$estimates["elpd_loo", "subsampling SE"], true_loo$estimates["elpd_loo", "Estimate"]) }) test_that("update.psis_loo_ss works as expected (compared with loo)", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(true_loo <- loo(llfun_test, draws = fake_posterior, data = fake_data, r_eff = rep(1, nrow(fake_data)))) expect_s3_class(true_loo, "psis_loo") expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 500, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_s3_class(loo_ss, "psis_loo_ss") # Check error when draws and data dimensions differ expect_error(loo_ss2 <- update(object = loo_ss, draws = cbind(fake_posterior, 1), data = fake_data, observations = 600, r_eff = rep(1, nrow(fake_data)))) expect_error(loo_ss2 <- update(object = loo_ss, draws = fake_posterior, data = fake_data[-1,], observations = 600, r_eff = rep(1, nrow(fake_data)))) # Add tests for adding observations expect_silent(loo_ss2 <- update(object = loo_ss, draws = fake_posterior, data = fake_data, observations = 600, r_eff = rep(1, nrow(fake_data)))) expect_equal(dim(loo_ss2)[2] - dim(loo_ss)[2], expected = 100) expect_equal(dim(loo_ss2)[2], expected = dim(loo_ss2$pointwise)[1]) expect_length(loo_ss2$diagnostics$pareto_k, 600) expect_length(loo_ss2$diagnostics$n_eff, 600) for(i in 1:nrow(loo_ss2$estimates)) { expect_lt(loo_ss2$estimates[i, "subsampling SE"], loo_ss$estimates[i, "subsampling SE"]) } expect_silent(loo_ss2b <- update(object = loo_ss, draws = fake_posterior, data = fake_data)) expect_equal(loo_ss2b$estimates, loo_ss$estimates) expect_equal(loo_ss2b$pointwise, loo_ss$pointwise) expect_equal(loo_ss2b$diagnostics$pareto_k, loo_ss$diagnostics$pareto_k) expect_equal(loo_ss2b$diagnostics$n_eff, loo_ss$diagnostics$n_eff) expect_silent(loo_ss3 <- update(object = loo_ss2, draws = fake_posterior, data = fake_data, observations = loo_ss)) expect_equal(loo_ss3$estimates, loo_ss$estimates) expect_equal(loo_ss3$pointwise, loo_ss$pointwise) expect_equal(loo_ss3$diagnostics$pareto_k, loo_ss$diagnostics$pareto_k) expect_equal(loo_ss3$diagnostics$n_eff, loo_ss$diagnostics$n_eff) expect_silent(loo_ss4 <- update(object = loo_ss, draws = fake_posterior, data = fake_data, observations = 1000, r_eff = rep(1, nrow(fake_data)))) expect_equal(loo_ss4$estimates[,1], true_loo$estimates[,1]) expect_equal(loo_ss4$estimates[,2], true_loo$estimates[,2], tol = 0.001) expect_silent(loo_ss5 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1000, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) ss4_order <- order(loo_ss4$pointwise[, "idx"]) expect_equal(loo_ss4$pointwise[ss4_order,c(1,3,4)], loo_ss5$pointwise[,c(1,3,4)]) expect_equal(loo_ss4$diagnostics$pareto_k[ss4_order], loo_ss5$diagnostics$pareto_k) expect_equal(loo_ss4$diagnostics$n_eff[ss4_order], loo_ss5$diagnostics$n_eff) expect_equal(loo_ss4$pointwise[ss4_order,c(1,3,4)], true_loo$pointwise[,c(1,3,4)]) expect_equal(loo_ss4$diagnostics$pareto_k[ss4_order], true_loo$diagnostics$pareto_k) expect_equal(loo_ss4$diagnostics$n_eff[ss4_order], true_loo$diagnostics$n_eff) expect_error(loo_ss_min <- update(object = loo_ss, draws = fake_posterior, data = fake_data, observations = 50, r_eff = rep(1, nrow(fake_data)))) expect_silent(true_loo_ss <- loo:::as.psis_loo_ss.psis_loo(true_loo)) expect_silent(loo_ss_subset0 <- update(true_loo_ss, observations = loo_ss, r_eff = rep(1, nrow(fake_data)))) expect_true(identical(obs_idx(loo_ss_subset0), obs_idx(loo_ss))) expect_silent(loo_ss_subset1 <- update(object = loo_ss, observations = loo_ss, r_eff = rep(1, nrow(fake_data)))) expect_message(loo_ss_subset2 <- update(object = loo_ss, observations = obs_idx(loo_ss)[1:10], r_eff = rep(1, nrow(fake_data)))) expect_equal(nobs(loo_ss_subset2), 10) expect_silent(true_loo_ss <- loo:::as.psis_loo_ss.psis_loo(true_loo)) set.seed(4711) expect_silent(loo_ss2 <- update(object = loo_ss, draws = fake_posterior, data = fake_data, observations = 600, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss2_subset0 <- update(object = true_loo_ss, observations = loo_ss2, r_eff = rep(1, nrow(fake_data)))) expect_true(setequal(obs_idx(loo_ss2), obs_idx(loo_ss2_subset0))) expect_true(identical(obs_idx(loo_ss2), obs_idx(loo_ss2_subset0))) expect_true(identical(loo_ss2$diagnostic, loo_ss2_subset0$diagnostic)) # Add tests for changing approx variable expect_silent(loo_ss_lpd <- update(object = loo_ss, draws = fake_posterior, data = fake_data, loo_approximation = "lpd", r_eff = rep(1, nrow(fake_data)))) expect_failure(expect_equal(loo_ss_lpd$loo_subsampling$elpd_loo_approx, loo_ss$loo_subsampling$elpd_loo_approx)) expect_equal(dim(loo_ss_lpd)[2], dim(loo_ss)[2]) expect_equal(dim(loo_ss_lpd)[2], dim(loo_ss_lpd$pointwise)[1]) expect_length(loo_ss_lpd$diagnostics$pareto_k, 500) expect_length(loo_ss_lpd$diagnostics$n_eff, 500) expect_failure(expect_equal(loo_ss_lpd$estimates[1, "subsampling SE"], loo_ss$estimates[1, "subsampling SE"])) expect_failure(expect_equal(loo_ss_lpd$estimates[3, "subsampling SE"], loo_ss$estimates[3, "subsampling SE"])) }) context("loo_subsampling_approximations") geterate_test_elpd_dataset <- function() { N <- 10; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- draws <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) list(fake_posterior = fake_posterior, fake_data = fake_data) } test_elpd_loo_approximation <- function(cores) { set.seed(123) test_data <- geterate_test_elpd_dataset() fake_posterior <- test_data$fake_posterior fake_data <- test_data$fake_data llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } # Compute plpd approximation expect_silent(pi_vals <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "plpd", cores = cores)) # Compute it manually point <- mean(fake_posterior) llik <- dbinom(fake_data$y, size = fake_data$K, prob = point, log = TRUE) abs_lliks <- abs(llik) man_elpd_loo_approximation <- abs_lliks/sum(abs_lliks) expect_equal(abs(pi_vals)/sum(abs(pi_vals)), man_elpd_loo_approximation, tol = 0.00001) # Compute lpd approximation expect_silent(pi_vals <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "lpd", cores = cores)) # Compute it manually llik <- numeric(10) for(i in seq_along(fake_data$y)){ llik[i] <- loo:::logMeanExp(dbinom(fake_data$y[i], size = fake_data$K, prob = fake_posterior, log = TRUE)) } abs_lliks <- abs(llik) man_approx_loo_variable <- abs_lliks/sum(abs_lliks) expect_equal(abs(pi_vals)/sum(abs(pi_vals)), man_approx_loo_variable, tol = 0.00001) # Compute waic approximation expect_silent(pi_vals_waic <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "waic", cores = cores)) expect_true(all(pi_vals > pi_vals_waic)) expect_true(sum(pi_vals) - sum(pi_vals_waic) < 1) # Compute tis approximation expect_silent(pi_vals_tis <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "tis", loo_approximation_draws = 100, cores = cores)) expect_true(all(pi_vals > pi_vals_tis)) expect_true(sum(pi_vals) - sum(pi_vals_tis) < 1) } test_that("elpd_loo_approximation works as expected", { test_elpd_loo_approximation(1) }) test_that("elpd_loo_approximation with multiple cores", { test_elpd_loo_approximation(2) }) test_that("Test loo_approximation_draws", { set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- draws <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(res1 <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "waic", loo_approximation_draws = NULL, cores = 1)) expect_silent(res2 <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior, loo_approximation = "waic", loo_approximation_draws = 10, cores = 1)) expect_silent(res3 <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior[1:10*100,], loo_approximation = "waic", loo_approximation_draws = NULL, cores = 1)) expect_silent(res4 <- loo:::elpd_loo_approximation(.llfun = llfun_test, data = fake_data, draws = fake_posterior[1:10*100,, drop = FALSE], loo_approximation = "waic", loo_approximation_draws = NULL, cores = 1)) expect_failure(expect_equal(res1, res3)) expect_equal(res2, res3) expect_silent(loo_ss1 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss2 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "plpd", loo_approximation_draws = 10, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss3 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "plpd", loo_approximation_draws = 31, r_eff = rep(1, nrow(fake_data)))) expect_error(loo_ss4 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "plpd", loo_approximation_draws = 3100, r_eff = rep(1, nrow(fake_data)))) expect_equal(names(loo_ss1$loo_subsampling), c("elpd_loo_approx", "loo_approximation", "loo_approximation_draws", "estimator", ".llfun", ".llgrad", ".llhess", "data_dim", "ndraws")) expect_null(loo_ss1$loo_subsampling$loo_approximation_draws) expect_equal(loo_ss2$loo_subsampling$loo_approximation_draws, 10L) expect_equal(loo_ss3$loo_subsampling$loo_approximation_draws, 31L) }) test_that("waic using delta method and gradient", { if (FALSE){ # Code to generate testdata - saved and loaded to avoid dependency of mvtnorm set.seed(123) N <- 400; beta <- c(1,2); X_full <- matrix(rep(1,N), ncol = 1); X_full <- cbind(X_full, runif(N)); S <- 1000 y_full <- rnorm(n = N, mean = X_full%*%beta, sd = 1) X <- X_full; y <- y_full Lambda_0 <- diag(length(beta)); mu_0 <- c(0,0) b_hat <- solve(t(X)%*%X)%*%t(X)%*%y mu_n <- solve(t(X)%*%X)%*%(t(X)%*%X%*%b_hat + Lambda_0%*%mu_0) Lambda_n <- t(X)%*%X + Lambda_0 # Uncomment row below when running. Commented out to remove CHECK warnings # fake_posterior <- mvtnorm::rmvnorm(n = S, mean = mu_n, sigma = solve(Lambda_n)) colnames(fake_posterior) <- c("a", "b") fake_data <- data.frame(y, X) save(fake_posterior, fake_data, file = test_path("data-for-tests/normal_reg_waic_test_example.rda")) } else { load(file = test_path("data-for-tests/normal_reg_waic_test_example.rda")) } .llfun <- function(data_i, draws) { # data_i: ith row of fdata (fake_data[i,, drop=FALSE]) # draws: entire fake_posterior matrix dnorm(data_i$y, mean = draws[, c("a", "b")] %*% t(as.matrix(data_i[, c("X1", "X2")])), sd = 1, log = TRUE) } .llgrad <- function(data_i, draws) { x_i <- data_i[, "X2"] gr <- cbind(data_i$y - draws[,"a"] - draws[,"b"]*x_i, (data_i$y - draws[,"a"] - draws[,"b"]*x_i) * x_i) colnames(gr) <- c("a", "b") gr } fake_posterior <- cbind(fake_posterior, runif(nrow(fake_posterior))) expect_silent(approx_loo_waic <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic")) expect_silent(approx_loo_waic_delta <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic_grad", .llgrad = .llgrad)) expect_silent(approx_loo_waic_delta_diag <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic_grad_marginal", .llgrad = .llgrad)) # Test that the approaches should not deviate too much diff_waic_delta <- mean(approx_loo_waic - approx_loo_waic_delta) diff_waic_delta_diag <- mean(approx_loo_waic - approx_loo_waic_delta_diag) expect_equal(approx_loo_waic,approx_loo_waic_delta_diag, tol = 0.1) expect_equal(approx_loo_waic,approx_loo_waic_delta, tol = 0.01) # Test usage in subsampling_loo expect_silent(loo_ss_waic <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic", observations = 50, llgrad = .llgrad)) expect_silent(loo_ss_waic_delta <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_grad", observations = 50, llgrad = .llgrad)) expect_silent(loo_ss_waic_delta_marginal <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_grad_marginal", observations = 50, llgrad = .llgrad)) expect_silent(loo_ss_plpd <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "plpd", observations = 50, llgrad = .llgrad)) expect_error(loo_ss_waic_delta <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_grad", observations = 50)) }) test_that("waic using delta 2nd order method", { if (FALSE){ # Code to generate testdata - saved and loaded to avoid dependency of MCMCPack set.seed(123) N <- 100; beta <- c(1,2); X_full <- matrix(rep(1,N), ncol = 1); X_full <- cbind(X_full, runif(N)); S <- 1000 y_full <- rnorm(n = N, mean = X_full%*%beta, sd = 0.5) X <- X_full; y <- y_full # Uncomment row below when running. Commented out to remove CHECK warnings # fake_posterior <- MCMCpack::MCMCregress(y~x, data = data.frame(y = y,x=X[,2]), thin = 10, mcmc = 10000) # Because Im lazy fake_posterior <- as.matrix(fake_posterior) fake_posterior[,"sigma2"] <- sqrt(fake_posterior[,"sigma2"]) colnames(fake_posterior) <- c("a", "b", "sigma") fake_data <- data.frame(y, X) save(fake_posterior, fake_data, file = test_path("data-for-tests/normal_reg_waic_test_example2.rda"), compression_level = 9) } else { load(file = test_path("data-for-tests/normal_reg_waic_test_example2.rda")) } .llfun <- function(data_i, draws) { # data_i: ith row of fdata (data_i <- fake_data[i,, drop=FALSE]) # draws: entire fake_posterior matrix dnorm(data_i$y, mean = draws[, c("a", "b")] %*% t(as.matrix(data_i[, c("X1", "X2")])), sd = draws[, c("sigma")], log = TRUE) } .llgrad <- function(data_i, draws) { sigma <- draws[,"sigma"] sigma2 <- sigma^2 b <- draws[,"b"] a <- draws[,"a"] x_i <- unlist(data_i[, c("X1", "X2")]) e <- (data_i$y - draws[,"a"] * x_i[1] - draws[,"b"] * x_i[2]) gr <- cbind(e * x_i[1] / sigma2, e * x_i[2] / sigma2, - 1 / sigma + e^2 / (sigma2 * sigma)) colnames(gr) <- c("a", "b", "sigma") gr } .llhess <- function(data_i, draws) { hess_array <- array(0, dim = c(ncol(draws), ncol(draws), nrow(draws)), dimnames = list(colnames(draws),colnames(draws),NULL)) sigma <- draws[,"sigma"] sigma2 <- sigma^2 sigma3 <- sigma2*sigma b <- draws[,"b"] a <- draws[,"a"] x_i <- unlist(data_i[, c("X1", "X2")]) e <- (data_i$y - draws[,"a"] * x_i[1] - draws[,"b"] * x_i[2]) hess_array[1,1,] <- - x_i[1]^2 / sigma2 hess_array[1,2,] <- hess_array[2,1,] <- - x_i[1] * x_i[2] / sigma2 hess_array[2,2,] <- - x_i[2]^2 / sigma2 hess_array[3,1,] <- hess_array[1,3,] <- -2 * x_i[1] * e / sigma3 hess_array[3,2,] <- hess_array[2,3,] <- -2 * x_i[2] * e / sigma3 hess_array[3,3,] <- 1 / sigma2 - 3 * e^2 / (sigma2^2) hess_array } #data <- fake_data fake_posterior <- cbind(fake_posterior, runif(nrow(fake_posterior))) #draws <- fake_posterior <- cbind(fake_posterior, runif(nrow(fake_posterior))) expect_silent(approx_loo_waic <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic")) expect_silent(approx_loo_waic_delta <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic_grad", .llgrad = .llgrad)) expect_silent(approx_loo_waic_delta2 <- loo:::elpd_loo_approximation(.llfun, data = fake_data, draws = fake_posterior, cores = 1, loo_approximation = "waic_hess", .llgrad = .llgrad, .llhess = .llhess)) # Test that the approaches should not deviate too much expect_equal(approx_loo_waic,approx_loo_waic_delta2, tol = 0.01) expect_equal(approx_loo_waic,approx_loo_waic_delta, tol = 0.01) expect_silent(test_loo_ss_waic <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic", observations = 50, llgrad = .llgrad)) expect_error(test_loo_ss_delta2 <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_hess", observations = 50, llgrad = .llgrad)) expect_silent(test_loo_ss_delta2 <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_hess", observations = 50, llgrad = .llgrad, llhess = .llhess)) expect_silent(test_loo_ss_delta <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "waic_grad", observations = 50, llgrad = .llgrad)) expect_silent(test_loo_ss_point <- loo_subsample(x = .llfun, data = fake_data, draws = fake_posterior, cores = 1, r_eff = rep(1, nrow(fake_data)), loo_approximation = "plpd", observations = 50, llgrad = .llgrad)) }) context("loo_subsampling_estimation") test_that("whhest works as expected", { N <- 100 m <- 10 z <- rep(1/N, m) y <- 1:10 m_i <- rep(1,m) expect_silent(whe <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) expect_equal(whe$y_hat_ppz, 550) man_var <- (sum((whe$y_hat_ppz - y/z)^2)/(m-1))/m expect_equal(whe$v_hat_y_ppz, man_var) z <- 1:10/(sum(1:10)*10) expect_silent(whe <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) expect_equal(whe$y_hat_ppz, 550) expect_equal(whe$v_hat_y_ppz, 0) # School book example # https://newonlinecourses.science.psu.edu/stat506/node/15/ z <- c(650/15650, 2840/15650, 3200/15650) y <- c(420, 1785, 2198) m_i <- c(1,1,1) N <- 10 expect_silent(whe <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) expect_equal(round(whe$y_hat_ppz, 2), 10232.75, tol = 0) expect_equal(whe$v_hat_y_ppz, 73125.74, tol = 0.01) # Double check that it is rounding error man_var_round <- (sum((round(y/z,2) - 10232.75)^2)) * (1/2) * (1/3) expect_equal(man_var_round, 73125.74, tol = 0.001) man_var_exact <- (sum((y/z - 10232.75)^2)) * (1/2) * (1/3) expect_equal(whe$v_hat_y_ppz, man_var_exact, tol = 0.001) # Add test for variance estimation N <- 100 m <- 10 y <- rep(1:10, 1) true_var <- var(rep(y, 10)) * (99) z <- rep(1/N, m) m_i <- rep(100000, m) expect_silent(whe <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) expect_equal(true_var, whe$hat_v_y_ppz, tol = 0.01) # Add tests for m_i N <- 100 y <- rep(1:10, 2) m <- length(y) z <- rep(1/N, m) m_i <- rep(1,m) expect_silent(whe1 <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) y <- rep(1:10) m <- length(y) z <- rep(1/N, m) m_i <- rep(2,m) expect_silent(whe2 <- loo:::whhest(z = z, m_i = m_i, y = y, N = N)) expect_equal(whe1$y_hat_ppz, whe2$y_hat_ppz) expect_equal(whe1$v_hat_y_ppz, whe2$v_hat_y_ppz) expect_equal(whe1$hat_v_y_ppz, whe1$hat_v_y_ppz) }) test_that("srs_diff_est works as expected", { set.seed(1234) N <- 1000 y_true <- 1:N sigma_hat_true <- sqrt(N * sum((y_true - mean(y_true))^2) / length(y_true)) y_approx <- rnorm(N, y_true, 0.1) m <- 100 sigma_hat <- y_hat <- se_y_hat <- numeric(10000) for(i in 1:10000){ y_idx <- sample(1:N, size = m) y <- y_true[y_idx] res <- loo:::srs_diff_est(y_approx, y, y_idx) y_hat[i] <- res$y_hat se_y_hat[i] <- sqrt(res$v_y_hat) sigma_hat[i] <- sqrt(res$hat_v_y) } expect_equal(mean(y_hat), sum(y_true), tol = 0.1) in_ki <- y_hat + 2 * se_y_hat > sum(y_true) & y_hat - 2*se_y_hat < sum(y_true) expect_equal(mean(in_ki), 0.95, tol = 0.01) # Should be unbiased expect_equal(mean(sigma_hat), sigma_hat_true, tol = 0.1) m <- N y_idx <- sample(1:N, size = m) y <- y_true[y_idx] res <- loo:::srs_diff_est(y_approx, y, y_idx) expect_equal(res$y_hat, 500500, tol = 0.0001) expect_equal(res$v_y_hat, 0, tol = 0.0001) expect_equal(sqrt(res$hat_v_y), sigma_hat_true, tol = 0.1) }) test_that("srs_est works as expected", { set.seed(1234) # Cochran 1976 example Table 2.2 y <- c(rep(42,23),rep(41,4), 36, 32, 29, 27, 27, 23, 19, 16, 16, 15, 15, 14, 11, 10, 9, 7, 6, 6, 6, 5, 5, 4, 3) expect_equal(sum(y), 1471) approx_loo <- rep(0L, 676) expect_equal(sum(y^2), 54497) res <- loo:::srs_est(y = y, approx_loo) expect_equal(res$y_hat, 19888, tol = 0.0001) expect_equal(res$v_y_hat, 676^2*229*(1-0.074)/50, tol = 0.0001) expect_equal(res$hat_v_y, 676 * var(y), tol = 0.0001) # Simulation example set.seed(1234) N <- 1000 y_true <- 1:N sigma_hat_true <- sqrt(N * sum((y_true - mean(y_true))^2) / length(y_true)) m <- 100 y_hat <- se_y_hat <- sigma_hat <- numeric(10000) for(i in 1:10000){ y_idx <- sample(1:N, size = m) y <- y_true[y_idx] res <- loo:::srs_est(y = y, y_approx = y_true) y_hat[i] <- res$y_hat se_y_hat[i] <- sqrt(res$v_y_hat) sigma_hat[i] <- sqrt(res$hat_v_y) } expect_equal(mean(y_hat), sum(y_true), tol = 0.1) in_ki <- y_hat + 2*se_y_hat > sum(y_true) & y_hat - 2*se_y_hat < sum(y_true) expect_equal(mean(in_ki), 0.95, tol = 0.01) # Should be unbiased expect_equal(mean(sigma_hat), sigma_hat_true, tol = 0.1) m <- N y_idx <- sample(1:N, size = m) y <- y_true[y_idx] res <- loo:::srs_est(y, y_true) expect_equal(res$y_hat, 500500, tol = 0.0001) expect_equal(res$v_y_hat, 0, tol = 0.0001) }) context("loo_subsampling cases") test_that("Test loo_subsampling and loo_approx with radon data", { skip_on_cran() # avoid going over time limit for tests load(test_path("data-for-tests/test_radon_laplace_loo.rda")) # Rename to spot variable leaking errors llfun_test <- llfun log_p_test <- log_p log_g_test <- log_q draws_test <- draws data_test <- data rm(llfun, log_p,log_q, draws, data) set.seed(134) expect_silent(full_loo <- loo(llfun_test, draws = draws_test, data = data_test, r_eff = rep(1, nrow(data_test)))) expect_s3_class(full_loo, "psis_loo") set.seed(134) expect_silent(loo_ss <- loo_subsample(x = llfun_test, draws = draws_test, data = data_test, observations = 200, loo_approximation = "plpd", r_eff = rep(1, nrow(data_test)))) expect_s3_class(loo_ss, "psis_loo_ss") set.seed(134) expect_silent(loo_ap_ss <- loo_subsample(x = llfun_test, draws = draws_test, data = data_test, log_p = log_p_test, log_g = log_g_test, observations = 200, loo_approximation = "plpd", r_eff = rep(1, nrow(data_test)))) expect_s3_class(loo_ap_ss, "psis_loo_ss") expect_s3_class(loo_ap_ss, "psis_loo_ap") expect_silent(loo_ap_ss_full <- loo_subsample(x = llfun_test, log_p = log_p_test, log_g = log_g_test, draws = draws_test, data = data_test, observations = NULL, loo_approximation = "plpd", r_eff = rep(1, nrow(data_test)))) expect_failure(expect_s3_class(loo_ap_ss_full, "psis_loo_ss")) expect_s3_class(loo_ap_ss_full, "psis_loo_ap") # Expect similar results z <- 2 expect_lte(loo_ss$estimates["elpd_loo", "Estimate"] - z * loo_ss$estimates["elpd_loo", "subsampling SE"], full_loo$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ss$estimates["elpd_loo", "Estimate"] + z * loo_ss$estimates["elpd_loo", "subsampling SE"], full_loo$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ss$estimates["p_loo", "Estimate"] - z * loo_ss$estimates["p_loo", "subsampling SE"], full_loo$estimates["p_loo", "Estimate"]) expect_gte(loo_ss$estimates["p_loo", "Estimate"] + z * loo_ss$estimates["p_loo", "subsampling SE"], full_loo$estimates["p_loo", "Estimate"]) expect_lte(loo_ss$estimates["looic", "Estimate"] - z * loo_ss$estimates["looic", "subsampling SE"], full_loo$estimates["looic", "Estimate"]) expect_gte(loo_ss$estimates["looic", "Estimate"] + z * loo_ss$estimates["looic", "subsampling SE"], full_loo$estimates["looic", "Estimate"]) expect_failure(expect_equal(full_loo$estimates["elpd_loo", "Estimate"], loo_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(full_loo$estimates["p_loo", "Estimate"], loo_ss$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(full_loo$estimates["looic", "Estimate"], loo_ss$estimates["looic", "Estimate"], tol = 0.00000001)) z <- 2 expect_lte(loo_ap_ss$estimates["elpd_loo", "Estimate"] - z * loo_ap_ss$estimates["elpd_loo", "subsampling SE"], loo_ap_ss_full$estimates["elpd_loo", "Estimate"]) expect_gte(loo_ap_ss$estimates["elpd_loo", "Estimate"] + z * loo_ap_ss$estimates["elpd_loo", "subsampling SE"], loo_ap_ss_full$estimates["elpd_loo", "Estimate"]) expect_lte(loo_ap_ss$estimates["p_loo", "Estimate"] - z * loo_ap_ss$estimates["p_loo", "subsampling SE"], loo_ap_ss_full$estimates["p_loo", "Estimate"]) expect_gte(loo_ap_ss$estimates["p_loo", "Estimate"] + z * loo_ap_ss$estimates["p_loo", "subsampling SE"], loo_ap_ss_full$estimates["p_loo", "Estimate"]) expect_lte(loo_ap_ss$estimates["looic", "Estimate"] - z * loo_ap_ss$estimates["looic", "subsampling SE"], loo_ap_ss_full$estimates["looic", "Estimate"]) expect_gte(loo_ap_ss$estimates["looic", "Estimate"] + z * loo_ap_ss$estimates["looic", "subsampling SE"], loo_ap_ss_full$estimates["looic", "Estimate"]) expect_failure(expect_equal(loo_ap_ss_full$estimates["elpd_loo", "Estimate"], loo_ap_ss$estimates["elpd_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(loo_ap_ss_full$estimates["p_loo", "Estimate"], loo_ap_ss$estimates["p_loo", "Estimate"], tol = 0.00000001)) expect_failure(expect_equal(loo_ap_ss_full$estimates["looic", "Estimate"], loo_ap_ss$estimates["looic", "Estimate"], tol = 0.00000001)) # Correct printout expect_failure(expect_output(print(full_loo), "Posterior approximation correction used\\.")) expect_failure(expect_output(print(full_loo), "subsampled log-likelihood\nvalues")) expect_failure(expect_output(print(loo_ss), "Posterior approximation correction used\\.")) expect_output(print(loo_ss), "subsampled log-likelihood\nvalues") expect_output(print(loo_ap_ss), "Posterior approximation correction used\\.") expect_output(print(loo_ap_ss), "subsampled log-likelihood\nvalues") expect_output(print(loo_ap_ss_full), "Posterior approximation correction used\\.") expect_failure(expect_output(print(loo_ap_ss_full), "subsampled log-likelihood\nvalues")) # Test conversion of objects expect_silent(loo_ap_full <- loo:::as.psis_loo.psis_loo(loo_ap_ss_full)) expect_s3_class(loo_ap_full, "psis_loo_ap") expect_silent(loo_ap_full_ss <- loo:::as.psis_loo_ss.psis_loo(loo_ap_full)) expect_s3_class(loo_ap_full_ss, "psis_loo_ss") expect_s3_class(loo_ap_full_ss, "psis_loo_ap") expect_silent(loo_ap_full2 <- loo:::as.psis_loo.psis_loo_ss(loo_ap_full_ss)) expect_s3_class(loo_ap_full2, "psis_loo_ap") expect_failure(expect_s3_class(loo_ap_full2, "psis_loo_ss")) expect_equal(loo_ap_full2,loo_ap_full) # Test update set.seed(4712) expect_silent(loo_ss2 <- update(loo_ss, draws = draws_test, data = data_test, observations = 1000, r_eff = rep(1, nrow(data_test)))) expect_gt(dim(loo_ss2)[2], dim(loo_ss)[2]) expect_gt(dim(loo_ss2$pointwise)[1], dim(loo_ss$pointwise)[1]) expect_equal(nobs(loo_ss), 200) expect_equal(nobs(loo_ss2), 1000) for(i in 1:nrow(loo_ss2$estimates)) { expect_lt(loo_ss2$estimates[i, "subsampling SE"], loo_ss$estimates[i, "subsampling SE"]) } set.seed(4712) expect_silent(loo_ap_ss2 <- update(object = loo_ap_ss, draws = draws_test, data = data_test, observations = 2000)) expect_gt(dim(loo_ap_ss2)[2], dim(loo_ap_ss)[2]) expect_gt(dim(loo_ap_ss2$pointwise)[1], dim(loo_ap_ss$pointwise)[1]) expect_equal(nobs(loo_ap_ss), 200) expect_equal(nobs(loo_ap_ss2), 2000) for(i in 1:nrow(loo_ap_ss2$estimates)) { expect_lt(loo_ap_ss2$estimates[i, "subsampling SE"], loo_ap_ss$estimates[i, "subsampling SE"]) } expect_equal(round(full_loo$estimates), round(loo_ap_ss_full$estimates)) expect_failure(expect_equal(full_loo$estimates, loo_ap_ss_full$estimates)) expect_equal(dim(full_loo), dim(loo_ap_ss_full)) expect_s3_class(loo_ap_ss_full, "psis_loo_ap") }) test_that("Test the vignette", { skip_on_cran() # NOTE # If any of these test fails, the vignette probably needs to be updated if (FALSE) { # Generate vignette test case library("rstan") stan_code <- " data { int N; // number of data points int P; // number of predictors (including intercept) matrix[N,P] X; // predictors (including 1s for intercept) int y[N]; // binary outcome } parameters { vector[P] beta; } model { beta ~ normal(0, 1); y ~ bernoulli_logit(X * beta); } " # logistic <- function(x) {1 / (1 + exp(-x))} # logit <- function(x) {log(x) - log(1-x)} llfun_logistic <- function(data_i, draws) { x_i <- as.matrix(data_i[, which(grepl(colnames(data_i), pattern = "X")), drop=FALSE]) y_pred <- draws %*% t(x_i) dbinom(x = data_i$y, size = 1, prob = 1 / (1 + exp(-y_pred)), log = TRUE) } # Prepare data url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" wells <- read.table(url) wells$dist100 <- with(wells, dist / 100) X <- model.matrix(~ dist100 + arsenic, wells) standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # Fit model set.seed(4711) fit_1 <- stan(model_code = stan_code, data = standata, seed = 4711) print(fit_1, pars = "beta") parameter_draws <- extract(fit_1)$beta stan_df <- as.data.frame(standata) loo_i(1, llfun_logistic, data = stan_df, draws = parameter_draws) sm <- stan_model(model_code = stan_code) set.seed(4711) fit_laplace <- optimizing(sm, data = standata, draws = 2000, seed = 42) parameter_draws_laplace <- fit_laplace$theta_tilde log_p <- fit_laplace$log_p # The log density of the posterior log_g <- fit_laplace$log_g # The log density of the approximation # For comparisons standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) stan_df2 <- as.data.frame(standata) set.seed(4711) fit_2 <- stan(fit = fit_1, data = standata, seed = 4711) parameter_draws_2 <- extract(fit_2)$beta save(llfun_logistic, stan_df, stan_df2, parameter_draws, parameter_draws_laplace, parameter_draws_2, log_p, log_g, file = test_path("data-for-tests/loo_subsample_vignette.rda"), compression_level = 9) } else { load(test_path("data-for-tests/loo_subsample_vignette.rda")) } set.seed(4711) expect_warning(looss_1 <- loo_subsample(llfun_logistic, draws = parameter_draws, data = stan_df, observations = 100)) expect_output(print(looss_1), "Computed from 4000 by 100 subsampled log-likelihood") expect_output(print(looss_1), "values from 3020 total observations.") expect_output(print(looss_1), "elpd_loo -1968.5 15.6 0.3") expect_output(print(looss_1), "p_loo 3.1 0.1 0.4") expect_s3_class(looss_1, c("psis_loo_ss", "psis_loo", "loo")) set.seed(4711) expect_warning(looss_1b <- update(looss_1, draws = parameter_draws, data = stan_df, observations = 200)) expect_output(print(looss_1b), "Computed from 4000 by 200 subsampled log-likelihood") expect_output(print(looss_1b), "values from 3020 total observations.") expect_output(print(looss_1b), "elpd_loo -1968.3 15.6 0.2") expect_output(print(looss_1b), "p_loo 3.2 0.1 0.4") expect_s3_class(looss_1b, c("psis_loo_ss", "psis_loo", "loo")) set.seed(4711) expect_warning(looss_2 <- loo_subsample(x = llfun_logistic, draws = parameter_draws, data = stan_df, observations = 100, estimator = "hh_pps", loo_approximation = "lpd", loo_approximation_draws = 100)) expect_output(print(looss_2), "Computed from 4000 by 100 subsampled log-likelihood") expect_output(print(looss_2), "values from 3020 total observations.") # Currently failing # expect_output(print(looss_2), "elpd_loo -1968.9 15.4 0.5") # expect_output(print(looss_2), "p_loo 3.5 0.2 0.5") expect_s3_class(looss_2, c("psis_loo_ss", "psis_loo", "loo")) set.seed(4711) expect_warning(aploo_1 <- loo_approximate_posterior(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g)) expect_output(print(aploo_1), "Computed from 2000 by 3020 log-likelihood matrix") expect_output(print(aploo_1), "elpd_loo -1968.4 15.6") expect_output(print(aploo_1), "p_loo 3.2 0.2") expect_output(print(aploo_1), "Posterior approximation correction used.") expect_output(print(aploo_1), "\\(-Inf, 0.5\\] \\(good\\) 2989 99.0") expect_output(print(aploo_1), "\\(0.5, 0.7\\] \\(ok\\) 31 1.0") expect_s3_class(aploo_1, c("psis_loo_ap", "psis_loo", "loo")) set.seed(4711) expect_warning(looapss_1 <- loo_subsample(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g, observations = 100)) expect_output(print(looapss_1), "Computed from 2000 by 100 subsampled log-likelihood") expect_output(print(looapss_1), "values from 3020 total observations.") expect_output(print(looapss_1), "elpd_loo -1968.2 15.6 0.4") expect_output(print(looapss_1), "p_loo 2.9 0.1 0.5") expect_output(print(looapss_1), "\\(-Inf, 0.5\\] \\(good\\) 97 97.0") expect_output(print(looapss_1), "\\(0.5, 0.7\\] \\(ok\\) 3 3.0") # Loo compare set.seed(4711) expect_warning(looss_1 <- loo_subsample(llfun_logistic, draws = parameter_draws, data = stan_df, observations = 100)) set.seed(4712) expect_warning(looss_2 <- loo_subsample(x = llfun_logistic, draws = parameter_draws_2, data = stan_df2, observations = 100)) expect_output(print(looss_2), "Computed from 4000 by 100 subsampled log-likelihood") expect_output(print(looss_2), "values from 3020 total observations.") expect_output(print(looss_2), "elpd_loo -1952.0 16.2 0.2") expect_output(print(looss_2), "p_loo 2.6 0.1 0.3") expect_warning(comp <- loo_compare(looss_1, looss_2), "Different subsamples in 'model2' and 'model1'. Naive diff SE is used.") expect_output(print(comp), "model1 16.5 22.5 0.4") set.seed(4712) expect_warning(looss_2_m <- loo_subsample(x = llfun_logistic, draws = parameter_draws_2, data = stan_df2, observations = looss_1)) expect_message(looss_2_m <- suppressWarnings(loo_subsample(x = llfun_logistic, draws = parameter_draws_2, data = stan_df2, observations = obs_idx(looss_1))), "Simple random sampling with replacement assumed.") expect_silent(comp <- loo_compare(looss_1, looss_2_m)) expect_output(print(comp), "model1 16.1 4.4 0.1") set.seed(4712) expect_warning(looss_1 <- update(looss_1, draws = parameter_draws, data = stan_df, observations = 200)) expect_warning(looss_2_m <- update(looss_2_m, draws = parameter_draws_2, data = stan_df2, observations = looss_1)) expect_silent(comp2 <- loo_compare(looss_1, looss_2_m)) expect_output(print(comp2), "model1 16.3 4.4 0.1") expect_warning(looss_2_full <- loo(x = llfun_logistic, draws = parameter_draws_2, data = stan_df2)) expect_message(comp3 <- loo_compare(x = list(looss_1, looss_2_full)), "Estimated elpd_diff using observations included in loo calculations for all models.") expect_output(print(comp3), "model1 16.5 4.4 0.3") }) context("loo_compare_subsample") test_that("loo_compare_subsample", { skip_on_cran() # to get under cran check time limit set.seed(123) N <- 1000 x1 <- rnorm(N) x2 <- rnorm(N) x3 <- rnorm(N) sigma <- 2 y <- rnorm(N, 1 + 2*x1 - 2*x2 - 1*x3, sd = sigma) X <- cbind("x0" = rep(1, N), x1, x2, x3) # Generate samples from posterior samples_blin <- function(X, y, sigma, draws = 1000){ XtX <- t(X)%*%X b_hat <- solve(XtX) %*% (t(X) %*% y) Lambda_n = XtX + diag(ncol(X)) mu_n <- solve(Lambda_n) %*% (XtX %*% b_hat + diag(ncol(X)) %*% rep(0,ncol(X))) L <- t(chol(sigma^2 * solve(Lambda_n))) draws_mat <- matrix(0, ncol=ncol(X), nrow = draws) for(i in 1:draws){ z <- rnorm(length(mu_n)) draws_mat[i,] <- L %*% z + mu_n } draws_mat } fake_posterior1 <- samples_blin(X[,1:2], y, sigma, draws = 1000) fake_posterior2 <- samples_blin(X[,1:3], y, sigma, draws = 1000) fake_posterior3 <- samples_blin(X, y, sigma, draws = 1000) fake_data1 <- data.frame(y, X[,1:2]) fake_data2 <- data.frame(y, X[,1:3]) fake_data3 <- data.frame(y, X) llfun_test <- function(data_i, draws) { dnorm(x = data_i$y, mean = draws %*% t(data_i[,-1, drop=FALSE]), sd = sigma, log = TRUE) } expect_silent(l1 <- loo(llfun_test, data = fake_data1, draws = fake_posterior1, r_eff = rep(1, N))) expect_silent(l2 <- loo(llfun_test, data = fake_data2, draws = fake_posterior2, r_eff = rep(1, N))) expect_silent(l3 <- loo(llfun_test, data = fake_data3, draws = fake_posterior3, r_eff = rep(1, N))) expect_silent(lss1 <- loo_subsample(llfun_test, data = fake_data1, draws = fake_posterior1, observations = 100, r_eff = rep(1, N))) expect_silent(lss2 <- loo_subsample(llfun_test, data = fake_data2, draws = fake_posterior2, observations = 100, r_eff = rep(1, N))) expect_silent(lss3 <- loo_subsample(llfun_test, data = fake_data3, draws = fake_posterior3, observations = 100, r_eff = rep(1, N))) expect_silent(lss2o1 <- loo_subsample(llfun_test, data = fake_data2, draws = fake_posterior2, observations = lss1, r_eff = rep(1, N))) expect_silent(lss3o1 <- loo_subsample(llfun_test, data = fake_data3, draws = fake_posterior3, observations = lss1, r_eff = rep(1, N))) expect_silent(lss2hh <- loo_subsample(llfun_test, data = fake_data2, draws = fake_posterior2, observations = 100, estimator = "hh_pps", r_eff = rep(1, N))) expect_warning(lcss <- loo:::loo_compare.psis_loo_ss_list(x = list(lss1, lss2, lss3))) expect_warning(lcss2 <- loo:::loo_compare.psis_loo_ss_list(x = list(lss1, lss2, lss3o1))) expect_silent(lcsso <- loo:::loo_compare.psis_loo_ss_list(x = list(lss1, lss2o1, lss3o1))) expect_warning(lcssohh <- loo:::loo_compare.psis_loo_ss_list(x = list(lss1, lss2hh, lss3o1))) expect_message(lcssf1 <- loo:::loo_compare.psis_loo_ss_list(x = list(loo:::as.psis_loo_ss.psis_loo(l1), lss2o1, lss3o1))) expect_message(lcssf2 <- loo:::loo_compare.psis_loo_ss_list(x = list(loo:::as.psis_loo_ss.psis_loo(l1), lss2o1, loo:::as.psis_loo_ss.psis_loo(l3)))) expect_equal(lcss[,1], lcsso[,1], tolerance = 1) expect_equal(lcss2[,1], lcsso[,1], tolerance = 1) expect_equal(lcssohh[,1], lcsso[,1], tolerance = 1) expect_equal(lcssf1[,1], lcsso[,1], tolerance = 1) expect_equal(lcssf2[,1], lcsso[,1], tolerance = 1) expect_gt(lcss[,2][2], lcsso[,2][2]) expect_gt(lcss[,2][3], lcsso[,2][3]) expect_gt(lcss2[,2][2], lcsso[,2][2]) expect_equal(lcss2[,2][3], lcsso[,2][3]) expect_gt(lcssohh[,2][2], lcsso[,2][2]) expect_equal(lcssohh[,2][3], lcsso[,2][3]) expect_silent(lcss2m <- loo:::loo_compare.psis_loo_ss_list(x = list(lss2o1, lss3o1))) expect_equal(unname(lcss2m[,]), unname(lcsso[1:2,])) expect_warning(lcssapi <- loo_compare(lss1, lss2, lss3)) expect_equal(lcssapi, lcss) expect_warning(lcssohhapi <- loo_compare(lss1, lss2hh, lss3o1)) expect_equal(lcssohhapi, lcssohh) expect_silent(lcss2mapi <- loo_compare(lss2o1, lss3o1)) expect_equal(lcss2mapi, lcss2m) }) context("subsample with tis, sis") test_that("Test 'tis' and 'sis'", { skip_on_cran() set.seed(123) N <- 1000; K <- 10; S <- 1000; a0 <- 3; b0 <- 2 p <- 0.7 y <- rbinom(N, size = K, prob = p) a <- a0 + sum(y); b <- b0 + N * K - sum(y) fake_posterior <- draws <- as.matrix(rbeta(S, a, b)) fake_data <- data.frame(y,K) rm(N, K, S, a0, b0, p, y, a, b) llfun_test <- function(data_i, draws) { dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) } expect_silent(loo_ss_full <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 1000, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_plpd <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "plpd", r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_tis_S1000 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "tis", r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_tis_S100 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "tis", loo_approximation_draws = 100, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_tis_S10 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "tis", loo_approximation_draws = 10, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_sis_S1000 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "sis", r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_sis_S100 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "sis", loo_approximation_draws = 100, r_eff = rep(1, nrow(fake_data)))) expect_silent(loo_ss_sis_S10 <- loo_subsample(x = llfun_test, draws = fake_posterior, data = fake_data, observations = 100, loo_approximation = "sis", loo_approximation_draws = 10, r_eff = rep(1, nrow(fake_data)))) SEs <- 4 expect_gt(loo_ss_tis_S1000$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_tis_S1000$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_tis_S1000$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_tis_S1000$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_gt(loo_ss_tis_S100$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_tis_S100$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_tis_S100$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_tis_S100$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_gt(loo_ss_tis_S10$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_tis_S10$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_tis_S10$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_tis_S10$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_gt(loo_ss_sis_S1000$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_sis_S1000$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_sis_S1000$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_sis_S1000$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_gt(loo_ss_sis_S100$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_sis_S100$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_sis_S100$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_sis_S100$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_gt(loo_ss_sis_S10$estimates["elpd_loo", "Estimate"] + SEs*loo_ss_sis_S10$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) expect_lt(loo_ss_sis_S10$estimates["elpd_loo", "Estimate"] - SEs*loo_ss_sis_S10$estimates["elpd_loo", "subsampling SE"], loo_ss_full$estimates["elpd_loo", "Estimate"]) }) loo/tests/testthat/data-for-tests/0000755000176200001440000000000014411130104016667 5ustar liggesusersloo/tests/testthat/data-for-tests/normal_reg_waic_test_example2.rda0000644000176200001440000005716414411130104025360 0ustar liggesusers?o?_]"B!)#J"%YEђʈ2Y)!NDDJ\˞ײ5zn_x?o<81tE_Ex8^#LJBY^ 8}>MD5Wh L2캫 2gRS02~jRs[`HVq茁I7j&Pßm1M 6F[0bW־h 86P K$֓@5&h1FuK0i֩8K$äR<[7Xɛ VwjIкc0QTn&U q^s`zT=&o% 1? b!2FVcjJr{0T_<}'f;{% j- 0f? 螧kaRxq7bH L;+d>ɥS3{nLz;&P9cA'-›]shoׯ`B $枽;GVep1{!aFQtXzj@;):L+ooHw 'h 7h0Rnt2-k+ym=Y ր,̧,`yμEQ3P/u wG}]L)#'L['܀`0y(Kd0S]fͥmI#[֩w> 7o`x@y=+_fK7C=zǀnE$yeyTmN`\vVɵ$nI@9=&~o;t;*[wiӷ]ZL =m&a3ӝ~2=bc{`b0Ё_\C:E"~FH;"ˢ}9Sɝ) Ux_Ftf*?]0(60B1s࿺a ?y+0%t`;\9 Gbc6(#A#8׸mfkS"`XƸXN#w6@]%1>lPcBZnjf Ru(NLfM{V'",c ji> -f=zQTS76d]R]"\;Wǽ¸'u60m=WU]KTs`<)5ޥ ¥'kt=%v 0fI*_IR*H/)OIig9y :`L>l0:o;![+9묹0($s$BS|Aen?>c oaDYkamFoC!0? @b j.MU Z)̄A)('ufEgQ8pXX `FٓC@;.vW`.*c\SuV{gDz3lv h!J.fp<) #śPכ}mz{Fh:УkWn C1Ji|@?ə0C?R0^G:3.d0:ߒ^` !=]8S)P s\tGHUH7_O؍prue|6fa!Vy 0׾s`jm;ޯbW*mkX!f =ҬD00.4& ĶXh*mu0:d|P9ydA!hu*Ns[U ů4SQ["' y9Đŗt4y LߍQrCc t7H/t[}0Ož1܀x=&FRB3&gsON}fZ_b4VXV#@p07t0\|0|ޅb![ZImX87oNF|h~ Cqul tF]@;RFP =af/`UPj[i§cY-0J\E'c {0VKo`TDv/ >鰶s6N?,Xq1usL3@ŅP>cpj5}G`(.yS:g`*_33As{VDb=ܒ ;s7p{f$9_"9\h{'&Oj킿?HgEږ^-0P< S_/B:ӈ7 E#ߍ? k~_nm@~Fso sߐh=3?@vmXlj10ݭ\{P  sOv*L DxJ[?NηEm@t=žsP4pD'SHoᲅi5@ N&f5R/!"ʳ=kO*霚㚁UμͶhgs.[WܒBn};z9?溲!}{ pfy8S6 :i"(]Q_Kbꐯ[ !=(=#/ ^"kz~6Hq~%6E^y OnF} K κh6Z`t&0\Q`Õ՚#`}Z[q~ s7'* ۀAx|<-r?3Tۀj0mg&&с#9K ՕoUM)A}|LÄs%02R]*xh:%jYho sp*^ԏNjL,015K~Fյ"¢Ɣⳋf~~`,hU T2BhjYg0ޭ+YHD>[<9|a(iJMpܮF GtTk/zA1LyAM h]B~hCh5keᤜQ[4nxOGdXEf}0kP] vթ5&37^N'~4XhgWw}q)%ز zo>Nm3  `R0^nz5L=0dxel#ihn1aM>0vo^V{сΛN9ljv~_ 0`hn{ZTBt;Kar[bUS->p&'b~]&5s%O)zG$zJG|Ѻ]qH;iRa֬ywngwdDE yŻb!\ˉ5Gf~<ꥎoDUAt10kp5ThD Pe66uA?-Dzп@yzvoP_<Lz?bb0m-awY`0n.B}1g(]+Tlơ3ٷÄ]IU.~.?E~O0iqlMC⫸~\lW;u!=7uuG>`>^qx'0e M=k Fʦ@u7Po;6D: *F ׈q nx:O-3+5GGmgAZ*Ƒ@)kTפֿFzíLI+|:7GԿE?zWK@6PY/ 'M%6V'-~Lh⧳/Xvҋfek=aRRV 7Ofl_e`|vŲ}ǥMy :NxƤ49`TfGٗK c#mi`lu]cYĢUV$L:8J4dEnf`Fuyƫn𿲫=`( y3Gx #ڜa"?~G `ګ &nsL?+Jſq``M=ѻo!Kv߉O?F뼬Zx:E}- MxQt+Tt;?+&cal_6*4O:!:h=뾊R6DٕpYoNxH =zP>&% 9`~&5OIc3}}\F8OY9QE8}g~!^:|0#WVFZpP7YKlJQ/=8ԧQ1[n=B8Z@oMC+Fx Ȓ0єW{hy@N{Q4Po}Y%-t{};ބtⅿVcmW 0湒9͏DSlDe_,L)k 4 Ε.E8{΢47o5B?H`]E~<犯@'MW%1请l sU'_=O0Oq2Lvp/In/0Mv2 :ZO#Y,2Zk' ~^qI>x]75:^gCbofypw߿>RsMB0gjJ&΅:0+?fsvp'ƌ?Ct 09HEatzK'`|mm0hs- Ư3꒮s^0VVmT T22$@*7Zq'#+Iv}S2wt;q`칁ɣ>sP0ˆI i@wQeFTWWNnV*]M8zY6~< LËjo"8.Qz&s{QW9wݫs-)<鏾tNӍOY9}O7U)50LF4>\:STׂ!{I* \#h}Ll;V ̕x/NZo]^aec+dUَkTGR=]v<>}{/J~̌NW)ȧ- h"0qȯ0Y/V5Cאzݹ ~̿{CpgP|:RTʻ%^`ki4:{5g*Nr` aݜk;n/_BEʞoRkJl+p[ȃ5Ϧ#Gn[bw$:'g{,Em8I?JGU4llF0\H`'= 1ĹUS? ~ vdyn' 1)}7Cs}Ǖ >4}m2Vn` I3pg.L ؃ۀI@6 zK#f+c|I\b]P~syN+ @=)a%X4~k`Wx}`7_Z_:smj`3b##&yS+, ?̼=߭n?6~S߾nxXE[z %7sYak+ҀlG1(6},KI.J3Cz#ɻ0奴K.zii\ O?cx\pm v DfwcxLO` %ԀpP_?Ւ3peSs͏`h25˟En*CHtNFBH"0{ᬜW .]2t /X,k6 }h|:pD-(_8 sTZK]>WaE}Keÿi]$kU>_ KMٜw8,`. 1tKEJ*|IiEH4O0^z pQgHaa]'1 3w6Nt5 KXj+}ow.݃#.p9D}sFTsʂ+85du:a /Ei9vn7D8#u=٫cp.wR1R''^vMN*c#$ǯ+B=[hNz1xbJ})ii%p?$WH`Yzَn/8n;#wgb ٟb(/rҊ]1<6t_(@8g{r>>Fۡ}jp%Π- pN&R0`7y`ap!=2;.z#ܖM(|~K_.m+8gi"epA_Opr/t wϢ|fwvP !H'T?aFGelai iyfGl0Jh?W7"KPQzoqXx =sD}e'AgGFe'nNiOF_[`T 8*gwѭfRDh^D$?v8:sƾy!8<"~ -i4Kt;Tl#]*r\Í (z`uB # wԯ8 1'cǧGπTYC}ޯoQ?STuG\Jpֺe4׶w"R ( t={R(S 05?'ՒON2+"Pk aÒݛf{`)^GNF p{xk+W7Þ'ruox6L~RA4_V l2蚻]dÍ<~ '2=;8A5\TP!,3aÿs][K^pJ\iEV0tZwoןX>4|֥;IF+l*! }58>HGR< ṌgSC f<_ZBq%R"u#kNup:i mnLM}jx¢^Qg g `1*y kPb,v|Q?Ssaq}/dY#L؎J1K:b[1 Qpq}x>ŇcL`QUrP kg;T\ s64{ " 9L2~d#;Q^γlw`:y؃)jZ {JpX<2PPp Rp:/wt2kK<~8~ΚQo=tR0K|ǚė2:]<ދpnS?ai}MJ_7 {_]жH7+o| 繊7x-鲡BVD8%Zo{ Q _˺; ȧ4E8y,.E+"ݮ }5;;붆AX[4r&! ؁G7cc%᷁{\l K8߀!>-R. 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r_eff_vec <- relative_eff(exp(LLvec), chain_id = chain_id) psis1 <- psis(log_ratios = -LLarr, r_eff = r_eff_arr) test_that("psis results haven't changed", { expect_equal_to_reference(psis1, "reference-results/psis.rds") }) test_that("psis returns object with correct structure", { expect_true(is.psis(psis1)) expect_false(is.loo(psis1)) expect_false(is.psis_loo(psis1)) expect_named(psis1, c("log_weights", "diagnostics")) expect_named(psis1$diagnostics, c("pareto_k", "n_eff")) expect_equal(dim(psis1), dim(LLmat)) expect_length(psis1$diagnostics$pareto_k, dim(psis1)[2]) expect_length(psis1$diagnostics$n_eff, dim(psis1)[2]) }) test_that("psis methods give same results", { psis2 <- suppressWarnings(psis(-LLmat, r_eff = r_eff_arr)) expect_identical(psis1, psis2) psisvec <- suppressWarnings(psis(-LLvec, r_eff = r_eff_vec)) psismat <- suppressWarnings(psis(-LLmat[, 1], r_eff = r_eff_vec)) expect_identical(psisvec, psismat) }) test_that("psis throws correct errors and warnings", { # r_eff=NULL warnings expect_warning(psis(-LLarr), "Relative effective sample sizes") expect_warning(psis(-LLmat), "Relative effective sample sizes") expect_warning(psis(-LLmat[, 1]), "Relative effective sample sizes") # r_eff=NA disables warnings expect_silent(psis(-LLarr, r_eff = NA)) expect_silent(psis(-LLmat, r_eff = NA)) expect_silent(psis(-LLmat[,1], r_eff = NA)) # r_eff=NULL and r_eff=NA give same answer expect_equal( suppressWarnings(psis(-LLarr)), psis(-LLarr, r_eff = NA) ) # r_eff wrong length is error expect_error(psis(-LLarr, r_eff = r_eff_arr[-1]), "one value per observation") # r_eff has some NA values causes error r_eff_arr[2] <- NA expect_error(psis(-LLarr, r_eff = r_eff_arr), "mix NA and not NA values") # tail length warnings expect_warning( psis(-LLarr[1:5,, ]), "Not enough tail samples to fit the generalized Pareto distribution" ) # no NAs or non-finite values allowed LLmat[1,1] <- NA expect_error(psis(-LLmat), "NAs not allowed in input") LLmat[1,1] <- 1 LLmat[10, 2] <- Inf expect_error(psis(-LLmat), "All input values must be finite") # no lists allowed expect_error(expect_warning(psis(as.list(-LLvec))), "List not allowed as input") # if array, must be 3-D array dim(LLarr) <- c(2, 250, 2, 32) expect_error( psis(-LLarr), "length(dim(log_ratios)) == 3 is not TRUE", fixed = TRUE ) }) test_that("throw_tail_length_warnings gives correct output", { expect_silent(throw_tail_length_warnings(10)) expect_equal(throw_tail_length_warnings(10), 10) expect_warning(throw_tail_length_warnings(1), "Not enough tail samples") expect_warning(throw_tail_length_warnings(c(1, 10, 2)), "Skipping the following columns: 1, 3") expect_warning(throw_tail_length_warnings(rep(1, 21)), "11 more not printed") }) test_that("weights method returns correct output", { # default arguments expect_identical(weights(psis1), weights(psis1, normalize = TRUE, log = TRUE)) # unnormalized log-weights same as in psis object expect_equal(psis1$log_weights, weights(psis1, normalize = FALSE)) # normalized weights sum to 1 expect_equal( colSums(weights(psis1, normalize = TRUE, log = FALSE)), rep(1, ncol(psis1$log_weights)) ) }) test_that("psis_n_eff methods works properly", { w <- weights(psis1, normalize = TRUE, log = FALSE) expect_equal(psis_n_eff.default(w[, 1], r_eff = 1), 1 / sum(w[, 1]^2)) expect_equal(psis_n_eff.default(w[, 1], r_eff = 2), 2 / sum(w[, 1]^2)) expect_equal( psis_n_eff.default(w[, 1], r_eff = 2), psis_n_eff.matrix(w, r_eff = rep(2, ncol(w)))[1] ) expect_warning(psis_n_eff.default(w[, 1]), "not adjusted based on MCMC n_eff") expect_warning(psis_n_eff.matrix(w), "not adjusted based on MCMC n_eff") }) test_that("do_psis_i throws warning if all tail values the same", { xx <- c(1,2,3,4,4,4,4,4,4,4,4) val <- expect_warning(do_psis_i(xx, tail_len_i = 6), "all tail values are the same") expect_equal(val$pareto_k, Inf) }) test_that("psis_smooth_tail returns original tail values if k is infinite", { # skip on M1 Mac until we figure out why this test fails only on M1 Mac skip_if(Sys.info()[["sysname"]] == "Darwin" && R.version$arch == "aarch64") xx <- c(1,2,3,4,4,4,4,4,4,4,4) val <- suppressWarnings(psis_smooth_tail(xx, 3)) expect_equal(val$tail, xx) expect_equal(val$k, Inf) }) loo/tests/testthat/test_deprecated_extractors.R0000644000176200001440000001275713267637066021641 0ustar liggesuserslibrary(loo) options(mc.cores = 1) set.seed(123) context("Depracted extractors") LLarr <- example_loglik_array() r_eff_arr <- relative_eff(exp(LLarr)) loo1 <- suppressWarnings(loo(LLarr, r_eff = r_eff_arr)) waic1 <- suppressWarnings(waic(LLarr)) expect_warning_fixed <- function(object, regexp = NULL) { expect_warning(object, regexp = regexp, fixed = TRUE) } test_that("extracting estimates by name is deprecated for loo objects", { # $ method expect_equal( expect_warning_fixed(loo1$elpd_loo, "elpd_loo using '$' is deprecated"), loo1$estimates["elpd_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1$se_elpd_loo, "se_elpd_loo using '$' is deprecated"), loo1$estimates["elpd_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1$p_loo, "p_loo using '$' is deprecated"), loo1$estimates["p_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1$se_p_loo, "se_p_loo using '$' is deprecated"), loo1$estimates["p_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1$looic, "looic using '$' is deprecated"), loo1$estimates["looic", "Estimate"] ) expect_equal( expect_warning_fixed(loo1$se_looic, "se_looic using '$' is deprecated"), loo1$estimates["looic", "SE"] ) # [ method expect_equal( expect_warning_fixed(loo1["elpd_loo"], "elpd_loo using '[' is deprecated")[[1]], loo1$estimates["elpd_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1["se_elpd_loo"], "se_elpd_loo using '[' is deprecated")[[1]], loo1$estimates["elpd_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1["p_loo"], "p_loo using '[' is deprecated")[[1]], loo1$estimates["p_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1["se_p_loo"], "se_p_loo using '[' is deprecated")[[1]], loo1$estimates["p_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1["looic"], "looic using '[' is deprecated")[[1]], loo1$estimates["looic", "Estimate"] ) expect_equal( expect_warning_fixed(loo1["se_looic"], "se_looic using '[' is deprecated")[[1]], loo1$estimates["looic", "SE"] ) # [[ method expect_equal( expect_warning_fixed(loo1[["elpd_loo"]], "elpd_loo using '[[' is deprecated"), loo1$estimates["elpd_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1[["se_elpd_loo"]], "se_elpd_loo using '[[' is deprecated"), loo1$estimates["elpd_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1[["p_loo"]], "p_loo using '[[' is deprecated"), loo1$estimates["p_loo", "Estimate"] ) expect_equal( expect_warning_fixed(loo1[["se_p_loo"]], "se_p_loo using '[[' is deprecated"), loo1$estimates["p_loo", "SE"] ) expect_equal( expect_warning_fixed(loo1[["looic"]], "looic using '[[' is deprecated"), loo1$estimates["looic", "Estimate"] ) expect_equal( expect_warning_fixed(loo1[["se_looic"]], "se_looic using '[[' is deprecated"), loo1$estimates["looic", "SE"] ) }) test_that("extracting estimates by name is deprecated for waic objects", { expect_equal( expect_warning_fixed(waic1$elpd_waic, "elpd_waic using '$' is deprecated"), waic1$estimates["elpd_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1$se_elpd_waic, "se_elpd_waic using '$' is deprecated"), waic1$estimates["elpd_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1$p_waic, "p_waic using '$' is deprecated"), waic1$estimates["p_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1$se_p_waic, "se_p_waic using '$' is deprecated"), waic1$estimates["p_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1$waic, "waic using '$' is deprecated"), waic1$estimates["waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1$se_waic, "se_waic using '$' is deprecated"), waic1$estimates["waic", "SE"] ) expect_equal( expect_warning_fixed(waic1["elpd_waic"], "elpd_waic using '[' is deprecated")[[1]], waic1$estimates["elpd_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1["se_elpd_waic"], "se_elpd_waic using '[' is deprecated")[[1]], waic1$estimates["elpd_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1["p_waic"], "p_waic using '[' is deprecated")[[1]], waic1$estimates["p_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1["se_p_waic"], "se_p_waic using '[' is deprecated")[[1]], waic1$estimates["p_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1["waic"], "waic using '[' is deprecated")[[1]], waic1$estimates["waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1["se_waic"], "se_waic using '[' is deprecated")[[1]], waic1$estimates["waic", "SE"] ) expect_equal( expect_warning_fixed(waic1[["elpd_waic"]], "elpd_waic using '[[' is deprecated"), waic1$estimates["elpd_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1[["se_elpd_waic"]], "se_elpd_waic using '[[' is deprecated"), waic1$estimates["elpd_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1[["p_waic"]], "p_waic using '[[' is deprecated"), waic1$estimates["p_waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1[["se_p_waic"]], "se_p_waic using '[[' is deprecated"), waic1$estimates["p_waic", "SE"] ) expect_equal( expect_warning_fixed(waic1[["waic"]], "waic using '[[' is deprecated"), waic1$estimates["waic", "Estimate"] ) expect_equal( expect_warning_fixed(waic1[["se_waic"]], "se_waic using '[[' is deprecated"), waic1$estimates["waic", "SE"] ) }) loo/tests/testthat/test_extract_log_lik.R0000644000176200001440000000051614407123455020410 0ustar liggesuserslibrary(loo) context("extract_log_lik") test_that("extract_log_lik throws appropriate errors", { x1 <- rnorm(100) expect_error(extract_log_lik(x1), regexp = "Not a stanfit object") x2 <- structure(x1, class = "stanfit") expect_error(extract_log_lik(x2)) # not an S4 object OR no applicable method (depending on R version) }) loo/tests/testthat/test_0_helpers.R0000644000176200001440000000522213575772017017127 0ustar liggesuserslibrary(loo) context("helper functions and example data") LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() test_that("example_loglik_array and example_loglik_matrix dimensions ok", { dim_arr <- dim(LLarr) dim_mat <- dim(LLmat) expect_equal(dim_mat[1], dim_arr[1] * dim_arr[2]) expect_equal(dim_mat[2], dim_arr[3]) }) test_that("example_loglik_array and example_loglik_matrix contain same values", { expect_equal(LLmat[1:500, ], LLarr[, 1, ]) expect_equal(LLmat[501:1000, ], LLarr[, 2, ]) }) test_that("reshaping functions result in correct dimensions", { LLmat2 <- llarray_to_matrix(LLarr) expect_identical(LLmat2, LLmat) LLarr2 <- llmatrix_to_array(LLmat2, chain_id = rep(1:2, each = 500)) expect_identical(LLarr2, LLarr) }) test_that("reshaping functions throw correct errors", { expect_error(llmatrix_to_array(LLmat, chain_id = rep(1:2, times = c(400, 600))), regexp = "Not all chains have same number of iterations", fixed = TRUE) expect_error(llmatrix_to_array(LLmat, chain_id = rep(1:2, each = 400)), regexp = "Number of rows in matrix not equal to length(chain_id)", fixed = TRUE) expect_error(llmatrix_to_array(LLmat, chain_id = rep(2:3, each = 500)), regexp = "max(chain_id) not equal to the number of chains", fixed = TRUE) expect_error(llmatrix_to_array(LLmat, chain_id = rnorm(1000)), regexp = "all(chain_id == as.integer(chain_id)) is not TRUE", fixed = TRUE) }) test_that("colLogMeanExps(x) = log(colMeans(exp(x))) ", { expect_equal(colLogMeanExps(LLmat), log(colMeans(exp(LLmat)))) }) test_that("validating log-lik objects and functions works", { f_ok <- function(data_i, draws) return(NULL) f_bad1 <- function(data_i) return(NULL) f_bad2 <- function(data, draws) return(NULL) expect_equal(validate_llfun(f_ok), f_ok) bad_msg <- "Log-likelihood function must have at least the arguments 'data_i' and 'draws'" expect_error(validate_llfun(f_bad1), bad_msg) expect_error(validate_llfun(f_bad2), bad_msg) }) test_that("nlist works", { a <- 1; b <- 2; c <- 3; nlist_val <- list(nlist(a, b, c), nlist(a, b, c = "tornado")) nlist_ans <- list(list(a = 1, b = 2, c = 3), list(a = 1, b = 2, c = "tornado")) expect_equal(nlist_val, nlist_ans) expect_equal(nlist(a = 1, b = 2, c = 3), list(a = 1, b = 2, c = 3)) }) test_that("loo_cores works", { expect_equal(loo_cores(10), 10) options(mc.cores = 2) expect_equal(loo_cores(getOption("mc.cores", 1)), 2) options(mc.cores = 1) options(loo.cores = 2) expect_warning(expect_equal(loo_cores(10), 2), "deprecated") options(loo.cores=NULL) }) loo/tests/testthat/test_psis_approximate_posterior.R0000644000176200001440000001774114411130104022723 0ustar liggesuserslibrary(loo) context("psis_approximate_posterior") load(test_path("data-for-tests/test_data_psis_approximate_posterior.rda")) test_that("Laplace approximation, independent posterior", { log_p <- test_data_psis_approximate_posterior$laplace_independent$log_p log_g <- test_data_psis_approximate_posterior$laplace_independent$log_q ll <- test_data_psis_approximate_posterior$laplace_independent$log_liks expect_silent( psis_lap <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap, "psis") expect_lt(pareto_k_values(psis_lap), 0.7) expect_silent( psis_lap_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap_ll, "loo") expect_true(all(pareto_k_values(psis_lap_ll) < 0.7)) }) test_that("Laplace approximation, correlated posterior", { log_p <- test_data_psis_approximate_posterior$laplace_correlated$log_p log_g <- test_data_psis_approximate_posterior$laplace_correlated$log_q ll <- test_data_psis_approximate_posterior$laplace_correlated$log_liks expect_silent( psis_lap <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap, "psis") expect_lt(pareto_k_values(psis_lap), 0.7) expect_silent( psis_lap_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap_ll, "loo") expect_true(all(pareto_k_values(psis_lap_ll) < 0.7)) }) test_that("Laplace approximation, normal model", { log_p <- test_data_psis_approximate_posterior$laplace_normal$log_p log_g <- test_data_psis_approximate_posterior$laplace_normal$log_q ll <- test_data_psis_approximate_posterior$laplace_normal$log_liks expect_warning( psis_lap <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap, "psis") expect_gt(pareto_k_values(psis_lap), 0.5) expect_warning( psis_lap_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_lap_ll, "loo") expect_true(all(pareto_k_values(psis_lap_ll) > 0.5)) }) test_that("ADVI fullrank approximation, independent posterior", { log_p <- test_data_psis_approximate_posterior$fullrank_independent$log_p log_g <- test_data_psis_approximate_posterior$fullrank_independent$log_q ll <- test_data_psis_approximate_posterior$fullrank_independent$log_liks expect_silent( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_lt(pareto_k_values(psis_advi), 0.7) expect_silent( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) < 0.7)) }) test_that("ADVI fullrank approximation, correlated posterior", { log_p <- test_data_psis_approximate_posterior$fullrank_correlated$log_p log_g <- test_data_psis_approximate_posterior$fullrank_correlated$log_q ll <- test_data_psis_approximate_posterior$fullrank_correlated$log_liks expect_silent( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_lt(pareto_k_values(psis_advi), 0.7) expect_silent( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) < 0.7)) }) test_that("ADVI fullrank approximation, correlated posterior", { log_p <- test_data_psis_approximate_posterior$fullrank_normal$log_p log_g <- test_data_psis_approximate_posterior$fullrank_normal$log_q ll <- test_data_psis_approximate_posterior$fullrank_normal$log_liks expect_warning( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_gt(pareto_k_values(psis_advi), 0.7) expect_warning( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) > 0.7)) }) test_that("ADVI meanfield approximation, independent posterior", { log_p <- test_data_psis_approximate_posterior$meanfield_independent$log_p log_g <- test_data_psis_approximate_posterior$meanfield_independent$log_q ll <- test_data_psis_approximate_posterior$meanfield_independent$log_liks expect_silent( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_lt(pareto_k_values(psis_advi), 0.7) expect_silent( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) < 0.7)) }) test_that("ADVI meanfield approximation, correlated posterior", { log_p <- test_data_psis_approximate_posterior$meanfield_correlated$log_p log_g <- test_data_psis_approximate_posterior$meanfield_correlated$log_q ll <- test_data_psis_approximate_posterior$meanfield_correlated$log_liks expect_warning( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_gt(pareto_k_values(psis_advi), 0.7) expect_warning( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) > 0.5)) expect_true(any(pareto_k_values(psis_advi_ll) > 0.7)) }) test_that("ADVI meanfield approximation, normal model", { log_p <- test_data_psis_approximate_posterior$meanfield_normal$log_p log_g <- test_data_psis_approximate_posterior$meanfield_normal$log_q ll <- test_data_psis_approximate_posterior$meanfield_normal$log_liks expect_warning( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_gt(pareto_k_values(psis_advi), 0.7) expect_warning( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) > 0.7)) }) test_that("ADVI meanfield approximation, normal model", { log_p <- test_data_psis_approximate_posterior$meanfield_normal$log_p log_g <- test_data_psis_approximate_posterior$meanfield_normal$log_q ll <- test_data_psis_approximate_posterior$meanfield_normal$log_liks expect_warning( psis_advi <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi, "psis") expect_gt(pareto_k_values(psis_advi), 0.7) expect_warning( psis_advi_ll <- psis_approximate_posterior(log_p = log_p, log_g = log_g, log_liks = ll , cores = 1, save_psis = FALSE) ) expect_s3_class(psis_advi_ll, "loo") expect_true(all(pareto_k_values(psis_advi_ll) > 0.7)) }) test_that("Deprecation of log_q argument", { log_p <- test_data_psis_approximate_posterior$laplace_independent$log_p log_g <- test_data_psis_approximate_posterior$laplace_independent$log_q ll <- test_data_psis_approximate_posterior$laplace_independent$log_liks expect_warning( psis_lap <- loo:::psis_approximate_posterior(log_p = log_p, log_q = log_g, cores = 1, save_psis = FALSE) , regexp = "argument log_q has been changed to log_g" ) expect_s3_class(psis_lap, "psis") expect_lt(pareto_k_values(psis_lap), 0.7) }) loo/tests/testthat/test_E_loo.R0000644000176200001440000001303413575772017016303 0ustar liggesuserslibrary(loo) context("E_loo") LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() LLvec <- LLmat[, 1] chain_id <- rep(1:2, each = dim(LLarr)[1]) r_eff_mat <- relative_eff(exp(LLmat), chain_id) r_eff_vec <- relative_eff(exp(LLvec), chain_id = chain_id) psis_mat <- psis(-LLmat, r_eff = r_eff_mat, cores = 2) psis_vec <- psis(-LLvec, r_eff = r_eff_vec) set.seed(123) x <- matrix(rnorm(length(LLmat)), nrow = nrow(LLmat), ncol = ncol(LLmat)) log_rats <- -LLmat # matrix method E_test_mean <- E_loo(x, psis_mat, type = "mean", log_ratios = log_rats) E_test_var <- E_loo(x, psis_mat, type = "var", log_ratios = log_rats) E_test_quant <- E_loo(x, psis_mat, type = "quantile", probs = 0.5, log_ratios = log_rats) E_test_quant2 <- E_loo(x, psis_mat, type = "quantile", probs = c(0.1, 0.9), log_ratios = log_rats) # vector method E_test_mean_vec <- E_loo(x[, 1], psis_vec, type = "mean", log_ratios = log_rats[,1]) E_test_var_vec <- E_loo(x[, 1], psis_vec, type = "var", log_ratios = log_rats[,1]) E_test_quant_vec <- E_loo(x[, 1], psis_vec, type = "quant", probs = 0.5, log_ratios = log_rats[,1]) E_test_quant_vec2 <- E_loo(x[, 1], psis_vec, type = "quant", probs = c(0.1, 0.5, 0.9), log_ratios = log_rats[,1]) # E_loo_khat khat <- E_loo_khat(x, psis_mat, log_rats) test_that("E_loo return types correct for matrix method", { expect_type(E_test_mean, "list") expect_named(E_test_mean, c("value", "pareto_k")) expect_length(E_test_mean, 2) expect_length(E_test_mean$value, ncol(x)) expect_length(E_test_mean$pareto_k, ncol(x)) expect_type(E_test_var, "list") expect_named(E_test_var, c("value", "pareto_k")) expect_length(E_test_var, 2) expect_length(E_test_var$value, ncol(x)) expect_length(E_test_var$pareto_k, ncol(x)) expect_type(E_test_quant, "list") expect_named(E_test_quant, c("value", "pareto_k")) expect_length(E_test_quant, 2) expect_length(E_test_quant$value, ncol(x)) expect_length(E_test_quant$pareto_k, ncol(x)) expect_type(E_test_quant2, "list") expect_named(E_test_quant2, c("value", "pareto_k")) expect_length(E_test_quant2, 2) expect_equal(dim(E_test_quant2$value), c(2, ncol(x))) expect_length(E_test_quant2$pareto_k, ncol(x)) }) test_that("E_loo return types correct for default/vector method", { expect_type(E_test_mean_vec, "list") expect_named(E_test_mean_vec, c("value", "pareto_k")) expect_length(E_test_mean_vec, 2) expect_length(E_test_mean_vec$value, 1) expect_length(E_test_mean_vec$pareto_k, 1) expect_type(E_test_var_vec, "list") expect_named(E_test_var_vec, c("value", "pareto_k")) expect_length(E_test_var_vec, 2) expect_length(E_test_var_vec$value, 1) expect_length(E_test_var_vec$pareto_k, 1) expect_type(E_test_quant_vec, "list") expect_named(E_test_quant_vec, c("value", "pareto_k")) expect_length(E_test_quant_vec, 2) expect_length(E_test_quant_vec$value, 1) expect_length(E_test_quant_vec$pareto_k, 1) expect_type(E_test_quant_vec2, "list") expect_named(E_test_quant_vec2, c("value", "pareto_k")) expect_length(E_test_quant_vec2, 2) expect_length(E_test_quant_vec2$value, 3) expect_length(E_test_quant_vec2$pareto_k, 1) }) test_that("E_loo.default equal to reference", { expect_equal_to_reference(E_test_mean_vec, "reference-results/E_loo_default_mean.rds") expect_equal_to_reference(E_test_var_vec, "reference-results/E_loo_default_var.rds") expect_equal_to_reference(E_test_quant_vec, "reference-results/E_loo_default_quantile_50.rds") expect_equal_to_reference(E_test_quant_vec2, "reference-results/E_loo_default_quantile_10_50_90.rds") }) test_that("E_loo.matrix equal to reference", { expect_equal_to_reference(E_test_mean, "reference-results/E_loo_matrix_mean.rds") expect_equal_to_reference(E_test_var, "reference-results/E_loo_matrix_var.rds") expect_equal_to_reference(E_test_quant, "reference-results/E_loo_matrix_quantile_50.rds") expect_equal_to_reference(E_test_quant2, "reference-results/E_loo_matrix_quantile_10_90.rds") }) test_that("E_loo throws correct errors and warnings", { # warnings expect_warning(E_loo.matrix(x, psis_mat), "'log_ratios' not specified") expect_warning(E_test <- E_loo.default(x[, 1], psis_vec), "'log_ratios' not specified") expect_null(E_test$pareto_k) # errors expect_error(E_loo(x, 1), "is.psis") expect_error( E_loo(x, psis_mat, type = "quantile", probs = 2), "all(probs > 0 & probs < 1) is not TRUE", fixed = TRUE ) expect_error( E_loo(rep("a", nrow(x)), psis_vec), "is.numeric(x) is not TRUE", fixed = TRUE ) expect_error( E_loo(1:10, psis_vec), "length(x) == dim(psis_object)[1] is not TRUE", fixed = TRUE ) expect_error( E_loo(cbind(1:10, 1:10), psis_mat), "identical(dim(x), dim(psis_object)) is not TRUE", fixed = TRUE ) }) test_that("weighted quantiles work", { .wquant_rapprox <- function(x, w, probs) { stopifnot(all(probs > 0 & probs < 1)) ord <- order(x) d <- x[ord] ww <- w[ord] p <- cumsum(ww) / sum(ww) stats::approx(p, d, probs, rule = 2)$y } .wquant_sim <- function(x, w, probs, n_sims) { xx <- sample(x, size = n_sims, replace = TRUE, prob = w / sum(w)) quantile(xx, probs, names = FALSE) } set.seed(123) pr <- seq(0.025, 0.975, 0.025) x1 <- rnorm(100) w1 <- rlnorm(100) expect_equal( .wquant(x1, w1, pr), .wquant_rapprox(x1, w1, pr) ) x1 <- rnorm(1e4) w1 <- rlnorm(1e4) # expect_equal( # .wquant(x1, w1, pr), # .wquant_sim(x1, w1, pr, n_sim = 5e6), # tol = 0.005 # ) expect_equal( .wquant(x1, rep(1, length(x1)), pr), quantile(x1, probs = pr, names = FALSE) ) }) loo/tests/testthat/test_compare.R0000644000176200001440000001264613575772017016704 0ustar liggesuserslibrary(loo) set.seed(123) SW <- suppressWarnings context("compare models") LLarr <- example_loglik_array() LLarr2 <- array(rnorm(prod(dim(LLarr)), c(LLarr), 0.5), dim = dim(LLarr)) LLarr3 <- array(rnorm(prod(dim(LLarr)), c(LLarr), 1), dim = dim(LLarr)) w1 <- SW(waic(LLarr)) w2 <- SW(waic(LLarr2)) test_that("loo_compare throws appropriate errors", { w3 <- SW(waic(LLarr[,, -1])) w4 <- SW(waic(LLarr[,, -(1:2)])) expect_error(loo_compare(2, 3), "must be a list if not a 'loo' object") expect_error(loo_compare(w1, w2, x = list(w1, w2)), "If 'x' is a list then '...' should not be specified") expect_error(loo_compare(w1, list(1,2,3)), "class 'loo'") expect_error(loo_compare(w1), "requires at least two models") expect_error(loo_compare(x = list(w1)), "requires at least two models") expect_error(loo_compare(w1, w3), "same number of data points") expect_error(loo_compare(w1, w2, w3), "same number of data points") }) test_that("loo_compare throws appropriate warnings", { w3 <- w1; w4 <- w2 class(w3) <- class(w4) <- c("kfold", "loo") attr(w3, "K") <- 2 attr(w4, "K") <- 3 expect_warning(loo_compare(w3, w4), "Not all kfold objects have the same K value") class(w4) <- c("psis_loo", "loo") attr(w4, "K") <- NULL expect_warning(loo_compare(w3, w4), "Comparing LOO-CV to K-fold-CV") w3 <- w1; w4 <- w2 attr(w3, "yhash") <- "a" attr(w4, "yhash") <- "b" expect_warning(loo_compare(w3, w4), "Not all models have the same y variable") }) comp_colnames <- c( "elpd_diff", "se_diff", "elpd_waic", "se_elpd_waic", "p_waic", "se_p_waic", "waic", "se_waic" ) test_that("loo_compare returns expected results (2 models)", { comp1 <- loo_compare(w1, w1) expect_s3_class(comp1, "compare.loo") expect_equal(colnames(comp1), comp_colnames) expect_equal(rownames(comp1), c("model1", "model2")) expect_output(print(comp1), "elpd_diff") expect_equivalent(comp1[1:2,1], c(0, 0)) expect_equivalent(comp1[1:2,2], c(0, 0)) comp2 <- loo_compare(w1, w2) expect_s3_class(comp2, "compare.loo") # expect_equal_to_reference(comp2, "reference-results/loo_compare_two_models.rds") comp2_ref <- readRDS(test_path("reference-results/loo_compare_two_models.rds")) expect_equivalent(comp2, comp2_ref) expect_equal(colnames(comp2), comp_colnames) # specifying objects via ... and via arg x gives equal results expect_equal(comp2, loo_compare(x = list(w1, w2))) }) test_that("loo_compare returns expected result (3 models)", { w3 <- SW(waic(LLarr3)) comp1 <- loo_compare(w1, w2, w3) expect_equal(colnames(comp1), comp_colnames) expect_equal(rownames(comp1), c("model1", "model2", "model3")) expect_equal(comp1[1,1], 0) expect_s3_class(comp1, "compare.loo") expect_s3_class(comp1, "matrix") # expect_equal_to_reference(comp1, "reference-results/loo_compare_three_models.rds") comp1_ref <- readRDS(test_path("reference-results/loo_compare_three_models.rds")) expect_equivalent(comp1, comp1_ref) # specifying objects via '...' gives equivalent results (equal # except rownames) to using 'x' argument expect_equivalent(comp1, loo_compare(x = list(w1, w2, w3))) }) # Tests for deprecated compare() ------------------------------------------ test_that("compare throws deprecation warnings", { expect_warning(loo::compare(w1, w2), "Deprecated") expect_warning(loo::compare(w1, w1, w2), "Deprecated") }) test_that("compare returns expected result (2 models)", { comp1 <- expect_warning(loo::compare(w1, w1), "Deprecated") expect_output(print(comp1), "elpd_diff") expect_equal(comp1[1:2], c(elpd_diff = 0, se = 0)) comp2 <- expect_warning(loo::compare(w1, w2), "Deprecated") # expect_equal_to_reference(comp2, "reference-results/compare_two_models.rds") expect_named(comp2, c("elpd_diff", "se")) expect_s3_class(comp2, "compare.loo") # specifying objects via ... and via arg x gives equal results comp_via_list <- expect_warning(loo::compare(x = list(w1, w2)), "Deprecated") expect_equal(comp2, comp_via_list) }) test_that("compare returns expected result (3 models)", { w3 <- SW(waic(LLarr3)) comp1 <- expect_warning(loo::compare(w1, w2, w3), "Deprecated") expect_equal( colnames(comp1), c( "elpd_diff", "se_diff", "elpd_waic", "se_elpd_waic", "p_waic", "se_p_waic", "waic", "se_waic" )) expect_equal(rownames(comp1), c("w1", "w2", "w3")) expect_equal(comp1[1,1], 0) expect_s3_class(comp1, "compare.loo") expect_s3_class(comp1, "matrix") # expect_equal_to_reference(comp1, "reference-results/compare_three_models.rds") # specifying objects via '...' gives equivalent results (equal # except rownames) to using 'x' argument comp_via_list <- expect_warning(loo::compare(x = list(w1, w2, w3)), "Deprecated") expect_equivalent(comp1, comp_via_list) }) test_that("compare throws appropriate errors", { expect_error(suppressWarnings(loo::compare(w1, w2, x = list(w1, w2))), "should not be specified") expect_error(suppressWarnings(loo::compare(x = 2)), "must be a list") expect_error(suppressWarnings(loo::compare(x = list(2))), "should have class 'loo'") expect_error(suppressWarnings(loo::compare(x = list(w1))), "requires at least two models") w3 <- SW(waic(LLarr2[,,-1])) expect_error(suppressWarnings(loo::compare(x = list(w1, w3))), "same number of data points") expect_error(suppressWarnings(loo::compare(x = list(w1, w2, w3))), "same number of data points") }) loo/tests/testthat/reference-results/0000755000176200001440000000000014411130104017467 5ustar liggesusersloo/tests/testthat/reference-results/gpdfit_old.rds0000644000176200001440000000013213575772017022341 0ustar liggesusersb```b`fad`b1@P ^X , @5/17k2fd3st2]loo/tests/testthat/reference-results/loo_predictive_metric_mae_quant.rds0000644000176200001440000000014414411130104026604 0ustar liggesusersb```b`abb`b1|> fn_0UXj7r͓0cH%%%P>Sq*:bqqloo/tests/testthat/reference-results/loo_predictive_metric_bacc_quant.rds0000644000176200001440000000013014411130104026725 0ustar liggesusersb```b`abb`b1|> f`>0piּb CbX#$37$g*N qloo/tests/testthat/reference-results/loo_predictive_metric_mse_mean.rds0000644000176200001440000000014414411130104026416 0ustar liggesusersb```b`abb`b1|> fv0"j~8&cjX84k^bnj1!1 ,ȑZ\X 3!37'qloo/tests/testthat/reference-results/moment_match_loo_Stan_1.rds0000644000176200001440000000570313701164066024756 0ustar liggesusers{=qƭ7A *7Bп ->z qHh"'n*vJ_JsN^X} .zHzPz<8jc,>JQSGD?D1:`EO Zv'bYI y#njS zYhtĠ2w1)UxBd@6>6Ow[>*!ZnZd 4&1l=KOlnM nT^5=wH?/n4kWٽbޮD_[|D𶰂Ip>Vju1z*)z1h3GmK `G G]C ,5NvR'DO&?rKܡ n~ad'_RmUTuxm%U|ö3 Aw.uKt{a )i\V'䄴r;C: qsAx0LIxa \,#6Y_yFQRz'_Sx gµL̡lj~5n6-j2|YKCvBɫp[u5;fܫ>9°]: o\%;7B no%NoOМP}8vY1dW AZ̎8Nῄ ZEW& l=٪C@=6I b9ÝU@Gsl;ՓK2Ȱ\isI=!{- m>:·,u࡞eRH]|$KpK78f'dpN4' vs:_ < ;b @.!/XH"N` sjA]Kp\«| Tj|C&0ɺ %|Z'Z]hDžOK9W Zz ? K"Skee"A֖7g]60RQJJ7NBgOz3"F`7hF1W݊jKAKM݃cz00׼rs74*pJ5AT<\"nxe6-V(w8DFC-q.B:paH !:A"*a=#HM7ZTҗ˄t<(E~Lhimʂo|)JƦ-C,+ jUmëeSPy^nx5u쐻Lqb >. j9$HvZǵ4"0> yCkYGۄKnꮙN{ր3޸d m)JZ*G432HQeIQeHQϨ@4g%}FUg8_}Feg>3N%}F3*>#(KrH#}Fg%}F:3ʑ>#HQeHq23*>&3RIN*(Ktgd> 3RIARH%}F:3J>"3ʒ> 3IQiH#}FigB gT$}Fyg?w5~n0*^` ?m|dž X fƠmL6x~B&4}XѭCZliV޻ NGA;#,UPzWڨ ^t9_@uwN_=}.42'Gڨ7c\z7T(_S9UCKj@M{6oKKh+oMO~q_)BSXXiB2pJXWFx2;5Muݴ󢊠JHh6g` A-TwqZZ6P6PAum$bLnvx.&}W" Ԅ:-t&zS^ex%h\`+7h8df^hѵf-6wJPՐk>2thJk7XB 6MC_lgi+Ψs좻A^}D_vG1/Rz Ŷos3n=G:dؐ'H'ЅTy:'u2p n~U@ލJEh\mUe&\,22ґuC?S1W5t) ;rs^zY& \⇌ uM̨2n,ro߮K&ŐLΑ) ONX"+n2F9=9.G$Xʉ%d4qKlR˙7ҷS<>.'7R_woloo/tests/testthat/reference-results/loo_predictive_metric_acc_quant.rds0000644000176200001440000000014114411130104026565 0ustar liggesusersb```b`abb`b1|> fvyg`aҬy@0 GjqIfnbI*T $i3qloo/tests/testthat/reference-results/loo_predictive_metric_mse_quant.rds0000644000176200001440000000014514411130104026627 0ustar liggesusersb```b`abb`b1|> ft`sW wg`aҬy@0 GjqIfnbI*T $URqloo/tests/testthat/reference-results/moment_match_loo_3.rds0000644000176200001440000000327313701164066023773 0ustar liggesusers{TTE/ Gh}@2fLI1Ezw.-ݫ0i –@^(3RHfi(rZ ];Sw;3[HQDb{`;*sqGpcֿ\;}Y}8gpe+q6-?R%$v`aYn{ ~n}q6SZAq T+5"DW'2zVТ^)jb}[v7R١t2eсVu'Q}sCɹQftGzmSݩ@;GܗtK52qg\ydq o,~zAm =OyWQn,Ay+3W{#{1C?a߽sZwYrc-x!҆ =?E"淤Ko )}M2w E}yWQOw$g8>Yۏ-^[5 Jl/yutl7װƩ\%^:iRm+SG|5(iV`kD4I QQH5+ !Mx.gH!MxgH::!Mx4҄gDxngH'҄gDxgH :!Mx4sǪ~<'AY{s}q,EYVqC35zpD{[~]ˤnË߸ڂ#J%(<-<GoxoyNGAYm{7'oS>('kŽ0{#)w)\u풯on–}8X {7eiM7_4 QɋS!_Z+Ɏi)8~m/J'`]A(r7CةyGBye.A5>R<ڕ߱!K) us!Z~1y>oIF~1hn12%MUmT3/h́&[7 c k]}s?n;_{ (V'R h 'gc*Z]<ՅP}M9 p`@gj>IRjMa9Z'X?"B߯`=ezH}xFg=' ?]`:Ӯ3W$r jM3&&&(M~ loo/tests/testthat/reference-results/mcse_loo.rds0000644000176200001440000000005713701164066022022 0ustar liggesusersb```b`fcd`b2~Nӆ§}|loo/tests/testthat/reference-results/loo_predictive_metric_acc_mean.rds0000644000176200001440000000014014411130104026354 0ustar liggesusersb```b`abb`b1|> fk/-gjX84k^bnj1!1 ,ȑZ\X 3Hqloo/tests/testthat/reference-results/loo.rds0000644000176200001440000000344213701164066021014 0ustar liggesusers{TGƿ$XE {b,X]NY\ 1@ᖐr @d VDMfnjE\R@W`4XqϞgd 6(ILEwYcΣ[,=Aflծ(ByQDw!':eD|Y4ny-I;Mĉc%J/q_%1QL7ݕW0 +>T&Hc f~H?3>H 1͘U<u(fDW,ט`aaF-bTxN_e:s3\lYyp|8λon?Y}n)f؞ E`ꥪ2g\ 'vYY-a>/K`*@y-YMvaVw5?FWvA6i>o:E` n/X SsT'ѳkW1Qlz`#Sq1 3)}.$[%/!m4v5 ׁbe 'n@{WdlVBM!r C%ϟ"SzW_-MlŪА +!i e6ur S*n!upb)g bOИ^{JށqʶϜ^Mk6Z][=U]4'v,lgN?Tjϟ!%K]%*JTY)`=0f^I/WRT|M*B_}X)Psdt*?MLsc9?T ;z>KD  uhV=)Zip^X6 M< Sk6vמظ HNs hzGTMmSiPyʚm%:Sp*臲Rz_Ω`#za5_v>*+_]ORP`USI!>OGJ:UD⿌o݌q7|XwVĆ+% K"cvH%"X!IlpƠ=Gݷd5xs(VE(YKRz:ͽ}bF9ӑst%SD~hg\q[U'(;.'jrNXTP`"ir=rt`ŝ2}2DeoZO9W=RtGl`lH _d(I"#706:y!mH9gK00 IJC$DHdtY#WJ"YX$<>![Zf]x>W#Vu0{u_6ecqߘftg/j!p loo/tests/testthat/reference-results/E_loo_matrix_var.rds0000644000176200001440000000116113575772017023522 0ustar liggesusersb```b`feb`b1|> V|Q~ M}ix`bA݂O޴p]-E>iB}J%sjCUjtSO>X[W}xwk(unmR}=#}ȦiW_NT>x>fC`ՍaןXV?3&#T>ؾr3+k?w*hԳ|V}۲[~هM|g}mgWG&mgim} :%Ŭo(8;o:<hwi5~4O{[{te~eS0ϵ?nO;5/9{.{ٿrc|nM&pyǓ+g& VnзlI8G=W$_?n+ojc{+uc\Aiiӡ3: jS +pG^-N=R״?eB+Ǯ٭z̯u)NJJ wxϖޚQ~7jRJ , @5/17'`YbNi*QXZ ddloo/tests/testthat/reference-results/moment_match_var_and_cov.rds0000644000176200001440000000355413701164066025243 0ustar liggesusers}TSG h$h A XhuF.JaC +"TGPB|ZPF=ES@ ~ *⡅E7샗Xѳf̽s͝maS0 Rm1F"ڹڈ~} :Ss\R>/ Up=)caK 1v Pq<)ѷ5yI[ esɛsi _IeJ;e%֯mNu14sxBSt 9޺Zk M?a3¢p-ƺ:ف&;{XH&ȏz4iw23whcԷ٫w Z,0wZk[2~8 ݞ̣Hk +ip!ʽ}Ξ{H?>k猎pqG6(NQ&'O]=@6&]qzDǂb~$P ff=iO-$:n}3/ 9F;φ 7>; gH'yV$ϐN $КڐerT.q&܊:8"̗s oQGS o)c&4/!Jy!|Q 'n^˘HM3z-ҮQHS@ЩۏѦlQ{W_ +|dӳWitiKHXG/ewRX{|PoSn6.GjiwͫLAV=J 5FE>;lw΋C.;;JX _F4 )`!U/*}M Jyzٰh)~$ ChjW T.ݹ d~~scTU>U@Ž(uZ-T=s+.FM?OֹhεQM銡B[p;i>r[rI(6X U_c䅡ʊﳏ+HMGPoBxw gJ9B/Jzc:%b*a loo/tests/testthat/reference-results/loo_compare_two_models.rds0000644000176200001440000000042513575772017024766 0ustar liggesusersb```b`fed`b2 ƷߖE_X@wO:E}:M0NA͂ RSxo;piH3A-}k )oqvœc)PP ,L@X8AJR`B0b [FLCrSRs QxFP5PQԜ̴4{q*2"_ ʣ 8*PX \T&$;9? 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return((df*scale)/rchisq(n, df = df)) } dinvchisq <- function(x, df, scale=1/df, log = FALSE, ...) { if (df <= 0) stop("df must be greater than zero") if (scale <= 0) stop("scale must be greater than zero") nu <- df/2 if (log) return(ifelse(x > 0, nu*log(nu) - log(gamma(nu)) + nu*log(scale) - (nu + 1)*log(x) - (nu*scale/x), NA)) else return(ifelse(x > 0, (((nu)^(nu))/gamma(nu)) * (scale^nu) * (x^(-(nu + 1))) * exp(-nu*scale/x), NA)) } # generate toy data # normally distributed data with known variance data_sd <- 1.1 data_mean <- 1.3 n <- as.integer(30) y <- rnorm(n = n, mean = data_mean, sd = data_sd) y_tilde <- 11 y[1] <- y_tilde ymean <- mean(y) s2 <- sum((y - ymean)^2)/(n - 1) # draws from the posterior distribution when including all observations draws_full_posterior_sigma2 <- rinvchisq(S, n - 1, s2) draws_full_posterior_mu <- rnorm(S, ymean, sqrt(draws_full_posterior_sigma2/n)) # create a dummy model object x <- list() x$data <- list() x$data$y <- y x$data$n <- n x$data$ymean <- ymean x$data$s2 <- s2 x$draws <- data.frame( mu = draws_full_posterior_mu, sigma = sqrt(draws_full_posterior_sigma2) ) # implement functions for moment matching loo # extract original posterior draws post_draws_test <- function(x, ...) { as.matrix(x$draws) } # extract original log lik draws log_lik_i_test <- function(x, i, ...) { -0.5*log(2*pi) - log(x$draws$sigma) - 1.0/(2*x$draws$sigma^2)*(x$data$y[i] - x$draws$mu)^2 } loglik <- matrix(0,S,n) for (j in seq(n)) { loglik[,j] <- log_lik_i_test(x, j) } # mu, log(sigma) unconstrain_pars_test <- function(x, pars, ...) { upars <- as.matrix(pars) upars[,2] <- log(upars[,2]) upars } log_prob_upars_test <- function(x, upars, ...) { dinvchisq(exp(upars[,2])^2,x$data$n - 1,x$data$s2, log = TRUE) + dnorm(upars[,1],x$data$ymean,exp(upars[,2])/sqrt(x$data$n), log = TRUE) } # compute log_lik_i values based on the unconstrained parameters log_lik_i_upars_test <- function(x, upars, i, ...) { -0.5*log(2*pi) - upars[,2] - 1.0/(2*exp(upars[,2])^2)*(x$data$y[i] - upars[,1])^2 } upars <- unconstrain_pars_test(x, x$draws) lwi_1 <- -loglik[,1] lwi_1 <- lwi_1 - matrixStats::logSumExp(lwi_1) test_that("log_prob_upars_test works", { upars <- unconstrain_pars_test(x, x$draws) xloo <- list() xloo$data <- list() xloo$data$y <- y[-1] xloo$data$n <- n - 1 xloo$data$ymean <- mean(y[-1]) xloo$data$s2 <- sum((y[-1] - mean(y[-1]))^2)/(n - 2) post1 <- log_prob_upars_test(x,upars) post1 <- post1 - matrixStats::logSumExp(post1) post2 <- log_prob_upars_test(xloo,upars) + loglik[,1] post2 <- post2 - matrixStats::logSumExp(post2) expect_equal(post1,post2) }) test_that("loo_moment_match.default warnings work", { # loo object loo_manual <- suppressWarnings(loo(loglik)) loo_manual_tis <- suppressWarnings(loo(loglik, is_method = "tis")) expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 100, split = FALSE, cov = TRUE, cores = 1), "Some Pareto k") expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = FALSE, cov = TRUE, cores = 1), "The accuracy of self-normalized importance sampling") expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 1, k_thres = 0.5, split = TRUE, cov = TRUE, cores = 1), "The maximum number of moment matching iterations") expect_error(loo_moment_match(x, loo_manual_tis, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = TRUE, cov = TRUE, cores = 1), "loo_moment_match currently supports only") }) test_that("loo_moment_match.default works", { # loo object loo_manual <- suppressWarnings(loo(loglik)) loo_moment_match_object <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.8, split = FALSE, cov = TRUE, cores = 1)) # diagnostic pareto k decreases but influence pareto k stays the same expect_lt(loo_moment_match_object$diagnostics$pareto_k[1], loo_moment_match_object$pointwise[1,"influence_pareto_k"]) expect_equal(loo_moment_match_object$pointwise[,"influence_pareto_k"],loo_manual$pointwise[,"influence_pareto_k"]) expect_equal(loo_moment_match_object$pointwise[,"influence_pareto_k"],loo_manual$diagnostics$pareto_k) expect_equal_to_reference(loo_moment_match_object, "reference-results/moment_match_loo_1.rds") loo_moment_match_object2 <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = FALSE, cov = TRUE, cores = 1)) expect_equal_to_reference(loo_moment_match_object2, "reference-results/moment_match_loo_2.rds") loo_moment_match_object3 <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = TRUE, cov = TRUE, cores = 1)) expect_equal_to_reference(loo_moment_match_object3, "reference-results/moment_match_loo_3.rds") loo_moment_match_object4 <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 100, split = FALSE, cov = TRUE, cores = 1)) expect_equal(loo_manual,loo_moment_match_object4) loo_manual_with_psis <- suppressWarnings(loo(loglik, save_psis = TRUE)) loo_moment_match_object5 <- suppressWarnings(loo_moment_match(x, loo_manual_with_psis, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.8, split = FALSE, cov = TRUE, cores = 1)) expect_equal(loo_moment_match_object5$diagnostics,loo_moment_match_object5$psis_object$diagnostics) }) test_that("variance and covariance transformations work", { S <- 2000 set.seed(8493874) draws_full_posterior_sigma2 <- rinvchisq(S, n - 1, s2) draws_full_posterior_mu <- rnorm(S, ymean, sqrt(draws_full_posterior_sigma2/n)) x$draws <- data.frame( mu = draws_full_posterior_mu, sigma = sqrt(draws_full_posterior_sigma2) ) loglik <- matrix(0,S,n) for (j in seq(n)) { loglik[,j] <- log_lik_i_test(x, j) } upars <- unconstrain_pars_test(x, x$draws) lwi_1 <- -loglik[,1] lwi_1 <- lwi_1 - matrixStats::logSumExp(lwi_1) loo_manual <- suppressWarnings(loo(loglik)) loo_moment_match_object <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.0, split = FALSE, cov = TRUE, cores = 1)) expect_equal_to_reference(loo_moment_match_object, "reference-results/moment_match_var_and_cov.rds") }) test_that("loo_moment_match.default works with multiple cores", { # loo object loo_manual <- suppressWarnings(loo(loglik)) loo_moment_match_manual3 <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = FALSE, cov = TRUE, cores = 1)) loo_moment_match_manual4 <- suppressWarnings(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, k_thres = 0.5, split = FALSE, cov = TRUE, cores = 2)) expect_equal(loo_moment_match_manual3$diagnostics$pareto_k, loo_moment_match_manual4$diagnostics$pareto_k) expect_equal(loo_moment_match_manual3$diagnostics$n_eff, loo_moment_match_manual4$diagnostics$n_eff) expect_equal(loo_moment_match_manual3$estimates, loo_moment_match_manual4$estimates) expect_equal(loo_moment_match_manual3$pointwise, loo_moment_match_manual4$pointwise, tolerance=5e-4) }) test_that("loo_moment_match_split works", { # skip on M1 Mac until we figure out why this test fails only on M1 Mac skip_if(Sys.info()[["sysname"]] == "Darwin" && R.version$arch == "aarch64") is_obj_1 <- suppressWarnings(importance_sampling.default(lwi_1, method = "psis", r_eff = 1, cores = 1)) lwi_1_ps <- as.vector(weights(is_obj_1)) split <- loo_moment_match_split( x, upars, cov = FALSE, total_shift = c(0,0), total_scaling = c(1,1), total_mapping = diag(c(1,1)), i = 1, log_prob_upars = log_prob_upars_test, log_lik_i_upars = log_lik_i_upars_test, cores = 1, r_eff_i = 1, is_method = "psis") expect_named(split,c("lwi", "lwfi", "log_liki", "r_eff_i")) expect_equal(lwi_1_ps,split$lwi) split2 <- loo_moment_match_split( x, upars, cov = FALSE, total_shift = c(-0.1,-0.2), total_scaling = c(0.7,0.7), total_mapping = matrix(c(1,0.1,0.1,1),2,2), i = 1, log_prob_upars = log_prob_upars_test, log_lik_i_upars = log_lik_i_upars_test, cores = 1, r_eff_i = 1, is_method = "psis") expect_equal_to_reference(split2, "reference-results/moment_match_split.rds") }) test_that("passing arguments works", { log_lik_i_upars_test_additional_argument <- function(x, upars, i, passed_arg = FALSE, ...) { if (!passed_arg) { warning("passed_arg was not passed here") } -0.5*log(2*pi) - upars[,2] - 1.0/(2*exp(upars[,2])^2)*(x$data$y[i] - upars[,1])^2 } unconstrain_pars_test_additional_argument <- function(x, pars, passed_arg = FALSE, ...) { if (!passed_arg) { warning("passed_arg was not passed here") } upars <- as.matrix(pars) upars[,2] <- log(upars[,2]) upars } log_prob_upars_test_additional_argument <- function(x, upars, passed_arg = FALSE, ...) { if (!passed_arg) { warning("passed_arg was not passed here") } dinvchisq(exp(upars[,2])^2,x$data$n - 1,x$data$s2, log = TRUE) + dnorm(upars[,1],x$data$ymean,exp(upars[,2])/sqrt(x$data$n), log = TRUE) } post_draws_test_additional_argument <- function(x, passed_arg = FALSE, ...) { if (!passed_arg) { warning("passed_arg was not passed here") } as.matrix(x$draws) } log_lik_i_test_additional_argument <- function(x, i, passed_arg = FALSE, ...) { if (!passed_arg) { warning("passed_arg was not passed here") } -0.5*log(2*pi) - log(x$draws$sigma) - 1.0/(2*x$draws$sigma^2)*(x$data$y[i] - x$draws$mu)^2 } # loo object loo_manual <- suppressWarnings(loo(loglik)) expect_silent(loo_moment_match(x, loo_manual, post_draws_test_additional_argument, log_lik_i_test_additional_argument, unconstrain_pars_test_additional_argument, log_prob_upars_test_additional_argument, log_lik_i_upars_test_additional_argument, max_iters = 30L, k_thres = 0.5, split = TRUE, cov = TRUE, cores = 1, passed_arg = TRUE)) }) loo/tests/testthat/test_loo_and_waic.R0000644000176200001440000001504114411130104017633 0ustar liggesuserslibrary(loo) options(mc.cores = 1) set.seed(123) context("loo, waic and elpd") LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() LLvec <- LLmat[, 1] chain_id <- rep(1:2, each = nrow(LLarr)) r_eff_arr <- relative_eff(exp(LLarr)) r_eff_mat <- relative_eff(exp(LLmat), chain_id = chain_id) loo1 <- suppressWarnings(loo(LLarr, r_eff = r_eff_arr)) waic1 <- suppressWarnings(waic(LLarr)) elpd1 <- suppressWarnings(elpd(LLarr)) test_that("using loo.cores is deprecated", { options(mc.cores = NULL) options(loo.cores = 1) expect_warning(loo(LLarr, r_eff = r_eff_arr, cores = 2), "loo.cores") options(loo.cores = NULL) options(mc.cores = 1) }) test_that("loo, waic and elpd results haven't changed", { expect_equal_to_reference(loo1, "reference-results/loo.rds") expect_equal_to_reference(waic1, "reference-results/waic.rds") expect_equal_to_reference(elpd1, "reference-results/elpd.rds") }) test_that("loo with cores=1 and cores=2 gives same results", { loo2 <- suppressWarnings(loo(LLarr, r_eff = r_eff_arr, cores = 2)) expect_equal(loo1$estimates, loo2$estimates) }) test_that("waic returns object with correct structure", { expect_true(is.waic(waic1)) expect_true(is.loo(waic1)) expect_false(is.psis_loo(waic1)) expect_named( waic1, c( "estimates", "pointwise", # deprecated but still there "elpd_waic", "p_waic", "waic", "se_elpd_waic", "se_p_waic", "se_waic" ) ) est_names <- dimnames(waic1$estimates) expect_equal(est_names[[1]], c("elpd_waic", "p_waic", "waic")) expect_equal(est_names[[2]], c("Estimate", "SE")) expect_equal(colnames(waic1$pointwise), est_names[[1]]) expect_equal(dim(waic1), dim(LLmat)) }) test_that("loo returns object with correct structure", { expect_false(is.waic(loo1)) expect_true(is.loo(loo1)) expect_true(is.psis_loo(loo1)) expect_named( loo1, c( "estimates", "pointwise", "diagnostics", "psis_object", # deprecated but still there "elpd_loo", "p_loo", "looic", "se_elpd_loo", "se_p_loo", "se_looic" ) ) expect_named(loo1$diagnostics, c("pareto_k", "n_eff")) expect_equal(dimnames(loo1$estimates)[[1]], c("elpd_loo", "p_loo", "looic")) expect_equal(dimnames(loo1$estimates)[[2]], c("Estimate", "SE")) expect_equal(colnames(loo1$pointwise), c("elpd_loo", "mcse_elpd_loo", "p_loo", "looic", "influence_pareto_k")) expect_equal(dim(loo1), dim(LLmat)) }) test_that("elpd returns object with correct structure", { expect_true(is.loo(elpd1)) expect_named( elpd1, c( "estimates", "pointwise" ) ) est_names <- dimnames(elpd1$estimates) expect_equal(est_names[[1]], c("elpd", "ic")) expect_equal(est_names[[2]], c("Estimate", "SE")) expect_equal(colnames(elpd1$pointwise), est_names[[1]]) expect_equal(dim(elpd1), dim(LLmat)) }) test_that("two pareto k values are equal", { expect_identical(loo1$pointwise[,"influence_pareto_k"], loo1$diagnostics$pareto_k) }) test_that("loo.array and loo.matrix give same result", { l2 <- suppressWarnings(loo(LLmat, r_eff = r_eff_mat)) expect_identical(loo1$estimates, l2$estimates) expect_identical(loo1$diagnostics, l2$diagnostics) # the mcse_elpd_loo columns won't be identical because we use sampling expect_identical(loo1$pointwise[, -2], l2$pointwise[, -2]) expect_equal(loo1$pointwise[, 2], l2$pointwise[, 2], tol = 0.005) }) test_that("loo.array runs with multiple cores", { loo_with_arr1 <- loo(LLarr, cores = 1, r_eff = NA) loo_with_arr2 <- loo(LLarr, cores = 2, r_eff = NA) expect_identical(loo_with_arr1$estimates, loo_with_arr2$estimates) }) test_that("waic.array and waic.matrix give same result", { waic2 <- suppressWarnings(waic(LLmat)) expect_identical(waic1, waic2) }) test_that("elpd.array and elpd.matrix give same result", { elpd2 <- suppressWarnings(elpd(LLmat)) expect_identical(elpd1, elpd2) }) test_that("loo, waic, and elpd error with vector input", { expect_error(loo(LLvec), regexp = "no applicable method") expect_error(waic(LLvec), regexp = "no applicable method") expect_error(elpd(LLvec), regexp = "no applicable method") }) # testing function methods ------------------------------------------------ source(test_path("data-for-tests/function_method_stuff.R")) waic_with_fn <- waic(llfun, data = data, draws = draws) waic_with_mat <- waic(llmat_from_fn) loo_with_fn <- loo(llfun, data = data, draws = draws, r_eff = rep(1, nrow(data))) loo_with_mat <- loo(llmat_from_fn, r_eff = rep(1, ncol(llmat_from_fn)), save_psis = TRUE) test_that("loo.cores deprecation warning works with function method", { options(loo.cores = 1) expect_warning(loo(llfun, cores = 2, data = data, draws = draws, r_eff = rep(1, nrow(data))), "loo.cores") options(loo.cores=NULL) }) test_that("loo_i results match loo results for ith data point", { expect_warning( loo_i_val <- loo_i(i = 2, llfun = llfun, data = data, draws = draws), "Relative effective sample sizes" ) expect_equal(loo_i_val$pointwise[, "elpd_loo"], loo_with_fn$pointwise[2, "elpd_loo"]) expect_equal(loo_i_val$pointwise[, "p_loo"], loo_with_fn$pointwise[2, "p_loo"]) expect_equal(loo_i_val$diagnostics$pareto_k, loo_with_fn$diagnostics$pareto_k[2]) expect_equal(loo_i_val$diagnostics$n_eff, loo_with_fn$diagnostics$n_eff[2]) }) test_that("function and matrix methods return same result", { expect_equal(waic_with_mat, waic_with_fn) expect_identical(loo_with_mat$estimates, loo_with_fn$estimates) expect_identical(loo_with_mat$diagnostics, loo_with_fn$diagnostics) expect_identical(dim(loo_with_mat), dim(loo_with_fn)) }) test_that("loo.function runs with multiple cores", { loo_with_fn1 <- loo(llfun, data = data, draws = draws, r_eff = rep(1, nrow(data)), cores = 1) loo_with_fn2 <- loo(llfun, data = data, draws = draws, r_eff = rep(1, nrow(data)), cores = 2) expect_identical(loo_with_fn2$estimates, loo_with_fn1$estimates) }) test_that("save_psis option to loo.function makes correct psis object", { loo_with_fn2 <- loo.function(llfun, data = data, draws = draws, r_eff = rep(1, nrow(data)), save_psis = TRUE) expect_identical(loo_with_fn2$psis_object, loo_with_mat$psis_object) }) test_that("loo throws r_eff warnings", { expect_warning(loo(-LLarr), "MCSE estimates will be over-optimistic") expect_warning(loo(-LLmat), "MCSE estimates will be over-optimistic") expect_warning(loo(llfun, data = data, draws = draws), "MCSE estimates will be over-optimistic") }) loo/tests/testthat/test_tisis.R0000644000176200001440000001253513701164066016374 0ustar liggesuserslibrary(loo) options(mc.cores=1) options(loo.cores=NULL) set.seed(123) context("tis and is") LLarr <- example_loglik_array() LLmat <- example_loglik_matrix() LLvec <- LLmat[, 1] chain_id <- rep(1:2, each = dim(LLarr)[1]) r_eff_arr <- relative_eff(exp(LLarr)) r_eff_vec <- relative_eff(exp(LLvec), chain_id = chain_id) psis1 <- psis(log_ratios = -LLarr, r_eff = r_eff_arr) tis1 <- tis(log_ratios = -LLarr, r_eff = r_eff_arr) is1 <- sis(log_ratios = -LLarr, r_eff = r_eff_arr) test_that("tis and is runs", { LLvec[1] <- -10 expect_silent(tis1 <- tis(log_ratios = -LLvec, r_eff = r_eff_vec)) expect_silent(is1 <- sis(log_ratios = -LLvec, r_eff = r_eff_vec)) expect_failure(expect_equal(tis1$log_weights, is1$log_weights)) expect_failure(expect_equal(tis1$log_weights, psis1$log_weights)) }) test_that("tis() and sis() returns object with correct structure for tis/sis", { expect_false(is.psis(tis1)) expect_false(is.psis(is1)) expect_true(is.tis(tis1)) expect_false(is.tis(is1)) expect_false(is.sis(tis1)) expect_true(is.sis(is1)) expect_false(is.loo(tis1)) expect_false(is.loo(is1)) expect_false(is.psis_loo(tis1)) expect_false(is.psis_loo(is1)) expect_named(tis1, c("log_weights", "diagnostics")) expect_named(is1, c("log_weights", "diagnostics")) expect_named(tis1$diagnostics, c("pareto_k", "n_eff")) expect_named(is1$diagnostics, c("pareto_k", "n_eff")) expect_equal(dim(tis1), dim(LLmat)) expect_equal(dim(is1), dim(LLmat)) expect_length(tis1$diagnostics$pareto_k, dim(psis1)[2]) expect_length(is1$diagnostics$pareto_k, dim(psis1)[2]) expect_length(tis1$diagnostics$n_eff, dim(psis1)[2]) expect_length(is1$diagnostics$n_eff, dim(psis1)[2]) expect_equal(attr(psis1, "method")[1], "psis") expect_equal(attr(tis1, "method")[1], "tis") expect_equal(attr(is1, "method")[1], "sis") }) test_that("psis methods give same results", { tis2 <- suppressWarnings(tis(-LLmat, r_eff = r_eff_arr)) expect_identical(tis1, tis2) tisvec <- suppressWarnings(tis(-LLvec, r_eff = r_eff_vec)) tismat <- suppressWarnings(tis(-LLmat[, 1], r_eff = r_eff_vec)) expect_identical(tisvec, tismat) is2 <- suppressWarnings(sis(-LLmat, r_eff = r_eff_arr)) expect_identical(is1, is2) isvec <- suppressWarnings(sis(-LLvec, r_eff = r_eff_vec)) ismat <- suppressWarnings(sis(-LLmat[, 1], r_eff = r_eff_vec)) expect_identical(isvec, ismat) }) test_that("psis throws correct errors and warnings", { # r_eff=NULL warnings expect_warning(tis(-LLarr), "Relative effective sample sizes") expect_warning(tis(-LLmat), "Relative effective sample sizes") expect_warning(tis(-LLmat[, 1]), "Relative effective sample sizes") # r_eff=NA disables warnings expect_silent(tis(-LLarr, r_eff = NA)) expect_silent(tis(-LLmat, r_eff = NA)) expect_silent(tis(-LLmat[,1], r_eff = NA)) # r_eff=NULL and r_eff=NA give same answer expect_equal( suppressWarnings(tis(-LLarr)), tis(-LLarr, r_eff = NA) ) # r_eff wrong length is error expect_error(tis(-LLarr, r_eff = r_eff_arr[-1]), "one value per observation") # r_eff has some NA values causes error r_eff_arr[2] <- NA expect_error(tis(-LLarr, r_eff = r_eff_arr), "mix NA and not NA values") # no NAs or non-finite values allowed LLmat[1,1] <- NA expect_error(tis(-LLmat), "NAs not allowed in input") LLmat[1,1] <- 1 LLmat[10, 2] <- Inf expect_error(tis(-LLmat), "All input values must be finite") # no lists allowed expect_error(expect_warning(tis(as.list(-LLvec)), "List not allowed as input")) # if array, must be 3-D array dim(LLarr) <- c(2, 250, 2, 32) expect_error( tis(-LLarr), "length(dim(log_ratios)) == 3 is not TRUE", fixed = TRUE ) }) test_that("explict test of values for 'sis' and 'tis'", { lw <- 1:16 expect_silent(tis_true <- tis(log_ratios = lw, r_eff = NA)) expect_equal(as.vector(weights(tis_true, log = TRUE, normalize = FALSE)), c(-14.0723, -13.0723, -12.0723, -11.0723, -10.0723, -9.0723, -8.0723, -7.0723, -6.0723, -5.0723, -4.0723, -3.0723, -2.0723, -1.0723, -0.0723, 0.) + 15.07238, tol = 0.001) expect_silent(is_true <- sis(log_ratios = lw, r_eff = NA)) expect_equal(as.vector(weights(is_true, log = TRUE, normalize = FALSE)), lw, tol = 0.00001) lw <- c(0.7609420, 1.3894140, 0.4158346, 2.5307927, 4.3379119, 2.4159240, 2.2462172, 0.8057697, 0.9333107, 1.5599302) expect_silent(tis_true <- tis(log_ratios = lw, r_eff = NA)) expect_equal(as.vector(weights(tis_true, log = TRUE, normalize = FALSE)), c(-2.931, -2.303, -3.276, -1.161, 0, -1.276, -1.446, -2.886, -2.759, -2.132) + 3.692668, tol = 0.001) expect_silent(is_true <- sis(log_ratios = lw, r_eff = NA)) expect_equal(as.vector(weights(is_true, log = TRUE, normalize = FALSE)), lw, tol = 0.00001) }) test_that("tis_loo and sis_loo are returned", { LLmat <- example_loglik_matrix() loo_psis <- suppressWarnings(loo(LLmat, r_eff = NA, is_method = "psis")) loo_tis <- suppressWarnings(loo(LLmat, r_eff = NA, is_method = "tis")) loo_sis <- suppressWarnings(loo(LLmat, r_eff = NA, is_method = "sis")) expect_s3_class(loo_tis, "tis_loo") expect_s3_class(loo_sis, "sis_loo") expect_s3_class(loo_tis, "importance_sampling_loo") expect_s3_class(loo_sis, "importance_sampling_loo") expect_output(print(loo_tis), regexp = "tis_loo") expect_output(print(loo_sis), regexp = "sis_loo") }) loo/tests/testthat/test_loo_predictive_metric.R0000644000176200001440000001377214411130104021600 0ustar liggesusersLL <- example_loglik_matrix() chain_id <- rep(1:2, each = dim(LL)[1] / 2) r_eff <- relative_eff(exp(LL), chain_id) psis_obj <- psis(-LL, r_eff = r_eff, cores = 2) set.seed(123) x <- matrix(rnorm(length(LL)), nrow = nrow(LL), ncol = ncol(LL)) x_prob <- 1 / (1 + exp(-x)) y <- rnorm(ncol(LL)) y_binary <- rbinom(ncol(LL), 1, 0.5) mae_mean <- loo_predictive_metric(x, y, LL, metric = 'mae', r_eff = r_eff) mae_quant <- loo_predictive_metric(x, y, LL, metric = 'mae', r_eff = r_eff, type = 'quantile', probs = 0.9) rmse_mean <- loo_predictive_metric(x, y, LL, metric = 'rmse', r_eff = r_eff) rmse_quant <- loo_predictive_metric(x, y, LL, metric = 'rmse', r_eff = r_eff, type = 'quantile', probs = 0.9) mse_mean <- loo_predictive_metric(x, y, LL, metric = 'mse', r_eff = r_eff) mse_quant <- loo_predictive_metric(x, y, LL, metric = 'mse', r_eff = r_eff, type = 'quantile', probs = 0.9) acc_mean <- loo_predictive_metric(x_prob, y_binary, LL, metric = 'acc', r_eff = r_eff) acc_quant <- loo_predictive_metric(x_prob, y_binary, LL, metric = 'acc', r_eff = r_eff, type = 'quantile', probs = 0.9) bacc_mean <- loo_predictive_metric(x_prob, y_binary, LL, metric = 'balanced_acc', r_eff = r_eff) bacc_quant <- loo_predictive_metric(x_prob, y_binary, LL, metric = 'balanced_acc', r_eff = r_eff, type = 'quantile', probs = 0.9) test_that('loo_predictive_metric stops with incorrect inputs', { expect_error(loo_predictive_metric(as.character(x), y, LL, r_eff = r_eff), 'no applicable method', fixed = TRUE) expect_error(loo_predictive_metric(x, as.character(y), LL, r_eff = r_eff), 'is.numeric(y) is not TRUE', fixed = TRUE) x_invalid <- matrix(rnorm(9), nrow = 3) expect_error(loo_predictive_metric(x_invalid, y, LL, r_eff = r_eff), 'identical(ncol(x), length(y)) is not TRUE', fixed = TRUE) x_invalid <- matrix(rnorm(64), nrow = 2) expect_error(loo_predictive_metric(x_invalid, y, LL, r_eff = r_eff), 'identical(dim(x), dim(log_lik)) is not TRUE', fixed = TRUE) }) test_that('loo_predictive_metric return types are correct', { # MAE expect_type(mae_mean, 'list') expect_type(mae_quant, 'list') expect_named(mae_mean, c('estimate', 'se')) expect_named(mae_quant, c('estimate', 'se')) # RMSE expect_type(rmse_mean, 'list') expect_type(rmse_quant, 'list') expect_named(rmse_mean, c('estimate', 'se')) expect_named(rmse_quant, c('estimate', 'se')) # MSE expect_type(mse_mean, 'list') expect_type(mse_quant, 'list') expect_named(mse_mean, c('estimate', 'se')) expect_named(mse_quant, c('estimate', 'se')) # Accuracy expect_type(acc_mean, 'list') expect_type(acc_quant, 'list') expect_named(acc_mean, c('estimate', 'se')) expect_named(acc_quant, c('estimate', 'se')) # Balanced accuracy expect_type(bacc_mean, 'list') expect_type(bacc_quant, 'list') expect_named(bacc_mean, c('estimate', 'se')) expect_named(bacc_quant, c('estimate', 'se')) }) test_that('loo_predictive_metric is equal to reference', { expect_equal_to_reference(mae_mean, 'reference-results/loo_predictive_metric_mae_mean.rds') expect_equal_to_reference(mae_quant, 'reference-results/loo_predictive_metric_mae_quant.rds') expect_equal_to_reference(rmse_mean, 'reference-results/loo_predictive_metric_rmse_mean.rds') expect_equal_to_reference(rmse_quant, 'reference-results/loo_predictive_metric_rmse_quant.rds') expect_equal_to_reference(mse_mean, 'reference-results/loo_predictive_metric_mse_mean.rds') expect_equal_to_reference(mse_quant, 'reference-results/loo_predictive_metric_mse_quant.rds') expect_equal_to_reference(acc_mean, 'reference-results/loo_predictive_metric_acc_mean.rds') expect_equal_to_reference(acc_quant, 'reference-results/loo_predictive_metric_acc_quant.rds') expect_equal_to_reference(bacc_mean, 'reference-results/loo_predictive_metric_bacc_mean.rds') expect_equal_to_reference(bacc_quant, 'reference-results/loo_predictive_metric_bacc_quant.rds') }) test_that('MAE computation is correct', { expect_equal( .mae(rep(0.5, 5), rep(1, 5))$estimate, 0.5) expect_equal( .mae(rep(0.5, 5), rep(1, 5))$se, 0.0) expect_error( .mae(rep(0.5, 5), rep(1, 3)), 'length(y) == length(yhat) is not TRUE', fixed = TRUE) }) test_that('MSE computation is correct', { expect_equal( .mse(rep(0.5, 5), rep(1, 5))$estimate, 0.25) expect_equal( .mse(rep(0.5, 5), rep(1, 5))$se, 0.0) expect_error( .mse(rep(0.5, 5), rep(1, 3)), 'length(y) == length(yhat) is not TRUE', fixed = TRUE) }) test_that('RMSE computation is correct', { expect_equal( .rmse(rep(0.5, 5), rep(1, 5))$estimate, sqrt(0.25)) expect_equal( .mse(rep(0.5, 5), rep(1, 5))$se, 0.0) expect_error( .mse(rep(0.5, 5), rep(1, 3)), 'length(y) == length(yhat) is not TRUE', fixed = TRUE) }) test_that('Accuracy computation is correct', { expect_equal( .accuracy(c(0, 0, 0, 1, 1, 1), c(0.2, 0.2, 0.2, 0.7, 0.7, 0.7))$estimate, 1.0 ) expect_error( .accuracy(c(1, 0), c(0.5)), 'length(y) == length(yhat) is not TRUE', fixed = TRUE) expect_error( .accuracy(c(2, 1), c(0.5, 0.5)), 'all(y <= 1 & y >= 0) is not TRUE', fixed = TRUE ) expect_error( .accuracy(c(1, 0), c(1.1, 0.5)), 'all(yhat <= 1 & yhat >= 0) is not TRUE', fixed = TRUE ) }) test_that('Balanced accuracy computation is correct', { expect_equal( .balanced_accuracy(c(0, 0, 1, 1, 1, 1), c(0.9, 0.9, 0.9, 0.9, 0.9, 0.9))$estimate, 0.5 ) expect_error( .balanced_accuracy(c(1, 0), c(0.5)), 'length(y) == length(yhat) is not TRUE', fixed = TRUE) expect_error( .balanced_accuracy(c(2, 1), c(0.5, 0.5)), 'all(y <= 1 & y >= 0) is not TRUE', fixed = TRUE ) expect_error( .balanced_accuracy(c(1, 0), c(1.1, 0.5)), 'all(yhat <= 1 & yhat >= 0) is not TRUE', fixed = TRUE ) }) loo/tests/testthat/test_model_weighting.R0000644000176200001440000001005214410420140020360 0ustar liggesuserslibrary(loo) context("loo_model_weights") # generate fake data set.seed(123) y<-rnorm(50,0,1) sd_sim1<- abs(rnorm(500,1.5, 0.1)) sd_sim2<- abs(rnorm(500,1.2, 0.1)) sd_sim3<- abs(rnorm(500,1, 0.05)) log_lik1 <- log_lik2 <- log_lik3 <- matrix(NA, 500, 50) for(s in 1:500) { log_lik1[s,] <- dnorm(y,-1,sd_sim1[s], log=T) log_lik2[s,] <- dnorm(y,0.7,sd_sim2[s], log=T) log_lik3[s,] <- dnorm(y,1,sd_sim3[s], log=T) } ll_list <- list(log_lik1, log_lik2,log_lik3) r_eff_list <- list(rep(0.9,50), rep(0.9,50), rep(0.9,50)) loo_list <- lapply(1:length(ll_list), function(j) { loo(ll_list[[j]], r_eff = r_eff_list[[j]]) }) tol <- 0.01 # absoulte tolerance of weights test_that("loo_model_weights throws correct errors and warnings", { expect_error(loo_model_weights(log_lik1), "list of matrices or a list of 'psis_loo' objects") expect_error(loo_model_weights(list(log_lik1)), "At least two models") expect_error(loo_model_weights(list(loo_list[[1]])), "At least two models") expect_error(loo_model_weights(list(log_lik1), method = "pseudobma"), "At least two models") expect_error(loo_model_weights(list(log_lik1, log_lik2[-1, ])), "same dimensions") expect_error(loo_model_weights(list(log_lik1, log_lik2, log_lik3[, -1])), "same dimensions") loo_list2 <- loo_list attr(loo_list2[[3]], "dims") <- c(10, 10) expect_error(loo_model_weights(loo_list2), "same dimensions") expect_error(loo_model_weights(ll_list, r_eff_list = r_eff_list[-1]), "one component for each model") r_eff_list[[3]] <- rep(0.9, 51) expect_error(loo_model_weights(ll_list, r_eff_list = r_eff_list), "same length as the number of columns") expect_error(loo_model_weights(list(loo_list[[1]], 2)), "List elements must all be 'psis_loo' objects or log-likelihood matrices", fixed = TRUE) expect_warning(loo_model_weights(ll_list), "Relative effective sample sizes") }) test_that("loo_model_weights (stacking and pseudo-BMA) gives expected result", { w1 <- loo_model_weights(ll_list, method = "stacking", r_eff_list = r_eff_list) expect_type(w1,"double") expect_s3_class(w1, "stacking_weights") expect_length(w1, 3) expect_named(w1, paste0("model" ,c(1:3))) expect_equal_to_reference(as.numeric(w1), "reference-results/model_weights_stacking.rds", tolerance = tol, scale=1) expect_output(print(w1), "Method: stacking") w1_b <- loo_model_weights(loo_list) expect_identical(w1, w1_b) w2 <- loo_model_weights(ll_list, r_eff_list=r_eff_list, method = "pseudobma", BB = TRUE) expect_type(w2, "double") expect_s3_class(w2, "pseudobma_bb_weights") expect_length(w2, 3) expect_named(w2, paste0("model", c(1:3))) expect_equal_to_reference(as.numeric(w2), "reference-results/model_weights_pseudobma.rds", tolerance = tol, scale=1) expect_output(print(w2), "Method: pseudo-BMA+") w3 <- loo_model_weights(ll_list, r_eff_list=r_eff_list, method = "pseudobma", BB = FALSE) expect_type(w3,"double") expect_length(w3, 3) expect_named(w3, paste0("model" ,c(1:3))) expect_equal(as.numeric(w3), c(5.365279e-05, 9.999436e-01, 2.707028e-06), tolerance = tol, scale = 1) expect_output(print(w3), "Method: pseudo-BMA") w3_b <- loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) expect_identical(w3, w3_b) }) test_that("stacking_weights and pseudobma_weights throw correct errors", { xx <- cbind(rnorm(10)) expect_error(stacking_weights(xx), "two models are required") expect_error(pseudobma_weights(xx), "two models are required") }) test_that("loo_model_weights uses correct names for list of loo objects", { loo1 <- loo_list[[1]] loo2 <- loo_list[[2]] loo3 <- loo_list[[3]] expect_named( loo_model_weights(list(loo1, loo2, loo3)), c("model1", "model2", "model3") ) expect_named( loo_model_weights(list("a" = loo1, loo2, "c" = loo3)), c("a", "model2", "c") ) expect_named( loo_model_weights(list(`a` = loo1, `b` = loo2, `c` = loo3)), c("a", "b", "c") ) }) loo/tests/testthat/test_crps.R0000644000176200001440000000436214411130104016170 0ustar liggesusersset.seed(123456789) n <- 10 S <- 100 y <- rnorm(n) x1 <- matrix(rnorm(n * S), nrow = S) x2 <- matrix(rnorm(n * S), nrow = S) ll <- matrix(rnorm(n * S) * 0.1 - 1, nrow = S) with_seed <- function(seed, code) { code <- substitute(code) orig.seed <- .Random.seed on.exit(.Random.seed <<- orig.seed) set.seed(seed) eval.parent(code) } test_that("crps computation is correct", { expect_equal(.crps_fun(2.0, 1.0), 0.0) expect_equal(.crps_fun(1.0, 2.0), -1.5) expect_equal(.crps_fun(pi, pi^2), 0.5 * pi - pi^2) expect_equal(.crps_fun(1.0, 0.0, scale = TRUE), 0.0) expect_equal(.crps_fun(1.0, 2.0, scale = TRUE), -2.0) expect_equal(.crps_fun(pi, pi^2, scale = TRUE), -pi^2/pi - 0.5 * log(pi)) }) test_that("crps matches references", { expect_equal_to_reference(with_seed(1, crps(x1, x2, y)), 'reference-results/crps.rds') expect_equal_to_reference(with_seed(1, scrps(x1, x2, y)), 'reference-results/scrps.rds') # Suppress warnings for the missing r_eff suppressWarnings(expect_equal_to_reference( with_seed(1, loo_crps(x1, x2, y, ll)), 'reference-results/loo_crps.rds')) suppressWarnings(expect_equal_to_reference( with_seed(1, loo_scrps(x1, x2, y, ll)), 'reference-results/loo_scrps.rds')) }) test_that("input validation throws correct errors", { expect_error(validate_crps_input(as.character(x1), x2, y), "is.numeric(x) is not TRUE", fixed = TRUE) expect_error(validate_crps_input(x1, as.character(x2), y), "is.numeric(x2) is not TRUE", fixed = TRUE) expect_error(validate_crps_input(x1, x2, c('a', 'b')), "is.numeric(y) is not TRUE", fixed = TRUE) expect_error(validate_crps_input(x1, t(x2), y), "identical(dim(x), dim(x2)) is not TRUE", fixed = TRUE) expect_error(validate_crps_input(x1, x2, c(1, 2)), "ncol(x) == length(y) is not TRUE", fixed = TRUE) expect_error(validate_crps_input(x1, x2, y, t(ll)), "ifelse(is.null(log_lik), TRUE, identical(dim(log_lik), dim(x))) is not TRUE", fixed = TRUE) }) test_that("methods for single data point don't error", { expect_silent(crps(x1[,1], x2[,1], y[1])) expect_silent(scrps(x1[,1], x2[,1], y[1])) }) loo/tests/testthat/test_print_plot.R0000644000176200001440000001317513701164066017434 0ustar liggesuserslibrary(loo) set.seed(1414) context("print, plot, diagnostics") LLarr <- example_loglik_array() waic1 <- suppressWarnings(waic(LLarr)) loo1 <- suppressWarnings(loo(LLarr)) psis1 <- suppressWarnings(psis(-LLarr)) # plotting ---------------------------------------------------------------- test_that("plot methods don't error", { expect_silent(plot(loo1, label_points = FALSE)) expect_silent(plot(psis1, label_points = TRUE)) expect_silent(plot(psis1, diagnostic = "n_eff", label_points = FALSE)) loo1$diagnostics$pareto_k[1] <- 10 expect_silent(plot(loo1, label_points = TRUE)) expect_output(print(loo1, plot_k = TRUE)) expect_output(print(psis1, plot_k = TRUE)) }) test_that("plot methods throw appropriate errors/warnings", { expect_error(plot(waic1), regexp = "No Pareto k estimates found") loo1$diagnostics$pareto_k[1:5] <- Inf psis1$diagnostics$pareto_k[1:5] <- Inf expect_warning(plot(loo1), regexp = "estimates are Inf/NA/NaN and not plotted.") expect_warning(plot(psis1), regexp = "estimates are Inf/NA/NaN and not plotted.") }) # printing ---------------------------------------------------------------- lldim_msg <- paste0("Computed from ", prod(dim(LLarr)[1:2]) , " by ", dim(LLarr)[3], " log-likelihood matrix") lwdim_msg <- paste0("Computed from ", prod(dim(LLarr)[1:2]) , " by ", dim(LLarr)[3], " log-weights matrix") test_that("print.waic output is ok",{ expect_output(print(waic1), lldim_msg) expect_output(print(waic1), "p_waic estimates greater than 0.4. We recommend trying loo instead." ) }) test_that("print.psis_loo and print.psis output ok",{ expect_output(print(psis1), lwdim_msg) expect_output(print(psis1), "Pareto k estimates are good") expect_output(print(loo1), lldim_msg) expect_output(print(loo1), "Pareto k estimates are good") loo1$diagnostics$pareto_k <- psis1$diagnostics$pareto_k <- runif(32, 0, .49) expect_output(print(loo1), regexp = "Pareto k estimates are good") expect_output(print(psis1), regexp = "Pareto k estimates are good") loo1$diagnostics$pareto_k[1] <- psis1$diagnostics$pareto_k[1] <- 0.71 expect_output(print(loo1), regexp = "Pareto k diagnostic") loo1$diagnostics$pareto_k[1] <- psis1$diagnostics$pareto_k[1] <- 1.1 expect_output(print(loo1), regexp = "Pareto k diagnostic") }) # pareto_k_[ids,values,table] --------------------------------------------- test_that("pareto_k_values works for psis_loo and psis objects, errors for waic", { kpsis <- pareto_k_values(psis1) kloo <- pareto_k_values(loo1) expect_identical(kpsis, kloo) expect_identical(kpsis, psis1$diagnostics$pareto_k) expect_error(pareto_k_values(waic1), "No Pareto k estimates found") }) test_that("pareto_k_influence_values works for psis_loo objects, errors for psis waic", { kloo <- pareto_k_influence_values(loo1) kloo2 <- pareto_k_values(loo1) expect_identical(kloo, kloo2) expect_error(pareto_k_influence_values(psis1), "No Pareto k influence estimates found") expect_error(pareto_k_influence_values(waic1), "No Pareto k influence estimates found") }) test_that("pareto_k_ids identifies correct observations", { for (j in 1:5) { loo1$diagnostics$pareto_k <- psis1$diagnostics$pareto_k <- runif(32, .25, 1.25) expect_identical( pareto_k_ids(loo1, threshold = 0.5), pareto_k_ids(psis1, threshold = 0.5) ) expect_identical( pareto_k_ids(loo1, threshold = 0.5), which(pareto_k_values(loo1) > 0.5) ) expect_identical( pareto_k_ids(psis1, threshold = 0.7), which(pareto_k_values(psis1) > 0.7) ) } }) test_that("pareto_k_table gives correct output", { psis1$diagnostics$pareto_k[1:10] <- runif(10, 0, 0.49) psis1$diagnostics$pareto_k[11:17] <- runif(7, 0.51, 0.69) psis1$diagnostics$pareto_k[18:20] <- runif(3, 0.71, 0.99) psis1$diagnostics$pareto_k[21:32] <- runif(12, 1, 10) k <- pareto_k_values(psis1) tab <- pareto_k_table(psis1) expect_output(print(tab), "Pareto k diagnostic values") expect_identical(colnames(tab), c("Count", "Proportion", "Min. n_eff")) expect_equal(sum(tab[, "Count"]), length(k)) expect_equal(sum(tab[, "Proportion"]), 1) expect_equal(sum(k <= 0.5), tab[1,1]) expect_equal(sum(k > 0.5 & k <= 0.7), tab[2,1]) expect_equal(sum(k > 0.7 & k <= 1), tab[3,1]) expect_equal(sum(k > 1), tab[4,1]) psis1$diagnostics$pareto_k[1:32] <- 0.4 expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are good (k < 0.5)", fixed = TRUE) psis1$diagnostics$pareto_k[1:32] <- 0.65 expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are ok (k < 0.7)", fixed = TRUE) # if n_eff is NULL psis1$diagnostics$n_eff <- NULL tab2 <- pareto_k_table(psis1) expect_output(print(tab2), "") expect_equal(unname(tab2[, "Min. n_eff"]), rep(NA_real_, 4)) }) # psis_neff and mcse_loo -------------------------------------------------- test_that("psis_n_eff_values extractor works", { n_eff_psis <- psis1$diagnostics$n_eff expect_type(n_eff_psis, "double") expect_identical(psis_n_eff_values(psis1), n_eff_psis) expect_identical(psis_n_eff_values(psis1), psis_n_eff_values(loo1)) psis1$diagnostics$n_eff <- NULL expect_error(psis_n_eff_values(psis1), "No PSIS n_eff estimates found") }) test_that("mcse_loo extractor gives correct value", { mcse <- mcse_loo(loo1) expect_type(mcse, "double") expect_equal_to_reference(mcse, "reference-results/mcse_loo.rds") }) test_that("mcse_loo returns NA when it should", { loo1$diagnostics$pareto_k[1] <- 1.5 mcse <- mcse_loo(loo1) expect_equal(mcse, NA) }) test_that("mcse_loo errors if not psis_loo object", { expect_error(mcse_loo(psis1), "psis_loo") }) loo/tests/testthat/test_gpdfit.R0000644000176200001440000000147313575772017016527 0ustar liggesuserslibrary(loo) context("generalized pareto") test_that("gpdfit returns correct result", { set.seed(123) x <- rexp(100) gpdfit_val_old <- unlist(gpdfit(x, wip=FALSE, min_grid_pts = 80)) expect_equal_to_reference(gpdfit_val_old, "reference-results/gpdfit_old.rds") gpdfit_val_wip <- unlist(gpdfit(x, wip=TRUE, min_grid_pts = 80)) expect_equal_to_reference(gpdfit_val_wip, "reference-results/gpdfit.rds") gpdfit_val_wip_default_grid <- unlist(gpdfit(x, wip=TRUE)) expect_equal_to_reference(gpdfit_val_wip_default_grid, "reference-results/gpdfit_default_grid.rds") }) test_that("qgpd returns the correct result ", { probs <- seq(from = 0, to = 1, by = 0.25) q1 <- qgpd(probs, k = 1, sigma = 1) expect_equal(q1, c(0, 1/3, 1, 3, Inf)) q2 <- qgpd(probs, k = 1, sigma = 0) expect_true(all(is.nan(q2))) }) loo/tests/testthat.R0000644000176200001440000000011413575010725014171 0ustar liggesuserslibrary(loo) library(testthat) Sys.setenv("R_TESTS" = "") test_check("loo") loo/vignettes/0000755000176200001440000000000014411465512013056 5ustar liggesusersloo/vignettes/loo2-weights.Rmd0000644000176200001440000003777114407123455016066 0ustar liggesusers--- title: "Bayesian Stacking and Pseudo-BMA weights using the loo package" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates the new functionality in __loo__ v2.0.0 for Bayesian stacking and Pseudo-BMA weighting. In this vignette we can't provide all of the necessary background on this topic, so we encourage readers to refer to the paper * Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, \doi:10.1214/17-BA1091. [Online](https://projecteuclid.org/euclid.ba/1516093227) which provides important details on the methods demonstrated in this vignette. Here we just quote from the abstract of the paper: > **Abstract**: Bayesian model averaging is flawed in the $\mathcal{M}$-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions. We extend the utility function to any proper scoring rule and use Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions. We compare stacking of predictive distributions to several alternatives: stacking of means, Bayesian model averaging (BMA), Pseudo-BMA, and a variant of Pseudo-BMA that is stabilized using the Bayesian bootstrap. Based on simulations and real-data applications, we recommend stacking of predictive distributions, with bootstrapped-Pseudo-BMA as an approximate alternative when computation cost is an issue. Ideally, we would avoid the Bayesian model combination problem by extending the model to include the separate models as special cases, and preferably as a continuous expansion of the model space. For example, instead of model averaging over different covariate combinations, all potentially relevant covariates should be included in a predictive model (for causal analysis more care is needed) and a prior assumption that only some of the covariates are relevant can be presented with regularized horseshoe prior (Piironen and Vehtari, 2017a). For variable selection we recommend projective predictive variable selection (Piironen and Vehtari, 2017a; [__projpred__ package](https://cran.r-project.org/package=projpred)). To demonstrate how to use __loo__ package to compute Bayesian stacking and Pseudo-BMA weights, we repeat two simple model averaging examples from Chapters 6 and 10 of _Statistical Rethinking_ by Richard McElreath. In _Statistical Rethinking_ WAIC is used to form weights which are similar to classical "Akaike weights". Pseudo-BMA weighting using PSIS-LOO for computation is close to these WAIC weights, but named after the Pseudo Bayes Factor by Geisser and Eddy (1979). As discussed below, in general we prefer using stacking rather than WAIC weights or the similar pseudo-BMA weights. # Setup In addition to the __loo__ package we will also load the __rstanarm__ package for fitting the models. ```{r setup, message=FALSE} library(rstanarm) library(loo) ``` # Example: Primate milk In _Statistical Rethinking_, McElreath describes the data for the primate milk example as follows: > A popular hypothesis has it that primates with larger brains produce more energetic milk, so that brains can grow quickly. ... The question here is to what extent energy content of milk, measured here by kilocalories, is related to the percent of the brain mass that is neocortex. ... We'll end up needing female body mass as well, to see the masking that hides the relationships among the variables. ```{r data} data(milk) d <- milk[complete.cases(milk),] d$neocortex <- d$neocortex.perc /100 str(d) ``` We repeat the analysis in Chapter 6 of _Statistical Rethinking_ using the following four models (here we use the default weakly informative priors in __rstanarm__, while flat priors were used in _Statistical Rethinking_). ```{r fits, results="hide"} fit1 <- stan_glm(kcal.per.g ~ 1, data = d, seed = 2030) fit2 <- update(fit1, formula = kcal.per.g ~ neocortex) fit3 <- update(fit1, formula = kcal.per.g ~ log(mass)) fit4 <- update(fit1, formula = kcal.per.g ~ neocortex + log(mass)) ``` McElreath uses WAIC for model comparison and averaging, so we'll start by also computing WAIC for these models so we can compare the results to the other options presented later in the vignette. The __loo__ package provides `waic` methods for log-likelihood arrays, matrices and functions. Since we fit our model with rstanarm we can use the `waic` method provided by the __rstanarm__ package (a wrapper around `waic` from the __loo__ package), which allows us to just pass in our fitted model objects instead of first extracting the log-likelihood values. ```{r waic} waic1 <- waic(fit1) waic2 <- waic(fit2) waic3 <- waic(fit3) waic4 <- waic(fit4) waics <- c( waic1$estimates["elpd_waic", 1], waic2$estimates["elpd_waic", 1], waic3$estimates["elpd_waic", 1], waic4$estimates["elpd_waic", 1] ) ``` We get some warnings when computing WAIC for models 3 and 4, indicating that we shouldn't trust the WAIC weights we will compute later. Following the recommendation in the warning, we next use the `loo` methods to compute PSIS-LOO instead. The __loo__ package provides `loo` methods for log-likelihood arrays, matrices, and functions, but since we fit our model with __rstanarm__ we can just pass the fitted model objects directly and __rstanarm__ will extract the needed values to pass to the __loo__ package. (Like __rstanarm__, some other R packages for fitting Stan models, e.g. __brms__, also provide similar methods for interfacing with the __loo__ package.) ```{r loo} # note: the loo function accepts a 'cores' argument that we recommend specifying # when working with bigger datasets loo1 <- loo(fit1) loo2 <- loo(fit2) loo3 <- loo(fit3) loo4 <- loo(fit4) lpd_point <- cbind( loo1$pointwise[,"elpd_loo"], loo2$pointwise[,"elpd_loo"], loo3$pointwise[,"elpd_loo"], loo4$pointwise[,"elpd_loo"] ) ``` With `loo` we don't get any warnings for models 3 and 4, but for illustration of good results, we display the diagnostic details for these models anyway. ```{r print-loo} print(loo3) print(loo4) ``` One benefit of PSIS-LOO over WAIC is better diagnostics. Here for both models 3 and 4 all $k<0.7$ and the Monte Carlo SE of `elpd_loo` is 0.1 or less, and we can expect the model comparison to be reliable. Next we compute and compare 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights. ```{r weights} waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ``` With all approaches Model 4 with `neocortex` and `log(mass)` gets most of the weight. Based on theory, Pseudo-BMA weights without Bayesian bootstrap should be close to WAIC weights, and we can also see that here. Pseudo-BMA+ weights with Bayesian bootstrap provide more cautious weights further away from 0 and 1 (see Yao et al. (2018) for a discussion of why this can be beneficial and results from related experiments). In this particular example, the Bayesian stacking weights are not much different from the other weights. One of the benefits of stacking is that it manages well if there are many similar models. Consider for example that there could be many irrelevant covariates that when included would produce a similar model to one of the existing models. To emulate this situation here we simply copy the first model a bunch of times, but you can imagine that instead we would have ten alternative models with about the same predictive performance. WAIC weights for such a scenario would be close to the following: ```{r waic_wts_demo} waic_wts_demo <- exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) / sum(exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)])) round(waic_wts_demo, 3) ``` Notice how much the weight for model 4 is lowered now that more models similar to model 1 (or in this case identical) have been added. Both WAIC weights and Pseudo-BMA approaches first estimate the predictive performance separately for each model and then compute weights based on estimated relative predictive performances. Similar models share similar weights so the weights of other models must be reduced for the total sum of the weights to remain the same. On the other hand, stacking optimizes the weights _jointly_, allowing for the very similar models (in this toy example repeated models) to share their weight while more unique models keep their original weights. In our example we can see this difference clearly: ```{r stacking_weights} stacking_weights(lpd_point[,c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) ``` Using stacking, the weight for the best model stays essentially unchanged. # Example: Oceanic tool complexity Another example we consider is the Kline oceanic tool complexity data, which McElreath describes as follows: >Different historical island populations possessed tool kits of different size. These kits include fish hooks, axes, boats, hand plows, and many other types of tools. A number of theories predict that larger populations will both develop and sustain more complex tool kits. ... It's also suggested that contact rates among populations effectively increases population [sic, probably should be tool kit] size, as it's relevant to technological evolution. We build models predicting the total number of tools given the log population size and the contact rate (high vs. low). ```{r Kline} data(Kline) d <- Kline d$log_pop <- log(d$population) d$contact_high <- ifelse(d$contact=="high", 1, 0) str(d) ``` We start with a Poisson regression model with the log population size, the contact rate, and an interaction term between them (priors are informative priors as in _Statistical Rethinking_). ```{r fit10, results="hide"} fit10 <- stan_glm( total_tools ~ log_pop + contact_high + log_pop * contact_high, family = poisson(link = "log"), data = d, prior = normal(0, 1, autoscale = FALSE), prior_intercept = normal(0, 100, autoscale = FALSE), seed = 2030 ) ``` Before running other models, we check whether Poisson is good choice as the conditional observation model. ```{r loo10} loo10 <- loo(fit10) print(loo10) ``` We get at least one observation with $k>0.7$ and the estimated effective number of parameters `p_loo` is larger than the total number of parameters in the model. This indicates that Poisson might be too narrow. A negative binomial model might be better, but with so few observations it is not so clear. We can compute LOO more accurately by running Stan again for the leave-one-out folds with high $k$ estimates. When using __rstanarm__ this can be done by specifying the `k_threshold` argument: ```{r loo10-threshold} loo10 <- loo(fit10, k_threshold=0.7) print(loo10) ``` In this case we see that there is not much difference, and thus it is relatively safe to continue. As a comparison we also compute WAIC: ```{r waic10} waic10 <- waic(fit10) print(waic10) ``` The WAIC computation is giving warnings and the estimated ELPD is slightly more optimistic. We recommend using the PSIS-LOO results instead. To assess whether the contact rate and interaction term are useful, we can make a comparison to models without these terms. ```{r contact_high, results="hide"} fit11 <- update(fit10, formula = total_tools ~ log_pop + contact_high) fit12 <- update(fit10, formula = total_tools ~ log_pop) ``` ```{r loo-contact_high} (loo11 <- loo(fit11)) (loo12 <- loo(fit12)) ``` ```{r relo-contact_high} loo11 <- loo(fit11, k_threshold=0.7) loo12 <- loo(fit12, k_threshold=0.7) lpd_point <- cbind( loo10$pointwise[, "elpd_loo"], loo11$pointwise[, "elpd_loo"], loo12$pointwise[, "elpd_loo"] ) ``` For comparison we'll also compute WAIC values for these additional models: ```{r waic-contact_high} waic11 <- waic(fit11) waic12 <- waic(fit12) waics <- c( waic10$estimates["elpd_waic", 1], waic11$estimates["elpd_waic", 1], waic12$estimates["elpd_waic", 1] ) ``` The WAIC computation again gives warnings, and we recommend using PSIS-LOO instead. Finally, we compute 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights. ```{r weights-contact_high} waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ``` All weights favor the second model with the log population and the contact rate. WAIC weights and Pseudo-BMA weights (without Bayesian bootstrap) are similar, while Pseudo-BMA+ is more cautious and closer to stacking weights. It may seem surprising that Bayesian stacking is giving zero weight to the first model, but this is likely due to the fact that the estimated effect for the interaction term is close to zero and thus models 1 and 2 give very similar predictions. In other words, incorporating the model with the interaction (model 1) into the model average doesn't improve the predictions at all and so model 1 is given a weight of 0. On the other hand, models 2 and 3 are giving slightly different predictions and thus their combination may be slightly better than either alone. This behavior is related to the repeated similar model illustration in the milk example above. # Simpler coding using `loo_model_weights` function Although in the examples above we called the `stacking_weights` and `pseudobma_weights` functions directly, we can also use the `loo_model_weights` wrapper, which takes as its input either a list of pointwise log-likelihood matrices or a list of precomputed loo objects. There are also `loo_model_weights` methods for stanreg objects (fitted model objects from __rstanarm__) as well as fitted model objects from other packages (e.g. __brms__) that do the preparation work for the user (see, e.g., the examples at `help("loo_model_weights", package = "rstanarm")`). ```{r loo_model_weights} # using list of loo objects loo_list <- list(loo10, loo11, loo12) loo_model_weights(loo_list) loo_model_weights(loo_list, method = "pseudobma") loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) ``` # References McElreath, R. (2016). _Statistical rethinking: A Bayesian course with examples in R and Stan_. Chapman & Hall/CRC. http://xcelab.net/rm/statistical-rethinking/ Piironen, J. and Vehtari, A. (2017a). Sparsity information and regularization in the horseshoe and other shrinkage priors. In Electronic Journal of Statistics, 11(2):5018-5051. [Online](https://projecteuclid.org/euclid.ejs/1513306866). Piironen, J. and Vehtari, A. (2017b). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3):711-735. \doi:10.1007/s11222-016-9649-y. [Online](https://link.springer.com/article/10.1007/s11222-016-9649-y). Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, \doi:10.1214/17-BA1091. [Online](https://projecteuclid.org/euclid.ba/1516093227). loo/vignettes/loo2-lfo.Rmd0000644000176200001440000006042614407123455015165 0ustar liggesusers--- title: "Approximate leave-future-out cross-validation for Bayesian time series models" author: "Paul Bürkner, Jonah Gabry, Aki Vehtari" date: "`r Sys.Date()`" output: html_vignette: toc: yes encoding: "UTF-8" params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r more-knitr-ops, include=FALSE} knitr::opts_chunk$set( cache = TRUE, message = FALSE, warning = FALSE ) ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` ## Introduction One of the most common goals of a time series analysis is to use the observed series to inform predictions for future observations. We will refer to this task of predicting a sequence of $M$ future observations as $M$-step-ahead prediction ($M$-SAP). Fortunately, once we have fit a model and can sample from the posterior predictive distribution, it is straightforward to generate predictions as far into the future as we want. It is also straightforward to evaluate the $M$-SAP performance of a time series model by comparing the predictions to the observed sequence of $M$ future data points once they become available. Unfortunately, we are often in the position of having to use a model to inform decisions _before_ we can collect the future observations required for assessing the predictive performance. If we have many competing models we may also need to first decide which of the models (or which combination of the models) we should rely on for predictions. In these situations the best we can do is to use methods for approximating the expected predictive performance of our models using only the observations of the time series we already have. If there were no time dependence in the data or if the focus is to assess the non-time-dependent part of the model, we could use methods like leave-one-out cross-validation (LOO-CV). For a data set with $N$ observations, we refit the model $N$ times, each time leaving out one of the $N$ observations and assessing how well the model predicts the left-out observation. LOO-CV is very expensive computationally in most realistic settings, but the Pareto smoothed importance sampling (PSIS, Vehtari et al, 2017, 2019) algorithm provided by the *loo* package allows for approximating exact LOO-CV with PSIS-LOO-CV. PSIS-LOO-CV requires only a single fit of the full model and comes with diagnostics for assessing the validity of the approximation. With a time series we can do something similar to LOO-CV but, except in a few cases, it does not make sense to leave out observations one at a time because then we are allowing information from the future to influence predictions of the past (i.e., times $t + 1, t+2, \ldots$ should not be used to predict for time $t$). To apply the idea of cross-validation to the $M$-SAP case, instead of leave-*one*-out cross-validation we need some form of leave-*future*-out cross-validation (LFO-CV). As we will demonstrate in this case study, LFO-CV does not refer to one particular prediction task but rather to various possible cross-validation approaches that all involve some form of prediction for new time series data. Like exact LOO-CV, exact LFO-CV requires refitting the model many times to different subsets of the data, which is computationally very costly for most nontrivial examples, in particular for Bayesian analyses where refitting the model means estimating a new posterior distribution rather than a point estimate. Although PSIS-LOO-CV provides an efficient approximation to exact LOO-CV, until now there has not been an analogous approximation to exact LFO-CV that drastically reduces the computational burden while also providing informative diagnostics about the quality of the approximation. In this case study we present PSIS-LFO-CV, an algorithm that typically only requires refitting the time-series model a small number times and will make LFO-CV tractable for many more realistic applications than previously possible. More details can be found in our paper about approximate LFO-CV (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1902.06281). ## $M$-step-ahead predictions Assume we have a time series of observations $y = (y_1, y_2, \ldots, y_N)$ and let $L$ be the _minimum_ number of observations from the series that we will require before making predictions for future data. Depending on the application and how informative the data is, it may not be possible to make reasonable predictions for $y_{i+1}$ based on $(y_1, \dots, y_{i})$ until $i$ is large enough so that we can learn enough about the time series to predict future observations. Setting $L=10$, for example, means that we will only assess predictive performance starting with observation $y_{11}$, so that we always have at least 10 previous observations to condition on. In order to assess $M$-SAP performance we would like to compute the predictive densities $$ p(y_{i+1:M} \,|\, y_{1:i}) = p(y_{i+1}, \ldots, y_{i + M} \,|\, y_{1},...,y_{i}) $$ for each $i \in \{L, \ldots, N - M\}$. The quantities $p(y_{i+1:M} \,|\, y_{1:i})$ can be computed with the help of the posterior distribution $p(\theta \,|\, y_{1:i})$ of the parameters $\theta$ conditional on only the first $i$ observations of the time-series: $$ p(y_{i+1:M} \,| \, y_{1:i}) = \int p(y_{i+1:M} \,| \, y_{1:i}, \theta) \, p(\theta\,|\,y_{1:i}) \,d\theta. $$ Having obtained $S$ draws $(\theta_{1:i}^{(1)}, \ldots, \theta_{1:i}^{(S)})$ from the posterior distribution $p(\theta\,|\,y_{1:i})$, we can estimate $p(y_{i+1:M} | y_{1:i})$ as $$ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{1}{S}\sum_{s=1}^S p(y_{i+1:M} \,|\, y_{1:i}, \theta_{1:i}^{(s)}). $$ ## Approximate $M$-SAP using importance-sampling {#approximate_MSAP} Unfortunately, the math above makes use of the posterior distributions from many different fits of the model to different subsets of the data. That is, to obtain the predictive density $p(y_{i+1:M} \,|\, y_{1:i})$ requires fitting a model to only the first $i$ data points, and we will need to do this for every value of $i$ under consideration (all $i \in \{L, \ldots, N - M\}$). To reduce the number of models that need to be fit for the purpose of obtaining each of the densities $p(y_{i+1:M} \,|\, y_{1:i})$, we propose the following algorithm. First, we refit the model using the first $L$ observations of the time series and then perform a single exact $M$-step-ahead prediction step for $p(y_{L+1:M} \,|\, y_{1:L})$. Recall that $L$ is the minimum number of observations we have deemed acceptable for making predictions (setting $L=0$ means the first data point will be predicted only based on the prior). We define $i^\star = L$ as the current point of refit. Next, starting with $i = i^\star + 1$, we approximate each $p(y_{i+1:M} \,|\, y_{1:i})$ via $$ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{ \sum_{s=1}^S w_i^{(s)}\, p(y_{i+1:M} \,|\, y_{1:i}, \theta^{(s)})} { \sum_{s=1}^S w_i^{(s)}}, $$ where $\theta^{(s)} = \theta^{(s)}_{1:i^\star}$ are draws from the posterior distribution based on the first $i^\star$ observations and $w_i^{(s)}$ are the PSIS weights obtained in two steps. First, we compute the raw importance ratios $$ r_i^{(s)} = \frac{f_{1:i}(\theta^{(s)})}{f_{1:i^\star}(\theta^{(s)})} \propto \prod_{j \in (i^\star + 1):i} p(y_j \,|\, y_{1:(j-1)}, \theta^{(s)}), $$ and then stabilize them using PSIS. The function $f_{1:i}$ denotes the posterior distribution based on the first $i$ observations, that is, $f_{1:i} = p(\theta \,|\, y_{1:i})$, with $f_{1:i^\star}$ defined analogously. The index set $(i^\star + 1):i$ indicates all observations which are part of the data for the model $f_{1:i}$ whose predictive performance we are trying to approximate but not for the actually fitted model $f_{1:i^\star}$. The proportional statement arises from the fact that we ignore the normalizing constants $p(y_{1:i})$ and $p(y_{1:i^\star})$ of the compared posteriors, which leads to a self-normalized variant of PSIS (see Vehtari et al, 2017). Continuing with the next observation, we gradually increase $i$ by $1$ (we move forward in time) and repeat the process. At some observation $i$, the variability of the importance ratios $r_i^{(s)}$ will become too large and importance sampling will fail. We will refer to this particular value of $i$ as $i^\star_1$. To identify the value of $i^\star_1$, we check for which value of $i$ does the estimated shape parameter $k$ of the generalized Pareto distribution first cross a certain threshold $\tau$ (Vehtari et al, 2019). Only then do we refit the model using the observations up to $i^\star_1$ and restart the process from there by setting $\theta^{(s)} = \theta^{(s)}_{1:i^\star_1}$ and $i^\star = i^\star_1$ until the next refit. In some cases we may only need to refit once and in other cases we will find a value $i^\star_2$ that requires a second refitting, maybe an $i^\star_3$ that requires a third refitting, and so on. We refit as many times as is required (only when $k > \tau$) until we arrive at observation $i = N - M$. For LOO, we recommend to use a threshold of $\tau = 0.7$ (Vehtari et al, 2017, 2019) and it turns out this is a reasonable threshold for LFO as well (Bürkner et al. 2020). ## Autoregressive models Autoregressive (AR) models are some of the most commonly used time-series models. An AR(p) model ---an autoregressive model of order $p$--- can be defined as $$ y_i = \eta_i + \sum_{k = 1}^p \varphi_k y_{i - k} + \varepsilon_i, $$ where $\eta_i$ is the linear predictor for the $i$th observation, $\phi_k$ are the autoregressive parameters and $\varepsilon_i$ are pairwise independent errors, which are usually assumed to be normally distributed with equal variance $\sigma^2$. The model implies a recursive formula that allows for computing the right-hand side of the above equation for observation $i$ based on the values of the equations for previous observations. ## Case Study: Annual measurements of the level of Lake Huron To illustrate the application of PSIS-LFO-CV for estimating expected $M$-SAP performance, we will fit a model for 98 annual measurements of the water level (in feet) of [Lake Huron](https://en.wikipedia.org/wiki/Lake_Huron) from the years 1875--1972. This data set is found in the **datasets** R package, which is installed automatically with **R**. In addition to the **loo** package, for this analysis we will use the **brms** interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting. ```{r pkgs, cache=FALSE} library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) CHAINS <- 4 SEED <- 5838296 set.seed(SEED) ``` Before fitting a model, we will first put the data into a data frame and then look at the time series. ```{r hurondata} N <- length(LakeHuron) df <- data.frame( y = as.numeric(LakeHuron), year = as.numeric(time(LakeHuron)), time = 1:N ) ggplot(df, aes(x = year, y = y)) + geom_point(size = 1) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)" ) ``` The above plot shows rather strong autocorrelation of the time-series as well as some trend towards lower levels for later points in time. We can specify an AR(4) model for these data using the **brms** package as follows: ```{r fit, results = "hide"} fit <- brm( y ~ ar(time, p = 4), data = df, prior = prior(normal(0, 0.5), class = "ar"), control = list(adapt_delta = 0.99), seed = SEED, chains = CHAINS ) ``` The model implied predictions along with the observed values can be plotted, which reveals a rather good fit to the data. ```{r plotpreds, cache = FALSE} preds <- posterior_predict(fit) preds <- cbind( Estimate = colMeans(preds), Q5 = apply(preds, 2, quantile, probs = 0.05), Q95 = apply(preds, 2, quantile, probs = 0.95) ) ggplot(cbind(df, preds), aes(x = year, y = Estimate)) + geom_smooth(aes(ymin = Q5, ymax = Q95), stat = "identity", size = 0.5) + geom_point(aes(y = y)) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)", subtitle = "Mean (blue) and 90% predictive intervals (gray) vs. observed data (black)" ) ``` To allow for reasonable predictions of future values, we will require at least $L = 20$ historical observations (20 years) to make predictions. ```{r setL} L <- 20 ``` We first perform approximate leave-one-out cross-validation (LOO-CV) for the purpose of later comparison with exact and approximate LFO-CV for the 1-SAP case. ```{r loo1sap, cache = FALSE} loo_cv <- loo(log_lik(fit)[, (L + 1):N]) print(loo_cv) ``` ## 1-step-ahead predictions leaving out all future values The most basic version of $M$-SAP is 1-SAP, in which we predict only one step ahead. In this case, $y_{i+1:M}$ simplifies to $y_{i}$ and the LFO-CV algorithm becomes considerably simpler than for larger values of $M$. ### Exact 1-step-ahead predictions Before we compute approximate LFO-CV using PSIS we will first compute exact LFO-CV for the 1-SAP case so we can use it as a benchmark later. The initial step for the exact computation is to calculate the log-predictive densities by refitting the model many times: ```{r exact_loglik, results="hide"} loglik_exact <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_i <- update(fit, newdata = df_past, recompile = FALSE) loglik_exact[, i + 1] <- log_lik(fit_i, newdata = df_oos, oos = oos)[, oos] } ``` Then we compute the exact expected log predictive density (ELPD): ```{r helpers} # some helper functions we'll use throughout # more stable than log(sum(exp(x))) log_sum_exp <- function(x) { max_x <- max(x) max_x + log(sum(exp(x - max_x))) } # more stable than log(mean(exp(x))) log_mean_exp <- function(x) { log_sum_exp(x) - log(length(x)) } # compute log of raw importance ratios # sums over observations *not* over posterior samples sum_log_ratios <- function(loglik, ids = NULL) { if (!is.null(ids)) loglik <- loglik[, ids, drop = FALSE] rowSums(loglik) } # for printing comparisons later rbind_print <- function(...) { round(rbind(...), digits = 2) } ``` ```{r exact1sap, cache = FALSE} exact_elpds_1sap <- apply(loglik_exact, 2, log_mean_exp) exact_elpd_1sap <- c(ELPD = sum(exact_elpds_1sap[-(1:L)])) rbind_print( "LOO" = loo_cv$estimates["elpd_loo", "Estimate"], "LFO" = exact_elpd_1sap ) ``` We see that the ELPD from LFO-CV for 1-step-ahead predictions is lower than the ELPD estimate from LOO-CV, which should be expected since LOO-CV is making use of more of the time series. That is, since the LFO-CV approach only uses observations from before the left-out data point but LOO-CV uses _all_ data points other than the left-out observation, we should expect to see the larger ELPD from LOO-CV. ### Approximate 1-step-ahead predictions We compute approximate 1-SAP with refit at observations where the Pareto $k$ estimate exceeds the threshold of $0.7$. ```{r setkthresh} k_thres <- 0.7 ``` The code becomes a little bit more involved as compared to the exact LFO-CV. Note that we can compute exact 1-SAP at the refitting points, which comes with no additional computational costs since we had to refit the model anyway. ```{r refit_loglik, results="hide"} approx_elpds_1sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- L + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[L + 1] <- log_mean_exp(loglik[, oos]) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[i + 1] <- log_mean_exp(loglik[, oos]) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] approx_elpds_1sap[i + 1] <- log_sum_exp(lw + loglik[, oos]) } } ``` We see that the final Pareto-$k$-estimates are mostly well below the threshold and that we only needed to refit the model a few times: ```{r plot_ks} plot_ks <- function(ks, ids, thres = 0.6) { dat_ks <- data.frame(ks = ks, ids = ids) ggplot(dat_ks, aes(x = ids, y = ks)) + geom_point(aes(color = ks > thres), shape = 3, show.legend = FALSE) + geom_hline(yintercept = thres, linetype = 2, color = "red2") + scale_color_manual(values = c("cornflowerblue", "darkblue")) + labs(x = "Data point", y = "Pareto k") + ylim(-0.5, 1.5) } ``` ```{r refitsummary1sap, cache=FALSE} cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - 1)) ``` The approximate 1-SAP ELPD is remarkably similar to the exact 1-SAP ELPD computed above, which indicates our algorithm to compute approximate 1-SAP worked well for the present data and model. ```{r lfosummary1sap, cache = FALSE} approx_elpd_1sap <- sum(approx_elpds_1sap, na.rm = TRUE) rbind_print( "approx LFO" = approx_elpd_1sap, "exact LFO" = exact_elpd_1sap ) ``` Plotting exact against approximate predictions, we see that no approximation value deviates far from its exact counterpart, providing further evidence for the good quality of our approximation. ```{r plot1sap, cache = FALSE} dat_elpd <- data.frame( approx_elpd = approx_elpds_1sap, exact_elpd = exact_elpds_1sap ) ggplot(dat_elpd, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ``` We can also look at the maximum difference and average difference between the approximate and exact ELPD calculations, which also indicate a ver close approximation: ```{r diffs1sap, cache=FALSE} max_diff <- with(dat_elpd, max(abs(approx_elpd - exact_elpd), na.rm = TRUE)) mean_diff <- with(dat_elpd, mean(abs(approx_elpd - exact_elpd), na.rm = TRUE)) rbind_print( "Max diff" = round(max_diff, 2), "Mean diff" = round(mean_diff, 3) ) ``` ## $M$-step-ahead predictions leaving out all future values To illustrate the application of $M$-SAP for $M > 1$, we next compute exact and approximate LFO-CV for the 4-SAP case. ### Exact $M$-step-ahead predictions The necessary steps are the same as for 1-SAP with the exception that the log-density values of interest are now the sums of the log predictive densities of four consecutive observations. Further, the stability of the PSIS approximation actually stays the same for all $M$ as it only depends on the number of observations we leave out, not on the number of observations we predict. ```{r exact_loglikm, results="hide"} M <- 4 loglikm <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm[, i + 1] <- rowSums(loglik[, oos]) } ``` ```{r exact4sap, cache = FALSE} exact_elpds_4sap <- apply(loglikm, 2, log_mean_exp) (exact_elpd_4sap <- c(ELPD = sum(exact_elpds_4sap, na.rm = TRUE))) ``` ### Approximate $M$-step-ahead predictions Computing the approximate PSIS-LFO-CV for the 4-SAP case is a little bit more involved than the approximate version for the 1-SAP case, although the underlying principles remain the same. ```{r refit_loglikm, results="hide"} approx_elpds_4sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- (L + 1):(L + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_1sap[L + 1] <- log_mean_exp(loglikm) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_mean_exp(loglikm) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_sum_exp(lw + loglikm) } } ``` Again, we see that the final Pareto-$k$-estimates are mostly well below the threshold and that we only needed to refit the model a few times: ```{r refitsummary4sap, cache = FALSE} cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - M)) ``` The approximate ELPD computed for the 4-SAP case is not as close to its exact counterpart as in the 1-SAP case. In general, the larger $M$, the larger the variation of the approximate ELPD around the exact ELPD. It turns out that the ELPD estimates of AR-models with $M>1$ show particular variation due to their predictions' dependency on other predicted values. In Bürkner et al. (2020) we provide further explanation and simulations for these cases. ```{r lfosummary4sap, cache = FALSE} approx_elpd_4sap <- sum(approx_elpds_4sap, na.rm = TRUE) rbind_print( "Approx LFO" = approx_elpd_4sap, "Exact LFO" = exact_elpd_4sap ) ``` Plotting exact against approximate pointwise predictions confirms that, for a few specific data points, the approximate predictions underestimate the exact predictions. ```{r plot4sap, cache = FALSE} dat_elpd_4sap <- data.frame( approx_elpd = approx_elpds_4sap, exact_elpd = exact_elpds_4sap ) ggplot(dat_elpd_4sap, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ``` ## Conclusion In this case study we have shown how to do carry out exact and approximate leave-future-out cross-validation for $M$-step-ahead prediction tasks. For the data and model used in our example, the PSIS-LFO-CV algorithm provides reasonably stable and accurate results despite not requiring us to refit the model nearly as many times. For more details on approximate LFO-CV, we refer to Bürkner et al. (2020).
## References Bürkner P. C., Gabry J., & Vehtari A. (2020). Approximate leave-future-out cross-validation for time series models. *Journal of Statistical Computation and Simulation*, 90(14):2499-2523. \doi:/10.1080/00949655.2020.1783262. [Online](https://www.tandfonline.com/doi/full/10.1080/00949655.2020.1783262). [arXiv preprint](https://arxiv.org/abs/1902.06281). Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646).
## Appendix ### Appendix: Session information ```{r sessioninfo} sessionInfo() ``` ### Appendix: Licenses * Code © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under BSD-3). * Text © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under CC-BY-NC 4.0). loo/vignettes/loo2-with-rstan.Rmd0000644000176200001440000002030714407123455016477 0ustar liggesusers--- title: "Writing Stan programs for use with the loo package" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to write a Stan program that computes and stores the pointwise log-likelihood required for using the __loo__ package. The other vignettes included with the package demonstrate additional functionality. Some sections from this vignette are excerpted from our papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.02646). which provide important background for understanding the methods implemented in the package. # Example: Well water in Bangladesh This example comes from a survey of residents from a small area in Bangladesh that was affected by arsenic in drinking water. Respondents with elevated arsenic levels in their wells were asked if they were interested in getting water from a neighbor's well, and a series of logistic regressions were fit to predict this binary response given various information about the households (Gelman and Hill, 2007). Here we fit a model for the well-switching response given two predictors: the arsenic level of the water in the resident's home, and the distance of the house from the nearest safe well. The sample size in this example is $N=3020$, which is not huge but is large enough that it is important to have a computational method for LOO that is fast for each data point. On the plus side, with such a large dataset, the influence of any given observation is small, and so the computations should be stable. ## Coding the Stan model Here is the Stan code for fitting the logistic regression model, which we save in a file called `logistic.stan`: ``` data { int N; // number of data points int P; // number of predictors (including intercept) matrix[N,P] X; // predictors (including 1s for intercept) int y[N]; // binary outcome } parameters { vector[P] beta; } model { beta ~ normal(0, 1); y ~ bernoulli_logit(X * beta); } generated quantities { vector[N] log_lik; for (n in 1:N) { log_lik[n] = bernoulli_logit_lpmf(y[n] | X[n] * beta); } } ``` We have defined the log likelihood as a vector named `log_lik` in the generated quantities block so that the individual terms will be saved by Stan. After running Stan, `log_lik` can be extracted (using the `extract_log_lik` function provided in the **loo** package) as an $S \times N$ matrix, where $S$ is the number of simulations (posterior draws) and $N$ is the number of data points. ## Fitting the model with RStan Next we fit the model in Stan using the **rstan** package: ```{r, eval=FALSE} library("rstan") # Prepare data url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" wells <- read.table(url) wells$dist100 <- with(wells, dist / 100) X <- model.matrix(~ dist100 + arsenic, wells) standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # Fit model fit_1 <- stan("logistic.stan", data = standata) print(fit_1, pars = "beta") ``` ``` mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat beta[1] 0.00 0 0.08 -0.16 -0.05 0.00 0.05 0.15 1964 1 beta[2] -0.89 0 0.10 -1.09 -0.96 -0.89 -0.82 -0.68 2048 1 beta[3] 0.46 0 0.04 0.38 0.43 0.46 0.49 0.54 2198 1 ``` ## Computing approximate leave-one-out cross-validation using PSIS-LOO We can then use the **loo** package to compute the efficient PSIS-LOO approximation to exact LOO-CV: ```{r, eval=FALSE} library("loo") # Extract pointwise log-likelihood # using merge_chains=FALSE returns an array, which is easier to # use with relative_eff() log_lik_1 <- extract_log_lik(fit_1, merge_chains = FALSE) # as of loo v2.0.0 we can optionally provide relative effective sample sizes # when calling loo, which allows for better estimates of the PSIS effective # sample sizes and Monte Carlo error r_eff <- relative_eff(exp(log_lik_1), cores = 2) # preferably use more than 2 cores (as many cores as possible) # will use value of 'mc.cores' option if cores is not specified loo_1 <- loo(log_lik_1, r_eff = r_eff, cores = 2) print(loo_1) ``` ``` Computed from 4000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1968.5 15.6 p_loo 3.2 0.1 looic 3937.0 31.2 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` The printed output from the `loo` function shows the estimates $\widehat{\mbox{elpd}}_{\rm loo}$ (expected log predictive density), $\widehat{p}_{\rm loo}$ (effective number of parameters), and ${\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}$ (the LOO information criterion). The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the $k$ parameter is explained in `help('pareto-k-diagnostic')` and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case the message tells us that all of the estimates for $k$ are fine. ## Comparing models To compare this model to an alternative model for the same data we can use the `loo_compare` function in the **loo** package. First we'll fit a second model to the well-switching data, using `log(arsenic)` instead of `arsenic` as a predictor: ```{r, eval=FALSE} standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) fit_2 <- stan(fit = fit_1, data = standata) log_lik_2 <- extract_log_lik(fit_2, merge_chains = FALSE) r_eff_2 <- relative_eff(exp(log_lik_2)) loo_2 <- loo(log_lik_2, r_eff = r_eff_2, cores = 2) print(loo_2) ``` ``` Computed from 4000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1952.3 16.2 p_loo 3.1 0.1 looic 3904.6 32.4 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` We can now compare the models on LOO using the `loo_compare` function: ```{r, eval=FALSE} # Compare comp <- loo_compare(loo_1, loo_2) ``` This new object, `comp`, contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error: ```{r, eval=FALSE} print(comp) # can set simplify=FALSE for more detailed print output ``` ``` elpd_diff se_diff model2 0.0 0.0 model1 -16.3 4.4 ``` The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in `elpd` and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (`model2`). # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2017). _The Stan C++ Library, Version 2.17.0._ https://mc-stan.org/ Stan Development Team (2018) _RStan: the R interface to Stan, Version 2.17.3._ https://mc-stan.org/ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/vignettes/loo2-example.Rmd0000644000176200001440000003015314407123455016032 0ustar liggesusers--- title: "Using the loo package (version >= 2.0.0)" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to use the __loo__ package to carry out Pareto smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) for purposes of model checking and model comparison. In this vignette we can't provide all necessary background information on PSIS-LOO and its diagnostics (Pareto $k$ and effective sample size), so we encourage readers to refer to the following papers for more details: * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). # Setup In addition to the __loo__ package, we'll also be using __rstanarm__ and __bayesplot__: ```{r setup, message=FALSE} library("rstanarm") library("bayesplot") library("loo") ``` # Example: Poisson vs negative binomial for the roaches dataset ## Background and model fitting The Poisson and negative binomial regression models used below in our example, as well as the `stan_glm` function used to fit the models, are covered in more depth in the __rstanarm__ vignette [_Estimating Generalized Linear Models for Count Data with rstanarm_](http://mc-stan.org/rstanarm/articles/count.html). In the rest of this vignette we will assume the reader is already familiar with these kinds of models. ### Roaches data The example data we'll use comes from Chapter 8.3 of [Gelman and Hill (2007)](http://www.stat.columbia.edu/~gelman/arm/). We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. Here is how Gelman and Hill describe the experiment and data (pg. 161): > the treatment and control were applied to 160 and 104 apartments, respectively, and the outcome measurement $y_i$ in each apartment $i$ was the number of roaches caught in a set of traps. Different apartments had traps for different numbers of days In addition to an intercept, the regression predictors for the model are `roach1`, the pre-treatment number of roaches (rescaled above to be in units of hundreds), the treatment indicator `treatment`, and a variable indicating whether the apartment is in a building restricted to elderly residents `senior`. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we use the `offset` argument to specify that `log(exposure2)` should be added to the linear predictor. ```{r data} # the 'roaches' data frame is included with the rstanarm package data(roaches) str(roaches) # rescale to units of hundreds of roaches roaches$roach1 <- roaches$roach1 / 100 ``` ### Fit Poisson model We'll fit a simple Poisson regression model using the `stan_glm` function from the __rstanarm__ package. ```{r count-roaches-mcmc, results="hide"} fit1 <- stan_glm( formula = y ~ roach1 + treatment + senior, offset = log(exposure2), data = roaches, family = poisson(link = "log"), prior = normal(0, 2.5, autoscale = TRUE), prior_intercept = normal(0, 5, autoscale = TRUE), seed = 12345 ) ``` Usually we would also run posterior predictive checks as shown in the __rstanarm__ vignette [Estimating Generalized Linear Models for Count Data with rstanarm](http://mc-stan.org/rstanarm/articles/count.html), but here we focus only on methods provided by the __loo__ package.
## Using the __loo__ package for model checking and comparison _Although cross-validation is mostly used for model comparison, it is also useful for model checking._ ### Computing PSIS-LOO and checking diagnostics We start by computing PSIS-LOO with the `loo` function. Since we fit our model using __rstanarm__ we can use the `loo` method for `stanreg` objects (fitted model objects from __rstanarm__), which doesn't require us to first extract the pointwise log-likelihood values. If we had written our own Stan program instead of using __rstanarm__ we would pass an array or matrix of log-likelihood values to the `loo` function (see, e.g. `help("loo.array", package = "loo")`). We'll also use the argument `save_psis = TRUE` to save some intermediate results to be re-used later. ```{r loo1} loo1 <- loo(fit1, save_psis = TRUE) ``` `loo` gives us warnings about the Pareto diagnostics, which indicate that for some observations the leave-one-out posteriors are different enough from the full posterior that importance-sampling is not able to correct the difference. We can see more details by printing the `loo` object. ```{r print-loo1} print(loo1) ``` The table shows us a summary of Pareto $k$ diagnostic, which is used to assess the reliability of the estimates. In addition to the proportion of leave-one-out folds with $k$ values in different intervals, the minimum of the effective sample sizes in that category is shown to give idea why higher $k$ values are bad. Since we have some $k>1$, we are not able to compute an estimate for the Monte Carlo standard error (SE) of the expected log predictive density (`elpd_loo`) and `NA` is displayed. (Full details on the interpretation of the Pareto $k$ diagnostics are available in the Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) papers referenced at the top of this vignette.) In this case the `elpd_loo` estimate should not be considered reliable. If we had a well-specified model we would expect the estimated effective number of parameters (`p_loo`) to be smaller than or similar to the total number of parameters in the model. Here `p_loo` is almost 300, which is about 70 times the total number of parameters in the model, indicating severe model misspecification. ### Plotting Pareto $k$ diagnostics Using the `plot` method on our `loo1` object produces a plot of the $k$ values (in the same order as the observations in the dataset used to fit the model) with horizontal lines corresponding to the same categories as in the printed output above. ```{r plot-loo1, out.width = "70%"} plot(loo1) ``` This plot is useful to quickly see the distribution of $k$ values, but it's often also possible to see structure with respect to data ordering. In our case this is mild, but there seems to be a block of data that is somewhat easier to predict (indices around 90--150). Unfortunately even for these data points we see some high $k$ values. ### Marginal posterior predictive checks The `loo` package can be used in combination with the `bayesplot` package for leave-one-out cross-validation marginal posterior predictive checks [Gabry et al (2018)](https://arxiv.org/abs/1709.01449). LOO-PIT values are cumulative probabilities for $y_i$ computed using the LOO marginal predictive distributions $p(y_i|y_{-i})$. For a good model, the distribution of LOO-PIT values should be uniform. In the following plot the distribution (smoothed density estimate) of the LOO-PIT values for our model (thick curve) is compared to many independently generated samples (each the same size as our dataset) from the standard uniform distribution (thin curves). ```{r ppc_loo_pit_overlay} yrep <- posterior_predict(fit1) ppc_loo_pit_overlay( y = roaches$y, yrep = yrep, lw = weights(loo1$psis_object) ) ``` The excessive number of values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion. ## Try alternative model with more flexibility Here we will try [negative binomial](https://en.wikipedia.org/wiki/Negative_binomial_distribution) regression, which is commonly used for overdispersed count data. Unlike the Poisson distribution, the negative binomial distribution allows the conditional mean and variance of $y$ to differ. ```{r count-roaches-negbin, results="hide"} fit2 <- update(fit1, family = neg_binomial_2) ``` ```{r loo2} loo2 <- loo(fit2, save_psis = TRUE, cores = 2) print(loo2) ``` ```{r plot-loo2} plot(loo2, label_points = TRUE) ``` Using the `label_points` argument will label any $k$ values larger than 0.7 with the index of the corresponding data point. These high values are often the result of model misspecification and frequently correspond to data points that would be considered ``outliers'' in the data and surprising according to the model [Gabry et al (2019)](https://arxiv.org/abs/1709.01449). Unfortunately, while large $k$ values are a useful indicator of model misspecification, small $k$ values are not a guarantee that a model is well-specified. If there are a small number of problematic $k$ values then we can use a feature in __rstanarm__ that lets us refit the model once for each of these problematic observations. Each time the model is refit, one of the observations with a high $k$ value is omitted and the LOO calculations are performed exactly for that observation. The results are then recombined with the approximate LOO calculations already carried out for the observations without problematic $k$ values: ```{r reloo} if (any(pareto_k_values(loo2) > 0.7)) { loo2 <- loo(fit2, save_psis = TRUE, k_threshold = 0.7) } print(loo2) ``` In the print output we can see that the Monte Carlo SE is small compared to the other uncertainties. On the other hand, `p_loo` is about 7 and still a bit higher than the total number of parameters in the model. This indicates that there is almost certainly still some degree of model misspecification, but this is much better than the `p_loo` estimate for the Poisson model. For further model checking we again examine the LOO-PIT values. ```{r ppc_loo_pit_overlay-negbin} yrep <- posterior_predict(fit2) ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2$psis_object)) ``` The plot for the negative binomial model looks better than the Poisson plot, but we still see that this model is not capturing all of the essential features in the data. ## Comparing the models on expected log predictive density We can use the `loo_compare` function to compare our two models on expected log predictive density (ELPD) for new data: ```{r loo_compare} loo_compare(loo1, loo2) ``` The difference in ELPD is much larger than several times the estimated standard error of the difference again indicating that the negative-binomial model is expected to have better predictive performance than the Poisson model. However, according to the LOO-PIT checks there is still some misspecification, and a reasonable guess is that a hurdle or zero-inflated model would be an improvement (we leave that for another case study).
# References Gabry, J., Simpson, D., Vehtari, A., Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. _J. R. Stat. Soc. A_, 182: 389-402. \doi:10.1111/rssa.12378. ([journal version](https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssa.12378), [arXiv preprint](https://arxiv.org/abs/1709.01449), [code on GitHub](https://github.com/jgabry/bayes-vis-paper)) Gelman, A. and Hill, J. (2007). _Data Analysis Using Regression and Multilevel/Hierarchical Models._ Cambridge University Press, Cambridge, UK. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/vignettes/loo2-moment-matching.Rmd0000644000176200001440000002774214407123455017500 0ustar liggesusers--- title: "Avoiding model refits in leave-one-out cross-validation with moment matching" author: "Topi Paananen, Paul Bürkner, Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to improve the Monte Carlo sampling accuracy of leave-one-out cross-validation with the __loo__ package and Stan. The __loo__ package automatically monitors the sampling accuracy using Pareto $k$ diagnostics for each observation. Here, we present a method for quickly improving the accuracy when the Pareto diagnostics indicate problems. This is done by performing some additional computations using the existing posterior sample. If successful, this will decrease the Pareto $k$ values, making the model assessment more reliable. __loo__ also stores the original Pareto $k$ values with the name `influence_pareto_k` which are not changed. They can be used as a diagnostic of how much each observation influences the posterior distribution. The methodology presented is based on the paper * Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2020). Implicitly Adaptive Importance Sampling. [arXiv preprint arXiv:1906.08850](https://arxiv.org/abs/1906.08850). More information about the Pareto $k$ diagnostics is given in the following papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). # Example: Eradication of Roaches We will use the same example as in the vignette [_Using the loo package (version >= 2.0.0)_](https://mc-stan.org/loo/articles/loo2-example.html). See the demo for a description of the problem and data. We will use the same Poisson regression model as in the case study. ## Coding the Stan model Here is the Stan code for fitting the Poisson regression model, which we will use for modeling the number of roaches. ```{r stancode} stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ``` Following the usual approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. ## Setup In addition to __loo__, we load the __rstan__ package for fitting the model, and the __rstanarm__ package for the data. ```{r setup, message=FALSE} library("rstan") library("loo") seed <- 9547 set.seed(seed) ``` ## Fitting the model with RStan Next we fit the model in Stan using the __rstan__ package: ```{r modelfit, message=FALSE} # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) y <- roaches$y x <- roaches[,c("roach1", "treatment", "senior")] offset <- log(roaches[,"exposure2"]) n <- dim(x)[1] k <- dim(x)[2] standata <- list(N = n, K = k, x = as.matrix(x), y = y, offset = offset, beta_prior_scale = 2.5, alpha_prior_scale = 5.0) # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = standata, seed = seed, refresh = 0) print(fit, pars = "beta") ``` Let us now evaluate the predictive performance of the model using `loo()`. ```{r loo1} loo1 <- loo(fit) loo1 ``` The `loo()` function output warnings that there are some observations which are highly influential, and thus the accuracy of importance sampling is compromised as indicated by the large Pareto $k$ diagnostic values (> 0.7). As discussed in the vignette [_Using the loo package (version >= 2.0.0)_](https://mc-stan.org/loo/articles/loo2-example.html), this may be an indication of model misspecification. Despite that, it is still beneficial to be able to evaluate the predictive performance of the model accurately. ## Moment matching correction for importance sampling To improve the accuracy of the `loo()` result above, we could perform leave-one-out cross-validation by explicitly leaving out single observations and refitting the model using MCMC repeatedly. However, the Pareto $k$ diagnostics indicate that there are 19 observations which are problematic. This would require 19 model refits which may require a lot of computation time. Instead of refitting with MCMC, we can perform a faster moment matching correction to the importance sampling for the problematic observations. This can be done with the `loo_moment_match()` function in the __loo__ package, which takes our existing `loo` object as input and modifies it. The moment matching requires some evaluations of the model posterior density. For models fitted with __rstan__, this can be conveniently done by using the existing `stanfit` object. First, we show how the moment matching can be used for a model fitted using __rstan__. It only requires setting the argument `moment_match` to `TRUE` in the `loo()` function. Optionally, you can also set the argument `k_threshold` which determines the Pareto $k$ threshold, above which moment matching is used. By default, it operates on all observations whose Pareto $k$ value is larger than 0.7. ```{r loo_moment_match} # available in rstan >= 2.21 loo2 <- loo(fit, moment_match = TRUE) loo2 ``` After the moment matching, all observations have the diagnostic Pareto $k$ less than 0.7, meaning that the estimates are now reliable. The total `elpd_loo` estimate also changed from `-5457.8` to `-5478.5`, showing that before moment matching, `loo()` overestimated the predictive performance of the model. The updated Pareto $k$ values stored in `loo2$diagnostics$pareto_k` are considered algorithmic diagnostic values that indicate the sampling accuracy. The original Pareto $k$ values are stored in `loo2$pointwise[,"influence_pareto_k"]` and these are not modified by the moment matching. These can be considered as diagnostics for how big influence each observation has on the posterior distribution. In addition to the Pareto $k$ diagnostics, moment matching also updates the effective sample size estimates. # Using `loo_moment_match()` directly The moment matching can also be performed by explicitly calling the function `loo_moment_match()`. This enables its use also for models that are not using __rstan__ or another package with built-in support for `loo_moment_match()`. To use `loo_moment_match()`, the user must give the model object `x`, the `loo` object, and 5 helper functions as arguments to `loo_moment_match()`. The helper functions are * `post_draws` + A function the takes `x` as the first argument and returns a matrix of posterior draws of the model parameters, `pars`. * `log_lik_i` + A function that takes `x` and `i` and returns a matrix (one column per chain) or a vector (all chains stacked) of log-likeliood draws of the ith observation based on the model `x`. If the draws are obtained using MCMC, the matrix with MCMC chains separated is preferred. * `unconstrain_pars` + A function that takes arguments `x` and `pars`, and returns posterior draws on the unconstrained space based on the posterior draws on the constrained space passed via `pars`. * `log_prob_upars` + A function that takes arguments `x` and `upars`, and returns a matrix of log-posterior density values of the unconstrained posterior draws passed via `upars`. * `log_lik_i_upars` + A function that takes arguments `x`, `upars`, and `i` and returns a vector of log-likelihood draws of the `i`th observation based on the unconstrained posterior draws passed via `upars`. Next, we show how the helper functions look like for RStan objects, and show an example of using `loo_moment_match()` directly. For stanfit objects from __rstan__ objects, the functions look like this: ```{r stanfitfuns} # create a named list of draws for use with rstan methods .rstan_relist <- function(x, skeleton) { out <- utils::relist(x, skeleton) for (i in seq_along(skeleton)) { dim(out[[i]]) <- dim(skeleton[[i]]) } out } # rstan helper function to get dims of parameters right .create_skeleton <- function(pars, dims) { out <- lapply(seq_along(pars), function(i) { len_dims <- length(dims[[i]]) if (len_dims < 1) return(0) return(array(0, dim = dims[[i]])) }) names(out) <- pars out } # extract original posterior draws post_draws_stanfit <- function(x, ...) { as.matrix(x) } # compute a matrix of log-likelihood values for the ith observation # matrix contains information about the number of MCMC chains log_lik_i_stanfit <- function(x, i, parameter_name = "log_lik", ...) { loo::extract_log_lik(x, parameter_name, merge_chains = FALSE)[, , i] } # transform parameters to the unconstraint space unconstrain_pars_stanfit <- function(x, pars, ...) { skeleton <- .create_skeleton(x@sim$pars_oi, x@par_dims[x@sim$pars_oi]) upars <- apply(pars, 1, FUN = function(theta) { rstan::unconstrain_pars(x, .rstan_relist(theta, skeleton)) }) # for one parameter models if (is.null(dim(upars))) { dim(upars) <- c(1, length(upars)) } t(upars) } # compute log_prob for each posterior draws on the unconstrained space log_prob_upars_stanfit <- function(x, upars, ...) { apply(upars, 1, rstan::log_prob, object = x, adjust_transform = TRUE, gradient = FALSE) } # compute log_lik values based on the unconstrained parameters log_lik_i_upars_stanfit <- function(x, upars, i, parameter_name = "log_lik", ...) { S <- nrow(upars) out <- numeric(S) for (s in seq_len(S)) { out[s] <- rstan::constrain_pars(x, upars = upars[s, ])[[parameter_name]][i] } out } ``` Using these function, we can call `loo_moment_match()` to update the existing `loo` object. ```{r loo_moment_match.default, message=FALSE} loo3 <- loo::loo_moment_match.default( x = fit, loo = loo1, post_draws = post_draws_stanfit, log_lik_i = log_lik_i_stanfit, unconstrain_pars = unconstrain_pars_stanfit, log_prob_upars = log_prob_upars_stanfit, log_lik_i_upars = log_lik_i_upars_stanfit ) loo3 ``` As expected, the result is identical to the previous result of `loo2 <- loo(fit, moment_match = TRUE)`. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2020) _RStan: the R interface to Stan, Version 2.21.1_ https://mc-stan.org Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2021). Implicitly adaptive importance sampling. _Statistics and Computing_, 31, 16. \doi:10.1007/s11222-020-09982-2. arXiv preprint arXiv:1906.08850. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/vignettes/loo2-non-factorized.Rmd0000644000176200001440000006631014407123455017325 0ustar liggesusers--- title: "Leave-one-out cross-validation for non-factorized models" author: "Aki Vehtari, Paul Bürkner and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes encoding: "UTF-8" params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r more-knitr-ops, include=FALSE} knitr::opts_chunk$set( cache=TRUE, message=FALSE, warning=FALSE ) ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction When computing ELPD-based LOO-CV for a Bayesian model we need to compute the log leave-one-out predictive densities $\log{p(y_i | y_{-i})}$ for every response value $y_i, \: i = 1, \ldots, N$, where $y_{-i}$ denotes all response values except observation $i$. To obtain $p(y_i | y_{-i})$, we need to have access to the pointwise likelihood $p(y_i\,|\, y_{-i}, \theta)$ and integrate over the model parameters $\theta$: $$ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\, y_{-i}, \theta) \, p(\theta\,|\, y_{-i}) \,d \theta $$ Here, $p(\theta\,|\, y_{-i})$ is the leave-one-out posterior distribution for $\theta$, that is, the posterior distribution for $\theta$ obtained by fitting the model while holding out the $i$th observation (we will later show how refitting the model to data $y_{-i}$ can be avoided). If the observation model is formulated directly as the product of the pointwise observation models, we call it a *factorized* model. In this case, the likelihood is also the product of the pointwise likelihood contributions $p(y_i\,|\, y_{-i}, \theta)$. To better illustrate possible structures of the observation models, we formally divide $\theta$ into two parts, observation-specific latent variables $f = (f_1, \ldots, f_N)$ and hyperparameters $\psi$, so that $p(y_i\,|\, y_{-i}, \theta) = p(y_i\,|\, y_{-i}, f_i, \psi)$. Depending on the model, one of the two parts of $\theta$ may also be empty. In very simple models, such as linear regression models, latent variables are not explicitly presented and response values are conditionally independent given $\psi$, so that $p(y_i\,|\, y_{-i}, f_i, \psi) = p(y_i \,|\, \psi)$. The full likelihood can then be written in the familiar form $$ p(y \,|\, \psi) = \prod_{i=1}^N p(y_i \,|\, \psi), $$ where $y = (y_1, \ldots, y_N)$ denotes the vector of all responses. When the likelihood factorizes this way, the conditional pointwise log-likelihood can be obtained easily by computing $p(y_i\,|\, \psi)$ for each $i$ with computational cost $O(n)$. Yet, there are several reasons why a *non-factorized* observation model may be necessary or preferred. In non-factorized models, the joint likelihood of the response values $p(y \,|\, \theta)$ is not factorized into observation-specific components, but rather given directly as one joint expression. For some models, an analytic factorized formulation is simply not available in which case we speak of a *non-factorizable* model. Even in models whose observation model can be factorized in principle, it may still be preferable to use a non-factorized form for reasons of efficiency and numerical stability (Bürkner et al. 2020). Whether a non-factorized model is used by necessity or for efficiency and stability, it comes at the cost of having no direct access to the leave-one-out predictive densities and thus to the overall leave-one-out predictive accuracy. In theory, we can express the observation-specific likelihoods in terms of the joint likelihood via $$ p(y_i \,|\, y_{i-1}, \theta) = \frac{p(y \,|\, \theta)}{p(y_{-i} \,|\, \theta)} = \frac{p(y \,|\, \theta)}{\int p(y \,|\, \theta) \, d y_i}, $$ but the expression on the right-hand side may not always have an analytical solution. Computing $\log p(y_i \,|\, y_{-i}, \theta)$ for non-factorized models is therefore often impossible, or at least inefficient and numerically unstable. However, there is a large class of multivariate normal and Student-$t$ models for which there are efficient analytical solutions available. More details can be found in our paper about LOO-CV for non-factorized models (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1810.10559). # LOO-CV for multivariate normal models In this vignette, we will focus on non-factorized multivariate normal models. Based on results of Sundararajan and Keerthi (2001), Bürkner et al. (2020) show that, for multivariate normal models with coriance matrix $C$, the LOO predictive mean and standard deviation can be computed as follows: \begin{align} \mu_{\tilde{y},-i} &= y_i-\bar{c}_{ii}^{-1} g_i \nonumber \\ \sigma_{\tilde{y},-i} &= \sqrt{\bar{c}_{ii}^{-1}}, \end{align} where $g_i$ and $\bar{c}_{ii}$ are \begin{align} g_i &= \left[C^{-1} y\right]_i \nonumber \\ \bar{c}_{ii} &= \left[C^{-1}\right]_{ii}. \end{align} Using these results, the log predictive density of the $i$th observation is then computed as $$ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{-i} - \frac{1}{2}\frac{(y_i-\mu_{-i})^2}{\sigma^2_{-i}}. $$ Expressing this same equation in terms of $g_i$ and $\bar{c}_{ii}$, the log predictive density becomes: $$ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) + \frac{1}{2}\log \bar{c}_{ii} - \frac{1}{2}\frac{g_i^2}{\bar{c}_{ii}}. $$ (Note that Vehtari et al. (2016) has a typo in the corresponding Equation 34.) From these equations we can now derive a recipe for obtaining the conditional pointwise log-likelihood for _all_ models that can be expressed conditionally in terms of a multivariate normal with invertible covariance matrix $C$. ## Approximate LOO-CV using integrated importance-sampling The above LOO equations for multivariate normal models are conditional on parameters $\theta$. Therefore, to obtain the leave-one-out predictive density $p(y_i \,|\, y_{-i})$ we need to integrate over $\theta$, $$ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\,y_{-i}, \theta) \, p(\theta\,|\,y_{-i}) \,d\theta. $$ Here, $p(\theta\,|\,y_{-i})$ is the leave-one-out posterior distribution for $\theta$, that is, the posterior distribution for $\theta$ obtained by fitting the model while holding out the $i$th observation. To avoid the cost of sampling from $N$ leave-one-out posteriors, it is possible to take the posterior draws $\theta^{(s)}, \, s=1,\ldots,S$, from the \emph{full} posterior $p(\theta\,|\,y)$, and then approximate the above integral using integrated importance sampling (Vehtari et al., 2016, Section 3.6.1): $$ p(y_i\,|\,y_{-i}) \approx \frac{ \sum_{s=1}^S p(y_i\,|\,y_{-i},\,\theta^{(s)}) \,w_i^{(s)}}{ \sum_{s=1}^S w_i^{(s)}}, $$ where $w_i^{(s)}$ are importance weights. First we compute the raw importance ratios $$ r_i^{(s)} \propto \frac{1}{p(y_i \,|\, y_{-i}, \,\theta^{(s)})}, $$ and then stabilize them using Pareto smoothed importance sampling (PSIS, Vehtari et al, 2019) to obtain the weights $w_i^{(s)}$. The resulting approximation is referred to as PSIS-LOO (Vehtari et al, 2017). ## Exact LOO-CV with re-fitting In order to validate the approximate LOO procedure, and also in order to allow exact computations to be made for a small number of leave-one-out folds for which the Pareto $k$ diagnostic (Vehtari et al, 2019) indicates an unstable approximation, we need to consider how we might to do _exact_ leave-one-out CV for a non-factorized model. In the case of a Gaussian process that has the marginalization property, we could just drop the one row and column of $C$ corresponding to the held out out observation. This does not hold in general for multivariate normal models, however, and to keep the original prior we may need to maintain the full covariance matrix $C$ even when one of the observations is left out. The solution is to model $y_i$ as a missing observation and estimate it along with all of the other model parameters. For a conditional multivariate normal model, $\log p(y_i\,|\,y_{-i})$ can be computed as follows. First, we model $y_i$ as missing and denote the corresponding parameter $y_i^{\mathrm{mis}}$. Then, we define $$ y_{\mathrm{mis}(i)} = (y_1, \ldots, y_{i-1}, y_i^{\mathrm{mis}}, y_{i+1}, \ldots, y_N). $$ to be the same as the full set of observations $y$, except replacing $y_i$ with the parameter $y_i^{\mathrm{mis}}$. Second, we compute the LOO predictive mean and standard deviations as above, but replace $y$ with $y_{\mathrm{mis}(i)}$ in the computation of $\mu_{\tilde{y},-i}$: $$ \mu_{\tilde{y},-i} = y_{{\mathrm{mis}}(i)}-\bar{c}_{ii}^{-1}g_i, $$ where in this case we have $$ g_i = \left[ C^{-1} y_{\mathrm{mis}(i)} \right]_i. $$ The conditional log predictive density is then computed with the above $\mu_{\tilde{y},-i}$ and the left out observation $y_i$: $$ \log p(y_i\,|\,y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{\tilde{y},-i} - \frac{1}{2}\frac{(y_i-\mu_{\tilde{y},-i})^2}{\sigma^2_{\tilde{y},-i}}. $$ Finally, the leave-one-out predictive distribution can then be estimated as $$ p(y_i\,|\,y_{-i}) \approx \sum_{s=1}^S p(y_i\,|\,y_{-i}, \theta_{-i}^{(s)}), $$ where $\theta_{-i}^{(s)}$ are draws from the posterior distribution $p(\theta\,|\,y_{\mathrm{mis}(i)})$. # Lagged SAR models A common non-factorized multivariate normal model is the simultaneously autoregressive (SAR) model, which is frequently used for spatially correlated data. The lagged SAR model is defined as $$ y = \rho Wy + \eta + \epsilon $$ or equivalently $$ (I - \rho W)y = \eta + \epsilon, $$ where $\rho$ is the spatial correlation parameter and $W$ is a user-defined weight matrix. The matrix $W$ has entries $w_{ii} = 0$ along the diagonal and the off-diagonal entries $w_{ij}$ are larger when areas $i$ and $j$ are closer to each other. In a linear model, the predictor term $\eta$ is given by $\eta = X \beta$ with design matrix $X$ and regression coefficients $\beta$. However, since the above equation holds for arbitrary $\eta$, these results are not restricted to linear models. If we have $\epsilon \sim {\mathrm N}(0, \,\sigma^2 I)$, it follows that $$ (I - \rho W)y \sim {\mathrm N}(\eta, \sigma^2 I), $$ which corresponds to the following log PDF coded in **Stan**: ```{r lpdf, eval=FALSE} /** * Normal log-pdf for spatially lagged responses * * @param y Vector of response values. * @param mu Mean parameter vector. * @param sigma Positive scalar residual standard deviation. * @param rho Positive scalar autoregressive parameter. * @param W Spatial weight matrix. * * @return A scalar to be added to the log posterior. */ real normal_lagsar_lpdf(vector y, vector mu, real sigma, real rho, matrix W) { int N = rows(y); real inv_sigma2 = 1 / square(sigma); matrix[N, N] W_tilde = -rho * W; vector[N] half_pred; for (n in 1:N) W_tilde[n,n] += 1; half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - 0.5 * dot_self(half_pred) * inv_sigma2; } ``` For the purpose of computing LOO-CV, it makes sense to rewrite the SAR model in slightly different form. Conditional on $\rho$, $\eta$, and $\sigma$, if we write \begin{align} y-(I-\rho W)^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(I-\rho W)^{-1}(I-\rho W)^{-T}), \end{align} or more compactly, with $\widetilde{W}=(I-\rho W)$, \begin{align} y-\widetilde{W}^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}), \end{align} then this has the same form as the zero mean Gaussian process from above. Accordingly, we can compute the leave-one-out predictive densities with the equations from Sundararajan and Keerthi (2001), replacing $y$ with $(y-\widetilde{W}^{-1}\eta)$ and taking the covariance matrix $C$ to be $\sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}$. ## Case Study: Neighborhood Crime in Columbus, Ohio In order to demonstrate how to carry out the computations implied by these equations, we will first fit a lagged SAR model to data on crime in 49 different neighborhoods of Columbus, Ohio during the year 1980. The data was originally described in Aneslin (1988) and ships with the **spdep** R package. In addition to the **loo** package, for this analysis we will use the **brms** interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting. ```{r setup, cache=FALSE} library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) SEED <- 10001 set.seed(SEED) # only sets seed for R (seed for Stan set later) # loads COL.OLD data frame and COL.nb neighbor list data(oldcol, package = "spdep") ``` The three variables in the data set relevant to this example are: * `CRIME`: the number of residential burglaries and vehicle thefts per thousand households in the neighbood * `HOVAL`: housing value in units of $1000 USD * `INC`: household income in units of $1000 USD ```{r data} str(COL.OLD[, c("CRIME", "HOVAL", "INC")]) ``` We will also use the object `COL.nb`, which is a list containing information about which neighborhoods border each other. From this list we will be able to construct the weight matrix to used to help account for the spatial dependency among the observations. ### Fit lagged SAR model A model predicting `CRIME` from `INC` and `HOVAL`, while accounting for the spatial dependency via an SAR structure, can be specified in **brms** as follows. ```{r fit, results="hide"} fit <- brm( CRIME ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 4, seed = SEED ) ``` The code above fits the model in **Stan** using a log PDF equivalent to the `normal_lagsar_lpdf` function we defined above. In the summary output below we see that both higher income and higher housing value predict lower crime rates in the neighborhood. Moreover, there seems to be substantial spatial correlation between adjacent neighborhoods, as indicated by the posterior distribution of the `lagsar` parameter. ```{r plot-lagsar, message=FALSE} lagsar <- as.matrix(fit, pars = "lagsar") estimates <- quantile(lagsar, probs = c(0.25, 0.5, 0.75)) mcmc_hist(lagsar) + vline_at(estimates, linetype = 2, size = 1) + ggtitle("lagsar: posterior median and 50% central interval") ``` ### Approximate LOO-CV After fitting the model, the next step is to compute the pointwise log-likelihood values needed for approximate LOO-CV. To do this we will use the recipe laid out in the previous sections. ```{r approx} posterior <- as.data.frame(fit) y <- fit$data$CRIME N <- length(y) S <- nrow(posterior) loglik <- yloo <- sdloo <- matrix(nrow = S, ncol = N) for (s in 1:S) { p <- posterior[s, ] eta <- p$b_Intercept + p$b_INC * fit$data$INC + p$b_HOVAL * fit$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y - solve(W_tilde, eta)) cbar <- diag(Cinv) yloo[s, ] <- y - g / cbar sdloo[s, ] <- sqrt(1 / cbar) loglik[s, ] <- dnorm(y, yloo[s, ], sdloo[s, ], log = TRUE) } # use loo for psis smoothing log_ratios <- -loglik psis_result <- psis(log_ratios) ``` The quality of the PSIS-LOO approximation can be investigated graphically by plotting the Pareto-k estimate for each observation. Ideally, they should not exceed $0.5$, but in practice the algorithm turns out to be robust up to values of $0.7$ (Vehtari et al, 2017, 2019). In the plot below, we see that the fourth observation is problematic and so may reduce the accuracy of the LOO-CV approximation. ```{r plot, cache = FALSE} plot(psis_result, label_points = TRUE) ``` We can also check that the conditional leave-one-out predictive distribution equations work correctly, for instance, using the last posterior draw: ```{r checklast, cache = FALSE} yloo_sub <- yloo[S, ] sdloo_sub <- sdloo[S, ] df <- data.frame( y = y, yloo = yloo_sub, ymin = yloo_sub - sdloo_sub * 2, ymax = yloo_sub + sdloo_sub * 2 ) ggplot(data=df, aes(x = y, y = yloo, ymin = ymin, ymax = ymax)) + geom_errorbar( width = 1, color = "skyblue3", position = position_jitter(width = 0.25) ) + geom_abline(color = "gray30", size = 1.2) + geom_point() ``` Finally, we use PSIS-LOO to approximate the expected log predictive density (ELPD) for new data, which we will validate using exact LOO-CV in the upcoming section. ```{r psisloo} (psis_loo <- loo(loglik)) ``` ### Exact LOO-CV Exact LOO-CV for the above example is somewhat more involved, as we need to re-fit the model $N$ times and each time model the held-out data point as a parameter. First, we create an empty dummy model that we will update below as we loop over the observations. ```{r fit_dummy, cache = TRUE} # see help("mi", "brms") for details on the mi() usage fit_dummy <- brm( CRIME | mi() ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 0 ) ``` Next, we fit the model $N$ times, each time leaving out a single observation and then computing the log predictive density for that observation. For obvious reasons, this takes much longer than the approximation we computed above, but it is necessary in order to validate the approximate LOO-CV method. Thanks to the PSIS-LOO approximation, in general doing these slow exact computations can be avoided. ```{r exact-loo-cv, results="hide", message=FALSE, warning=FALSE, cache = TRUE} S <- 500 res <- vector("list", N) loglik <- matrix(nrow = S, ncol = N) for (i in seq_len(N)) { dat_mi <- COL.OLD dat_mi$CRIME[i] <- NA fit_i <- update(fit_dummy, newdata = dat_mi, # just for vignette chains = 1, iter = S * 2) posterior <- as.data.frame(fit_i) yloo <- sdloo <- rep(NA, S) for (s in seq_len(S)) { p <- posterior[s, ] y_miss_i <- y y_miss_i[i] <- p$Ymi eta <- p$b_Intercept + p$b_INC * fit_i$data$INC + p$b_HOVAL * fit_i$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y_miss_i - solve(W_tilde, eta)) cbar <- diag(Cinv); yloo[s] <- y_miss_i[i] - g[i] / cbar[i] sdloo[s] <- sqrt(1 / cbar[i]) loglik[s, i] <- dnorm(y[i], yloo[s], sdloo[s], log = TRUE) } ypred <- rnorm(S, yloo, sdloo) res[[i]] <- data.frame(y = c(posterior$Ymi, ypred)) res[[i]]$type <- rep(c("pp", "loo"), each = S) res[[i]]$obs <- i } res <- do.call(rbind, res) ``` A first step in the validation of the pointwise predictive density is to compare the distribution of the implied response values for the left-out observation to the distribution of the $y_i^{\mathrm{mis}}$ posterior-predictive values estimated as part of the model. If the pointwise predictive density is correct, the two distributions should match very closely (up to sampling error). In the plot below, we overlay these two distributions for the first four observations and see that they match very closely (as is the case for all $49$ observations of in this example). ```{r yplots, cache = FALSE, fig.width=10, out.width="95%", fig.asp = 0.3} res_sub <- res[res$obs %in% 1:4, ] ggplot(res_sub, aes(y, fill = type)) + geom_density(alpha = 0.6) + facet_wrap("obs", scales = "fixed", ncol = 4) ``` In the final step, we compute the ELPD based on the exact LOO-CV and compare it to the approximate PSIS-LOO result computed earlier. ```{r loo_exact, cache=FALSE} log_mean_exp <- function(x) { # more stable than log(mean(exp(x))) max_x <- max(x) max_x + log(sum(exp(x - max_x))) - log(length(x)) } exact_elpds <- apply(loglik, 2, log_mean_exp) exact_elpd <- sum(exact_elpds) round(exact_elpd, 1) ``` The results of the approximate and exact LOO-CV are similar but not as close as we would expect if there were no problematic observations. We can investigate this issue more closely by plotting the approximate against the exact pointwise ELPD values. ```{r compare, fig.height=5} df <- data.frame( approx_elpd = psis_loo$pointwise[, "elpd_loo"], exact_elpd = exact_elpds ) ggplot(df, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + geom_point(data = df[4, ], size = 3, color = "red3") + xlab("Approximate elpds") + ylab("Exact elpds") + coord_fixed(xlim = c(-16, -3), ylim = c(-16, -3)) ``` In the plot above the fourth data point ---the observation flagged as problematic by the PSIS-LOO approximation--- is colored in red and is the clear outlier. Otherwise, the correspondence between the exact and approximate values is strong. In fact, summing over the pointwise ELPD values and leaving out the fourth observation yields practically equivalent results for approximate and exact LOO-CV: ````{r pt4} without_pt_4 <- c( approx = sum(psis_loo$pointwise[-4, "elpd_loo"]), exact = sum(exact_elpds[-4]) ) round(without_pt_4, 1) ``` From this we can conclude that the difference we found when including *all* observations does not indicate a bug in our implementation of the approximate LOO-CV but rather a violation of its assumptions. # Working with Stan directly So far, we have specified the models in brms and only used Stan implicitely behind the scenes. This allowed us to focus on the primary purpose of validating approximate LOO-CV for non-factorized models. However, we would also like to show how everything can be set up in Stan directly. The Stan code brms generates is human readable and so we can use it to learn some of the essential aspects of Stan and the particular model we are implementing. The Stan program below is a slightly modified version of the code extracted via `stancode(fit_dummy)`: ```{r brms-stan-code, eval=FALSE} // generated with brms 2.2.0 functions { /** * Normal log-pdf for spatially lagged responses * * @param y Vector of response values. * @param mu Mean parameter vector. * @param sigma Positive scalar residual standard deviation. * @param rho Positive scalar autoregressive parameter. * @param W Spatial weight matrix. * * @return A scalar to be added to the log posterior. */ real normal_lagsar_lpdf(vector y, vector mu, real sigma, real rho, matrix W) { int N = rows(y); real inv_sigma2 = 1 / square(sigma); matrix[N, N] W_tilde = -rho * W; vector[N] half_pred; for (n in 1:N) W_tilde[n, n] += 1; half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - 0.5 * dot_self(half_pred) * inv_sigma2; } } data { int N; // total number of observations vector[N] Y; // response variable int Nmi; // number of missings int Jmi[Nmi]; // positions of missings int K; // number of population-level effects matrix[N, K] X; // population-level design matrix matrix[N, N] W; // spatial weight matrix int prior_only; // should the likelihood be ignored? } transformed data { int Kc = K - 1; matrix[N, K - 1] Xc; // centered version of X vector[K - 1] means_X; // column means of X before centering for (i in 2:K) { means_X[i - 1] = mean(X[, i]); Xc[, i - 1] = X[, i] - means_X[i - 1]; } } parameters { vector[Nmi] Ymi; // estimated missings vector[Kc] b; // population-level effects real temp_Intercept; // temporary intercept real sigma; // residual SD real lagsar; // SAR parameter } transformed parameters { } model { vector[N] Yl = Y; vector[N] mu = Xc * b + temp_Intercept; Yl[Jmi] = Ymi; // priors including all constants target += student_t_lpdf(temp_Intercept | 3, 34, 17); target += student_t_lpdf(sigma | 3, 0, 17) - 1 * student_t_lccdf(0 | 3, 0, 17); // likelihood including all constants if (!prior_only) { target += normal_lagsar_lpdf(Yl | mu, sigma, lagsar, W); } } generated quantities { // actual population-level intercept real b_Intercept = temp_Intercept - dot_product(means_X, b); } ``` Here we want to focus on two aspects of the Stan code. First, because there is no built-in function in Stan that calculates the log-likelihood for the lag-SAR model, we define a new `normal_lagsar_lpdf` function in the `functions` block of the Stan program. This is the same function we showed earlier in the vignette and it can be used to compute the log-likelihood in an efficient and numerically stable way. The `_lpdf` suffix used in the function name informs Stan that this is a log probability density function. Second, this Stan program nicely illustrates how to set up missing value imputation. Instead of just computing the log-likelihood for the observed responses `Y`, we define a new variable `Yl` which is equal to `Y` if the reponse is observed and equal to `Ymi` if the response is missing. The latter is in turn defined as a parameter and thus estimated along with all other paramters of the model. More details about missing value imputation in Stan can be found in the *Missing Data & Partially Known Parameters* section of the [Stan manual](https://mc-stan.org/users/documentation/index.html). The Stan code extracted from brms is not only helpful when learning Stan, but can also drastically speed up the specification of models that are not support by brms. If brms can fit a model similar but not identical to the desired model, we can let brms generate the Stan program for the similar model and then mold it into the program that implements the model we actually want to fit. Rather than calling `stancode()`, which requires an existing fitted model object, we recommend using `make_stancode()` and specifying the `save_model` argument to write the Stan program to a file. The corresponding data can be prepared with `make_standata()` and then manually amended if needed. Once the code and data have been edited, they can be passed to RStan's `stan()` function via the `file` and `data` arguments. # Conclusion In summary, we have shown how to set up and validate approximate and exact LOO-CV for non-factorized multivariate normal models using Stan with the **brms** and **loo** packages. Although we focused on the particular example of a spatial SAR model, the presented recipe applies more generally to models that can be expressed in terms of a multivariate normal likelihood.
# References Anselin L. (1988). *Spatial econometrics: methods and models*. Dordrecht: Kluwer Academic. Bürkner P. C., Gabry J., & Vehtari A. (2020). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-t models. *Computational Statistics*, \doi:10.1007/s00180-020-01045-4. [ArXiv preprint](https://arxiv.org/abs/1810.10559). Sundararajan S. & Keerthi S. S. (2001). Predictive approaches for choosing hyperparameters in Gaussian processes. *Neural Computation*, 13(5), 1103--1118. Vehtari A., Mononen T., Tolvanen V., Sivula T., & Winther O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. *Journal of Machine Learning Research*, 17(103), 1--38. [Online](https://jmlr.org/papers/v17/14-540.html). Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/vignettes/loo2-mixis.Rmd0000644000176200001440000002204314407123455015527 0ustar liggesusers--- title: "Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models" author: "Luca Silva and Giacomo Zanella" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette shows how to perform Bayesian leave-one-out cross-validation (LOO-CV) using the mixture estimators proposed in the paper [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-$k$ diagnostics and potential instability of PSIS-LOO estimators. For this illustration we consider a high-dimensional Bayesian Logistic regression model applied to the _Voice_ dataset. ## Setup: load packages and set seed ```{r, warnings=FALSE, message=FALSE} library("rstan") library("loo") library("matrixStats") options(mc.cores = parallel::detectCores(), parallel=FALSE) set.seed(24877) ``` ## Model This is the Stan code for a logistic regression model with regularized horseshoe prior. The code includes an if statement to include a code line needed later for the MixIS approach. ```{r stancode_horseshoe} stancode_horseshoe <- " data { int n; int p; int y[n]; matrix [n,p] X; real scale_global; int mixis; } transformed data { real nu_global=1; // degrees of freedom for the half-t priors for tau real nu_local=1; // degrees of freedom for the half-t priors for lambdas // (nu_local = 1 corresponds to the horseshoe) real slab_scale=2;// for the regularized horseshoe real slab_df=100; // for the regularized horseshoe } parameters { vector[p] z; // for non-centered parameterization real tau; // global shrinkage parameter vector [p] lambda; // local shrinkage parameter real caux; } transformed parameters { vector[p] beta; { vector[p] lambda_tilde; // 'truncated' local shrinkage parameter real c = slab_scale * sqrt(caux); // slab scale lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda))); beta = z .* lambda_tilde*tau; } } model { vector[n] means=X*beta; vector[n] log_lik; target += std_normal_lpdf(z); target += student_t_lpdf(lambda | nu_local, 0, 1); target += student_t_lpdf(tau | nu_global, 0, scale_global); target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df); for (index in 1:n) { log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]); } target += sum(log_lik); if (mixis) { target += log_sum_exp(-log_lik); } } generated quantities { vector[n] means=X*beta; vector[n] log_lik; for (index in 1:n) { log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]); } } " ``` ## Dataset The _LSVT Voice Rehabilitation Data Set_ (see [link](https://archive.ics.uci.edu/ml/datasets/LSVT+Voice+Rehabilitation) for details) has $p=312$ covariates and $n=126$ observations with binary response. We construct data list for Stan. ```{r, results='hide', warning=FALSE, message=FALSE, error=FALSE} data(voice) y <- voice$y X <- voice[2:length(voice)] n <- dim(X)[1] p <- dim(X)[2] p0 <- 10 scale_global <- 2*p0/(p-p0)/sqrt(n-1) standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) ``` Note that in our prior specification we divide the prior variance by the number of covariates $p$. This is often done in high-dimensional contexts to have a prior variance for the linear predictors $X\beta$ that remains bounded as $p$ increases. ## PSIS estimators and Pareto-$k$ diagnostics LOO-CV computations are challenging in this context due to high-dimensionality of the parameter space. To show that, we compute PSIS-LOO estimators, which require sampling from the posterior distribution, and inspect the associated Pareto-$k$ diagnostics. ```{r, results='hide', warning=FALSE} chains <- 4 n_iter <- 2000 warm_iter <- 1000 stanmodel <- stan_model(model_code = stancode_horseshoe) fit_post <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0) loo_post <-loo(fit_post) ``` ```{r} print(loo_post) ``` As we can see the diagnostics signal either "bad" or "very bad" Pareto-$k$ values for roughly $15-30\%$ of the observations which is a significant portion of the dataset. ## Mixture estimators We now compute the mixture estimators proposed in Silva and Zanella (2022). These require to sample from the following mixture of leave-one-out posteriors \begin{equation} q_{mix}(\theta) = \frac{\sum_{i=1}^n p(y_{-i}|\theta)p(\theta)}{\sum_{i=1}^np(y_{-i})}\propto p(\theta|y)\cdot \left(\sum_{i=1}^np(y_i|\theta)^{-1}\right). \end{equation} The code to generate a Stan model for the above mixture distribution is the same to the one for the posterior, just enabling one line of code with a _LogSumExp_ contribution to account for the last term in the equation above. ``` if (mixis) { target += log_sum_exp(-log_lik); } ``` We sample from the mixture and collect the log-likelihoods term. ```{r, results='hide', warnings=FALSE} standata$mixis <- 1 fit_mix <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0, pars = "log_lik") log_lik_mix <- extract(fit_mix)$log_lik ``` We now compute the mixture estimators, following the numerically stable implementation in Appendix A.2 of [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). The code below makes use of the package "matrixStats". ```{r} l_common_mix <- rowLogSumExps(-log_lik_mix) log_weights <- -log_lik_mix - l_common_mix elpd_mixis <- logSumExp(-l_common_mix) - rowLogSumExps(t(log_weights)) ``` ## Comparison with benchmark values obtained with long simulations To evaluate the performance of mixture estimators (MixIS) we also generate _benchmark values_, i.e.\ accurate approximations of the LOO predictives $\{p(y_i|y_{-i})\}_{i=1,\dots,n}$, obtained by brute-force sampling from the leave-one-out posteriors directly, getting $90k$ samples from each and discarding the first $10k$ as warmup. This is computationally heavy, hence we have saved the results and we just load them in the current vignette. ```{r} data(voice_loo) elpd_loo <- voice_loo$elpd_loo ``` We can then compute the root mean squared error (RMSE) of the PSIS and mixture estimators relative to such benchmark values. ```{r} elpd_psis <- loo_post$pointwise[,1] print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2))) print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2))) ``` Here mixture estimator provides a reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results). We then compare the overall ELPD estimates with the brute force one. ```{r} elpd_psis <- loo_post$pointwise[,1] print(paste("ELPD (PSIS)=",round(sum(elpd_psis),2))) print(paste("ELPD (MixIS)=",round(sum(elpd_mixis),2))) print(paste("ELPD (brute force)=",round(sum(elpd_loo),2))) ``` In this example, MixIS provides a more accurate ELPD estimate closer to the brute force estimate, while PSIS severely overestimates the ELPD. Note that low accuracy of the PSIS ELPD estimate is expected in this example given the large number of large Pareto-$k$ values. In this example, the accuracy of MixIS estimate will also improve with bigger MCMC sample size. More generally, mixture estimators can be useful in situations where standard PSIS estimators struggle and return many large Pareto-$k$ values. In these contexts MixIS often provides more accurate LOO-CV and ELPD estimates with a single sampling routine (i.e. with a cost comparable to sampling from the original posterior). ## References Silva L. and Zanella G. (2022). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Preprint at [arXiv:2209.09190](https://arxiv.org/abs/2209.09190) Vehtari A., Gelman A., and Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. Preprint at [arXiv:1507.04544](https://arxiv.org/abs/1507.04544) Vehtari A., Simpson D., Gelman A., Yao Y., and Gabry J. (2022). Pareto smoothed importance sampling. Preprint at [arXiv:1507.02646](https://arxiv.org/abs/1507.02646) loo/vignettes/loo2-large-data.Rmd0000644000176200001440000005036613764522153016413 0ustar liggesusers--- title: "Using Leave-one-out cross-validation for large data" author: "Mans Magnusson, Paul Bürkner, Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to do leave-one-out cross-validation for large data using the __loo__ package and Stan. There are two approaches covered: LOO with subsampling and LOO using approximations to posterior distributions. Some sections from this vignette are excerpted from the papers * Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. [arXiv preprint arXiv:2001.00980](https://arxiv.org/abs/2001.00980). * Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 [online](http://proceedings.mlr.press/v97/magnusson19a.html), [arXiv preprint arXiv:1904.10679](https://arxiv.org/abs/1904.10679). * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). which provide important background for understanding the methods implemented in the package. # Setup In addition to the __loo__ package, we'll also be using __rstan__: ```{r setup, message=FALSE} library("rstan") library("loo") set.seed(4711) ``` # Example: Well water in Bangladesh We will use the same example as in the vignette [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html). See that vignette for a description of the problem and data. The sample size in this example is only $N=3020$, which is not large enough to _require_ the special methods for large data described in this vignette, but is sufficient for demonstration purposes in this tutorial. ## Coding the Stan model Here is the Stan code for fitting the logistic regression model, which we save in a file called `logistic.stan`: ``` // save in `logistic.stan` data { int N; // number of data points int P; // number of predictors (including intercept) matrix[N,P] X; // predictors (including 1s for intercept) int y[N]; // binary outcome } parameters { vector[P] beta; } model { beta ~ normal(0, 1); y ~ bernoulli_logit(X * beta); } ``` Importantly, unlike the general approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we do _not_ compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. Here we are assuming we have a large data set (larger than the one we're actually using in this demonstration) and so it is preferable to instead define a function in R to compute the log-likelihood for each data point when needed rather than storing all of the log-likelihood values in memory. The log-likelihood in R can be coded as follows: ```{r llfun_logistic} # we'll add an argument log to toggle whether this is a log-likelihood or # likelihood function. this will be useful later in the vignette. llfun_logistic <- function(data_i, draws, log = TRUE) { x_i <- as.matrix(data_i[, which(grepl(colnames(data_i), pattern = "X")), drop=FALSE]) logit_pred <- draws %*% t(x_i) dbinom(x = data_i$y, size = 1, prob = 1/(1 + exp(-logit_pred)), log = log) } ``` The function `llfun_logistic()` needs to have arguments `data_i` and `draws`. Below we will test that the function is working by using the `loo_i()` function. ## Fitting the model with RStan Next we fit the model in Stan using the **rstan** package: ```{r, eval=FALSE} # Prepare data url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" wells <- read.table(url) wells$dist100 <- with(wells, dist / 100) X <- model.matrix(~ dist100 + arsenic, wells) standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # Compile stan_mod <- stan_model("logistic.stan") # Fit model fit_1 <- sampling(stan_mod, data = standata, seed = 4711) print(fit_1, pars = "beta") ``` ``` mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat beta[1] 0.00 0 0.08 -0.15 -0.05 0.00 0.06 0.16 1933 1 beta[2] -0.89 0 0.10 -1.09 -0.96 -0.89 -0.82 -0.69 2332 1 beta[3] 0.46 0 0.04 0.38 0.43 0.46 0.49 0.54 2051 1 ``` Before we move on to computing LOO we can now test that the log-likelihood function we wrote is working as it should. The `loo_i()` function is a helper function that can be used to test a log-likelihood function on a single observation. ```{r, eval=FALSE} # used for draws argument to loo_i parameter_draws_1 <- extract(fit_1)$beta # used for data argument to loo_i stan_df_1 <- as.data.frame(standata) # compute relative efficiency (this is slow and optional but is recommended to allow # for adjusting PSIS effective sample size based on MCMC effective sample size) r_eff <- relative_eff(llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_1, draws = parameter_draws_1, cores = 2) loo_i(i = 1, llfun_logistic, r_eff = r_eff, data = stan_df_1, draws = parameter_draws_1) ``` ``` $pointwise elpd_loo mcse_elpd_loo p_loo looic influence_pareto_k 1 -0.3314552 0.0002887608 0.0003361772 0.6629103 -0.05679886 ... ``` # Approximate LOO-CV using PSIS-LOO and subsampling We can then use the `loo_subsample()` function to compute the efficient PSIS-LOO approximation to exact LOO-CV using subsampling: ```{r, eval=FALSE} set.seed(4711) loo_ss_1 <- loo_subsample( llfun_logistic, observations = 100, # take a subsample of size 100 cores = 2, # these next objects were computed above r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1 ) print(loo_ss_1) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.5 15.6 0.3 p_loo 3.1 0.1 0.4 looic 3936.9 31.2 0.6 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` The `loo_subsample()` function creates an object of class `psis_loo_ss`, that inherits from `psis_loo, loo` (the classes of regular `loo` objects). The printed output above shows the estimates $\widehat{\mbox{elpd}}_{\rm loo}$ (expected log predictive density), $\widehat{p}_{\rm loo}$ (effective number of parameters), and ${\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}$ (the LOO information criterion). Unlike when using `loo()`, when using `loo_subsample()` there is an additional column giving the "subsampling SE", which reflects the additional uncertainty due to the subsampling used. The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the $k$ parameter is explained in `help('pareto-k-diagnostic')` and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case, the message tells us that all of the estimates for $k$ are fine _for this given subsample_. ## Adding additional subsamples If we are not satisfied with the subsample size (i.e., the accuracy) we can simply add more samples until we are satisfied using the `update()` method. ```{r, eval=FALSE} set.seed(4711) loo_ss_1b <- update( loo_ss_1, observations = 200, # subsample 200 instead of 100 r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1 ) print(loo_ss_1b) ``` ``` Computed from 4000 by 200 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.3 15.6 0.2 p_loo 3.2 0.1 0.4 looic 3936.7 31.2 0.5 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` ## Specifying estimator and sampling method The performance relies on two components: the estimation method and the approximation used for the elpd. See the documentation for `loo_subsample()` more information on which estimators and approximations are implemented. The default implementation is using the point log predictive density evaluated at the mean of the posterior (`loo_approximation="plpd"`) and the difference estimator (`estimator="diff_srs"`). This combination has a focus on fast inference. But we can easily use other estimators as well as other elpd approximations, for example: ```{r, eval=FALSE} set.seed(4711) loo_ss_1c <- loo_subsample( x = llfun_logistic, r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1, observations = 100, estimator = "hh_pps", # use Hansen-Hurwitz loo_approximation = "lpd", # use lpd instead of plpd loo_approximation_draws = 100, cores = 2 ) print(loo_ss_1c) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.9 15.4 0.5 p_loo 3.5 0.2 0.5 looic 3937.9 30.7 1.1 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` See the documentation and references for `loo_subsample()` for details on the implemented approximations. # Approximate LOO-CV using PSIS-LOO with posterior approximations Using posterior approximations, such as variational inference and Laplace approximations, can further speed-up LOO-CV for large data. Here we demonstrate using a Laplace approximation in Stan. ```{r, eval=FALSE} fit_laplace <- optimizing(stan_mod, data = standata, draws = 2000, importance_resampling = TRUE) parameter_draws_laplace <- fit_laplace$theta_tilde # draws from approximate posterior log_p <- fit_laplace$log_p # log density of the posterior log_g <- fit_laplace$log_g # log density of the approximation ``` Using the posterior approximation we can then do LOO-CV by correcting for the posterior approximation when we compute the elpd. To do this we use the `loo_approximate_posterior()` function. ```{r, eval=FALSE} set.seed(4711) loo_ap_1 <- loo_approximate_posterior( x = llfun_logistic, draws = parameter_draws_laplace, data = stan_df_1, log_p = log_p, log_g = log_g, cores = 2 ) print(loo_ap_1) ``` The function creates a class, `psis_loo_ap` that inherits from `psis_loo, loo`. ``` Computed from 2000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1968.4 15.6 p_loo 3.2 0.2 looic 3936.8 31.2 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. Pareto k diagnostic values: Count Pct. Min. n_eff (-Inf, 0.5] (good) 2989 99.0% 1827 (0.5, 0.7] (ok) 31 1.0% 1996 (0.7, 1] (bad) 0 0.0% (1, Inf) (very bad) 0 0.0% All Pareto k estimates are ok (k < 0.7). See help('pareto-k-diagnostic') for details. ``` ## Combining the posterior approximation method with subsampling The posterior approximation correction can also be used together with subsampling: ```{r, eval=FALSE} set.seed(4711) loo_ap_ss_1 <- loo_subsample( x = llfun_logistic, draws = parameter_draws_laplace, data = stan_df_1, log_p = log_p, log_g = log_g, observations = 100, cores = 2 ) print(loo_ap_ss_1) ``` ``` Computed from 2000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.2 15.6 0.4 p_loo 2.9 0.1 0.5 looic 3936.4 31.1 0.8 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. Pareto k diagnostic values: Count Pct. Min. n_eff (-Inf, 0.5] (good) 97 97.0% 1971 (0.5, 0.7] (ok) 3 3.0% 1997 (0.7, 1] (bad) 0 0.0% (1, Inf) (very bad) 0 0.0% All Pareto k estimates are ok (k < 0.7). See help('pareto-k-diagnostic') for details. ``` The object created is of class `psis_loo_ss`, which inherits from the `psis_loo_ap` class previously described. ## Comparing models To compare this model to an alternative model for the same data we can use the `loo_compare()` function just as we would if using `loo()` instead of `loo_subsample()` or `loo_approximate_posterior()`. First we'll fit a second model to the well-switching data, using `log(arsenic)` instead of `arsenic` as a predictor: ```{r, eval=FALSE} standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) fit_2 <- sampling(stan_mod, data = standata) parameter_draws_2 <- extract(fit_2)$beta stan_df_2 <- as.data.frame(standata) # recompute subsampling loo for first model for demonstration purposes # compute relative efficiency (this is slow and optional but is recommended to allow # for adjusting PSIS effective sample size based on MCMC effective sample size) r_eff_1 <- relative_eff( llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_1, draws = parameter_draws_1, cores = 2 ) set.seed(4711) loo_ss_1 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_1, draws = parameter_draws_1, data = stan_df_1, observations = 200, cores = 2 ) # compute subsampling loo for a second model (with log-arsenic) r_eff_2 <- relative_eff( llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_2, draws = parameter_draws_2, cores = 2 ) loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = 200, cores = 2 ) print(loo_ss_2) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1952.0 16.2 0.2 p_loo 2.6 0.1 0.3 looic 3903.9 32.4 0.4 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` We can now compare the models on LOO using the `loo_compare` function: ```{r, eval=FALSE} # Compare comp <- loo_compare(loo_ss_1, loo_ss_2) print(comp) ``` ``` Warning: Different subsamples in 'model2' and 'model1'. Naive diff SE is used. elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.5 22.5 0.4 ``` This new object `comp` contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error. As the warning indicates, because different subsamples were used the comparison will not take the correlations between different observations into account. Here we see that the naive SE is 22.5 and we cannot see any difference in performance between the models. To force subsampling to use the same observations for each of the models we can simply extract the observations used in `loo_ss_1` and use them in `loo_ss_2` by supplying the `loo_ss_1` object to the `observations` argument. ```{r, eval=FALSE} loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = loo_ss_1, cores = 2 ) ``` We could also supply the subsampling indices using the `obs_idx()` helper function: ```{r, eval=FALSE} idx <- obs_idx(loo_ss_1) loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = idx, cores = 2 ) ``` ``` Simple random sampling with replacement assumed. ``` This results in a message indicating that we assume these observations to have been sampled with simple random sampling, which is true because we had used the default `"diff_srs"` estimator for `loo_ss_1`. We can now compare the models and estimate the difference based on the same subsampled observations. ```{r, eval=FALSE} comp <- loo_compare(loo_ss_1, loo_ss_2) print(comp) ``` ``` elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.1 4.4 0.1 ``` First, notice that now the `se_diff` is now around 4 (as opposed to 20 when using different subsamples). The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in `elpd` and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (`model2`). Since the subsampling uncertainty is so small in this case it can effectively be ignored. If we need larger subsamples we can simply add samples using the `update()` method demonstrated earlier. It is also possible to compare a subsampled loo computation with a full loo object. ```{r, eval=FALSE} # use loo() instead of loo_subsample() to compute full PSIS-LOO for model 2 loo_full_2 <- loo( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, cores = 2 ) loo_compare(loo_ss_1, loo_full_2) ``` ``` Estimated elpd_diff using observations included in loo calculations for all models. ``` Because we are comparing a non-subsampled loo calculation to a subsampled calculation we get the message that only the observations that are included in the loo calculations for both `model1` and `model2` are included in the computations for the comparison. ``` elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.3 4.4 0.3 ``` Here we actually see an increase in `subsampling_se_diff`, but this is due to a technical detail not elaborated here. In general, the difference should be better or negligible. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2017). _The Stan C++ Library, Version 2.17.0._ https://mc-stan.org/ Stan Development Team (2018) _RStan: the R interface to Stan, Version 2.17.3._ https://mc-stan.org/ Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. [arXiv preprint arXiv:2001.00980](https://arxiv.org/abs/2001.00980). Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 [online](http://proceedings.mlr.press/v97/magnusson19a.html), [arXiv preprint arXiv:1904.10679](https://arxiv.org/abs/1904.10679). Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/vignettes/children/0000755000176200001440000000000013761524476014662 5ustar liggesusersloo/vignettes/children/SETTINGS-knitr.txt0000644000176200001440000000041613703433563020001 0ustar liggesusers```{r SETTINGS-knitr, include=FALSE} stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ``` loo/vignettes/children/SEE-ONLINE.txt0000644000176200001440000000016713761524476017025 0ustar liggesusers**NOTE: We recommend viewing the fully rendered version of this vignette online at https://mc-stan.org/loo/articles/** loo/vignettes/loo2-elpd.Rmd0000644000176200001440000001723514407123455015331 0ustar liggesusers--- title: "Holdout validation and K-fold cross-validation of Stan programs with the loo package" author: "Bruno Nicenboim" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to do holdout validation and K-fold cross-validation with __loo__ for a Stan program. # Example: Eradication of Roaches using holdout validation approach This vignette uses the same example as in the vignettes [_Using the loo package (version >= 2.0.0)_](http://mc-stan.org/loo/articles/loo2-example.html) and [_Avoiding model refits in leave-one-out cross-validation with moment matching_](https://mc-stan.org/loo/articles/loo2-moment-matching.html). ## Coding the Stan model Here is the Stan code for fitting a Poisson regression model: ```{r stancode} stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ``` Following the usual approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. ## Setup In addition to __loo__, we load the __rstan__ package for fitting the model. We will also need the __rstanarm__ package for the data. ```{r setup, message=FALSE} library("rstan") library("loo") seed <- 9547 set.seed(seed) ``` # Holdout validation For this approach, the model is first fit to the "train" data and then is evaluated on the held-out "test" data. ## Splitting the data between train and test The data is divided between train (80% of the data) and test (20%): ```{r modelfit-holdout, message=FALSE} # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) roaches$offset <- log(roaches[,"exposure2"]) # 20% of the data goes to the test set: roaches$test <- 0 roaches$test[sample(.2 * seq_len(nrow(roaches)))] <- 1 # data to "train" the model data_train <- list(y = roaches$y[roaches$test == 0], x = as.matrix(roaches[roaches$test == 0, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 0,]), K = 3, offset = roaches$offset[roaches$test == 0], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) # data to "test" the model data_test <- list(y = roaches$y[roaches$test == 1], x = as.matrix(roaches[roaches$test == 1, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 1,]), K = 3, offset = roaches$offset[roaches$test == 1], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) ``` ## Fitting the model with RStan Next we fit the model to the "test" data in Stan using the __rstan__ package: ```{r fit-train} # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) ``` We recompute the generated quantities using the posterior draws conditional on the training data, but we now pass in the held-out data to get the log predictive densities for the test data. Because we are using independent data, the log predictive density coincides with the log likelihood of the test data. ```{r gen-test} gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd <- extract_log_lik(gen_test) ``` ## Computing holdout elpd: Now we evaluate the predictive performance of the model on the test data using `elpd()`. ```{r elpd-holdout} (elpd_holdout <- elpd(log_pd)) ``` When one wants to compare different models, the function `loo_compare()` can be used to assess the difference in performance. # K-fold cross validation For this approach the data is divided into folds, and each time one fold is tested while the rest of the data is used to fit the model (see Vehtari et al., 2017). ## Splitting the data in folds We use the data that is already pre-processed and we divide it in 10 random folds using `kfold_split_random` ```{r prepare-folds, message=FALSE} # Prepare data roaches$fold <- kfold_split_random(K = 10, N = nrow(roaches)) ``` ## Fitting and extracting the log pointwise predictive densities for each fold We now loop over the 10 folds. In each fold we do the following. First, we fit the model to all the observations except the ones belonging to the left-out fold. Second, we compute the log pointwise predictive densities for the left-out fold. Last, we store the predictive density for the observations of the left-out fold in a matrix. The output of this loop is a matrix of the log pointwise predictive densities of all the observations. ```{r} # Prepare a matrix with the number of post-warmup iterations by number of observations: log_pd_kfold <- matrix(nrow = 4000, ncol = nrow(roaches)) # Loop over the folds for(k in 1:10){ data_train <- list(y = roaches$y[roaches$fold != k], x = as.matrix(roaches[roaches$fold != k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold != k,]), K = 3, offset = roaches$offset[roaches$fold != k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) data_test <- list(y = roaches$y[roaches$fold == k], x = as.matrix(roaches[roaches$fold == k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold == k,]), K = 3, offset = roaches$offset[roaches$fold == k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd_kfold[, roaches$fold == k] <- extract_log_lik(gen_test) } ``` ## Computing K-fold elpd: Now we evaluate the predictive performance of the model on the 10 folds using `elpd()`. ```{r elpd-kfold} (elpd_kfold <- elpd(log_pd_kfold)) ``` If one wants to compare several models (with `loo_compare`), one should use the same folds for all the different models. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2020) _RStan: the R interface to Stan, Version 2.21.1_ https://mc-stan.org Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). loo/R/0000755000176200001440000000000014411465511011246 5ustar liggesusersloo/R/sis.R0000644000176200001440000001134113701164066012171 0ustar liggesusers#' Standard importance sampling (SIS) #' #' Implementation of standard importance sampling (SIS). #' #' @param log_ratios An array, matrix, or vector of importance ratios on the log #' scale (for Importance sampling LOO, these are *negative* log-likelihood #' values). See the **Methods (by class)** section below for a detailed #' description of how to specify the inputs for each method. #' @template cores #' @param ... Arguments passed on to the various methods. #' @param r_eff Vector of relative effective sample size estimates containing #' one element per observation. The values provided should be the relative #' effective sample sizes of `1/exp(log_ratios)` (i.e., `1/ratios`). #' This is related to the relative efficiency of estimating the normalizing #' term in self-normalizing importance sampling. See the [relative_eff()] #' helper function for computing `r_eff`. If using `psis` with #' draws of the `log_ratios` not obtained from MCMC then the warning #' message thrown when not specifying `r_eff` can be disabled by #' setting `r_eff` to `NA`. #' #' @return The `sis()` methods return an object of class `"sis"`, #' which is a named list with the following components: #' #' \describe{ #' \item{`log_weights`}{ #' Vector or matrix of smoothed but *unnormalized* log #' weights. To get normalized weights use the #' [`weights()`][weights.importance_sampling] method provided for objects of #' class `sis`. #' } #' \item{`diagnostics`}{ #' A named list containing one vector: #' * `pareto_k`: Not used in `sis`, all set to 0. #' * `n_eff`: effective sample size estimates. #' } #' } #' #' Objects of class `"sis"` also have the following [attributes][attributes()]: #' \describe{ #' \item{`norm_const_log`}{ #' Vector of precomputed values of `colLogSumExps(log_weights)` that are #' used internally by the `weights` method to normalize the log weights. #' } #' \item{`r_eff`}{ #' If specified, the user's `r_eff` argument. #' } #' \item{`tail_len`}{ #' Not used for `sis`. #' } #' \item{`dims`}{ #' Integer vector of length 2 containing `S` (posterior sample size) #' and `N` (number of observations). #' } #' \item{`method`}{ #' Method used for importance sampling, here `sis`. #' } #' } #' #' @seealso #' * [psis()] for approximate LOO-CV using PSIS. #' * [loo()] for approximate LOO-CV. #' * [pareto-k-diagnostic] for PSIS diagnostics. #' #' @template loo-and-psis-references #' #' @examples #' log_ratios <- -1 * example_loglik_array() #' r_eff <- relative_eff(exp(-log_ratios)) #' sis_result <- sis(log_ratios, r_eff = r_eff) #' str(sis_result) #' #' # extract smoothed weights #' lw <- weights(sis_result) # default args are log=TRUE, normalize=TRUE #' ulw <- weights(sis_result, normalize=FALSE) # unnormalized log-weights #' #' w <- weights(sis_result, log=FALSE) # normalized weights (not log-weights) #' uw <- weights(sis_result, log=FALSE, normalize = FALSE) # unnormalized weights #' #' @export sis <- function(log_ratios, ...) UseMethod("sis") #' @export #' @templateVar fn sis #' @template array #' sis.array <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.array(log_ratios = log_ratios, ..., r_eff = r_eff, cores = cores, method = "sis") } #' @export #' @templateVar fn sis #' @template matrix #' sis.matrix <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.matrix(log_ratios, ..., r_eff = r_eff, cores = cores, method = "sis") } #' @export #' @templateVar fn sis #' @template vector #' sis.default <- function(log_ratios, ..., r_eff = NULL) { importance_sampling.default(log_ratios = log_ratios, ..., r_eff = r_eff, method = "sis") } #' @rdname psis #' @export is.sis <- function(x) { inherits(x, "sis") && is.list(x) } # internal ---------------------------------------------------------------- #' Standard IS on a single vector #' #' @noRd #' @param log_ratios_i A vector of log importance ratios (for `loo()`, negative #' log likelihoods). #' @param ... Not used. Included to conform to PSIS API. #' #' @details Implementation standard importance sampling. #' @return A named list containing: #' * `lw`: vector of unnormalized log weights #' * `pareto_k`: scalar Pareto k estimate. For IS, this defaults to 0. do_sis_i <- function(log_ratios_i, ...) { S <- length(log_ratios_i) list(log_weights = log_ratios_i, pareto_k = 0) } loo/R/loo_moment_matching.R0000644000176200001440000005527114407123455015430 0ustar liggesusers#' Moment matching for efficient approximate leave-one-out cross-validation (LOO) #' #' Moment matching algorithm for updating a loo object when Pareto k estimates #' are large. #' #' @export loo_moment_match loo_moment_match.default #' @param x A fitted model object. #' @param loo A loo object to be modified. #' @param post_draws A function the takes `x` as the first argument and returns #' a matrix of posterior draws of the model parameters. #' @param log_lik_i A function that takes `x` and `i` and returns a matrix (one #' column per chain) or a vector (all chains stacked) of log-likelihood draws #' of the `i`th observation based on the model `x`. If the draws are obtained #' using MCMC, the matrix with MCMC chains separated is preferred. #' @param unconstrain_pars A function that takes arguments `x`, and `pars` and #' returns posterior draws on the unconstrained space based on the posterior #' draws on the constrained space passed via `pars`. #' @param log_prob_upars A function that takes arguments `x` and `upars` and #' returns a matrix of log-posterior density values of the unconstrained #' posterior draws passed via `upars`. #' @param log_lik_i_upars A function that takes arguments `x`, `upars`, and `i` #' and returns a vector of log-likelihood draws of the `i`th observation based #' on the unconstrained posterior draws passed via `upars`. #' @param max_iters Maximum number of moment matching iterations. Usually this #' does not need to be modified. If the maximum number of iterations is #' reached, there will be a warning, and increasing `max_iters` may improve #' accuracy. #' @param k_threshold Threshold value for Pareto k values above which the moment #' matching algorithm is used. The default value is 0.5. #' @param split Logical; Indicate whether to do the split transformation or not #' at the end of moment matching for each LOO fold. #' @param cov Logical; Indicate whether to match the covariance matrix of the #' samples or not. If `FALSE`, only the mean and marginal variances are #' matched. #' @template cores #' @param ... Further arguments passed to the custom functions documented above. #' #' @return The `loo_moment_match()` methods return an updated `loo` object. The #' structure of the updated `loo` object is similar, but the method also #' stores the original Pareto k diagnostic values in the diagnostics field. #' #' @details The `loo_moment_match()` function is an S3 generic and we provide a #' default method that takes as arguments user-specified functions #' `post_draws`, `log_lik_i`, `unconstrain_pars`, `log_prob_upars`, and #' `log_lik_i_upars`. All of these functions should take `...`. as an argument #' in addition to those specified for each function. #' #' @seealso [loo()], [loo_moment_match_split()] #' @template moment-matching-references #' #' @examples #' # See the vignette for loo_moment_match() #' @export loo_moment_match <- function(x, ...) { UseMethod("loo_moment_match") } #' @describeIn loo_moment_match A default method that takes as arguments a #' user-specified model object `x`, a `loo` object and user-specified #' functions `post_draws`, `log_lik_i`, `unconstrain_pars`, `log_prob_upars`, #' and `log_lik_i_upars`. #' @export loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, unconstrain_pars, log_prob_upars, log_lik_i_upars, max_iters = 30L, k_threshold = 0.7, split = TRUE, cov = TRUE, cores = getOption("mc.cores", 1), ...) { # input checks checkmate::assertClass(loo,classes = "loo") checkmate::assertFunction(post_draws) checkmate::assertFunction(log_lik_i) checkmate::assertFunction(unconstrain_pars) checkmate::assertFunction(log_prob_upars) checkmate::assertFunction(log_lik_i_upars) checkmate::assertNumber(max_iters) checkmate::assertNumber(k_threshold) checkmate::assertLogical(split) checkmate::assertLogical(cov) checkmate::assertNumber(cores) if ("psis_loo" %in% class(loo)) { is_method <- "psis" } else { stop("loo_moment_match currently supports only the \"psis\" importance sampling class.") } S <- dim(loo)[1] N <- dim(loo)[2] pars <- post_draws(x, ...) # transform the model parameters to unconstrained space upars <- unconstrain_pars(x, pars = pars, ...) # number of parameters in the **parameters** block only npars <- dim(upars)[2] # if more parameters than samples, do not do Cholesky transformation cov <- cov && S >= 10 * npars # compute log-probabilities of the original parameter values orig_log_prob <- log_prob_upars(x, upars = upars, ...) # loop over all observations whose Pareto k is high ks <- loo$diagnostics$pareto_k kfs <- rep(0,N) I <- which(ks > k_threshold) loo_moment_match_i_fun <- function(i) { loo_moment_match_i(i = i, x = x, log_lik_i = log_lik_i, unconstrain_pars = unconstrain_pars, log_prob_upars = log_prob_upars, log_lik_i_upars = log_lik_i_upars, max_iters = max_iters, k_threshold = k_threshold, split = split, cov = cov, N = N, S = S, upars = upars, orig_log_prob = orig_log_prob, k = ks[i], is_method = is_method, npars = npars, ...) } if (cores == 1) { mm_list <- lapply(X = I, FUN = function(i) loo_moment_match_i_fun(i)) } else { if (!os_is_windows()) { mm_list <- parallel::mclapply(X = I, mc.cores = cores, FUN = function(i) loo_moment_match_i_fun(i)) } else { cl <- parallel::makePSOCKcluster(cores) on.exit(parallel::stopCluster(cl)) mm_list <- parallel::parLapply(cl = cl, X = I, fun = function(i) loo_moment_match_i_fun(i)) } } # update results for (ii in seq_along(I)) { i <- mm_list[[ii]]$i loo$pointwise[i, "elpd_loo"] <- mm_list[[ii]]$elpd_loo_i loo$pointwise[i, "p_loo"] <- mm_list[[ii]]$p_loo loo$pointwise[i, "mcse_elpd_loo"] <- mm_list[[ii]]$mcse_elpd_loo loo$pointwise[i, "looic"] <- mm_list[[ii]]$looic loo$diagnostics$pareto_k[i] <- mm_list[[ii]]$k loo$diagnostics$n_eff[i] <- mm_list[[ii]]$n_eff kfs[i] <- mm_list[[ii]]$kf if (!is.null(loo$psis_object)) { loo$psis_object$log_weights[, i] <- mm_list[[ii]]$lwi } } if (!is.null(loo$psis_object)) { attr(loo$psis_object, "norm_const_log") <- matrixStats::colLogSumExps(loo$psis_object$log_weights) loo$psis_object$diagnostics <- loo$diagnostics } # combined estimates cols_to_summarize <- !(colnames(loo$pointwise) %in% c("mcse_elpd_loo", "influence_pareto_k")) loo$estimates <- table_of_estimates(loo$pointwise[, cols_to_summarize, drop = FALSE]) # these will be deprecated at some point loo$elpd_loo <- loo$estimates["elpd_loo","Estimate"] loo$p_loo <- loo$estimates["p_loo","Estimate"] loo$looic <- loo$estimates["looic","Estimate"] loo$se_elpd_loo <- loo$estimates["elpd_loo","SE"] loo$se_p_loo <- loo$estimates["p_loo","SE"] loo$se_looic <- loo$estimates["looic","SE"] # Warn if some Pareto ks are still high psislw_warnings(loo$diagnostics$pareto_k) # if we don't split, accuracy may be compromised if (!split) { throw_large_kf_warning(kfs) } loo } # Internal functions --------------- #' Do moment matching for a single observation. #' #' @noRd #' @param i observation number. #' @param x A fitted model object. #' @param log_lik_i A function that takes `x` and `i` and returns a matrix (one #' column per chain) or a vector (all chains stacked) of log-likeliood draws #' of the `i`th observation based on the model `x`. If the draws are obtained #' using MCMC, the matrix with MCMC chains separated is preferred. #' @param unconstrain_pars A function that takes arguments `x`, and `pars` and #' returns posterior draws on the unconstrained space based on the posterior #' draws on the constrained space passed via `pars`. #' @param log_prob_upars A function that takes arguments `x` and `upars` and #' returns a matrix of log-posterior density values of the unconstrained #' posterior draws passed via `upars`. #' @param log_lik_i_upars A function that takes arguments `x`, `upars`, and `i` #' and returns a vector of log-likelihood draws of the `i`th observation based #' on the unconstrained posterior draws passed via `upars`. #' @param max_iters Maximum number of moment matching iterations. Usually this #' does not need to be modified. If the maximum number of iterations is #' reached, there will be a warning, and increasing `max_iters` may improve #' accuracy. #' @param k_threshold Threshold value for Pareto k values above which the moment #' matching algorithm is used. The default value is 0.5. #' @param split Logical; Indicate whether to do the split transformation or not #' at the end of moment matching for each LOO fold. #' @param cov Logical; Indicate whether to match the covariance matrix of the #' samples or not. If `FALSE`, only the mean and marginal variances are #' matched. #' @param N Number of observations. #' @param S number of MCMC draws. #' @param upars A matrix representing a sample of vector-valued parameters in #' the unconstrained space. #' @param orig_log_prob log probability densities of the original draws from the #' model `x`. #' @param k Pareto k value before moment matching #' @template is_method #' @param npars Number of parameters in the model #' @param ... Further arguments passed to the custom functions documented above. #' @return List with the updated elpd values and diagnostics #' loo_moment_match_i <- function(i, x, log_lik_i, unconstrain_pars, log_prob_upars, log_lik_i_upars, max_iters, k_threshold, split, cov, N, S, upars, orig_log_prob, k, is_method, npars, ...) { # initialize values for this LOO-fold uparsi <- upars ki <- k kfi <- 0 log_liki <- log_lik_i(x, i, ...) S_per_chain <- NROW(log_liki) N_chains <- NCOL(log_liki) dim(log_liki) <- c(S_per_chain, N_chains, 1) r_eff_i <- loo::relative_eff(exp(log_liki), cores = 1) dim(log_liki) <- NULL is_obj <- suppressWarnings(importance_sampling.default(-log_liki, method = is_method, r_eff = r_eff_i, cores = 1)) lwi <- as.vector(weights(is_obj)) lwfi <- rep(-matrixStats::logSumExp(rep(0, S)),S) # initialize objects that keep track of the total transformation total_shift <- rep(0, npars) total_scaling <- rep(1, npars) total_mapping <- diag(npars) # try several transformations one by one # if one does not work, do not apply it and try another one # to accept the transformation, Pareto k needs to improve # when transformation succeeds, start again from the first one iterind <- 1 while (iterind <= max_iters && ki > k_threshold) { if (iterind == max_iters) { throw_moment_match_max_iters_warning() } # 1. match means trans <- shift(x, uparsi, lwi) # gather updated quantities quantities_i <- update_quantities_i(x, trans$upars, i = i, orig_log_prob = orig_log_prob, log_prob_upars = log_prob_upars, log_lik_i_upars = log_lik_i_upars, r_eff_i = r_eff_i, cores = 1, is_method = is_method, ...) if (quantities_i$ki < ki) { uparsi <- trans$upars total_shift <- total_shift + trans$shift lwi <- quantities_i$lwi lwfi <- quantities_i$lwfi ki <- quantities_i$ki kfi <- quantities_i$kfi log_liki <- quantities_i$log_liki iterind <- iterind + 1 next } # 2. match means and marginal variances trans <- shift_and_scale(x, uparsi, lwi) # gather updated quantities quantities_i <- update_quantities_i(x, trans$upars, i = i, orig_log_prob = orig_log_prob, log_prob_upars = log_prob_upars, log_lik_i_upars = log_lik_i_upars, r_eff_i = r_eff_i, cores = 1, is_method = is_method, ...) if (quantities_i$ki < ki) { uparsi <- trans$upars total_shift <- total_shift + trans$shift total_scaling <- total_scaling * trans$scaling lwi <- quantities_i$lwi lwfi <- quantities_i$lwfi ki <- quantities_i$ki kfi <- quantities_i$kfi log_liki <- quantities_i$log_liki iterind <- iterind + 1 next } # 3. match means and covariances if (cov) { trans <- shift_and_cov(x, uparsi, lwi) # gather updated quantities quantities_i <- update_quantities_i(x, trans$upars, i = i, orig_log_prob = orig_log_prob, log_prob_upars = log_prob_upars, log_lik_i_upars = log_lik_i_upars, r_eff_i = r_eff_i, cores = 1, is_method = is_method, ...) if (quantities_i$ki < ki) { uparsi <- trans$upars total_shift <- total_shift + trans$shift total_mapping <- trans$mapping %*% total_mapping lwi <- quantities_i$lwi lwfi <- quantities_i$lwfi ki <- quantities_i$ki kfi <- quantities_i$kfi log_liki <- quantities_i$log_liki iterind <- iterind + 1 next } } # none of the transformations improved khat # so there is no need to try further break } # transformations are now done # if we don't do split transform, or # if no transformations were successful # stop and collect values if (split && (iterind > 1)) { # compute split transformation split_obj <- loo_moment_match_split( x, upars, cov, total_shift, total_scaling, total_mapping, i, log_prob_upars = log_prob_upars, log_lik_i_upars = log_lik_i_upars, cores = 1, r_eff_i = r_eff_i, is_method = is_method, ... ) log_liki <- split_obj$log_liki lwi <- split_obj$lwi lwfi <- split_obj$lwfi r_eff_i <- split_obj$r_eff_i } else { dim(log_liki) <- c(S_per_chain, N_chains, 1) r_eff_i <- loo::relative_eff(exp(log_liki), cores = 1) dim(log_liki) <- NULL } # pointwise estimates elpd_loo_i <- matrixStats::logSumExp(log_liki + lwi) lpd <- matrixStats::logSumExp(log_liki) - log(length(log_liki)) mcse_elpd_loo <- mcse_elpd( ll = as.matrix(log_liki), lw = as.matrix(lwi), E_elpd = exp(elpd_loo_i), r_eff = r_eff_i ) list(elpd_loo_i = elpd_loo_i, p_loo = lpd - elpd_loo_i, mcse_elpd_loo = mcse_elpd_loo, looic = -2 * elpd_loo_i, k = ki, kf = kfi, n_eff = min(1.0 / sum(exp(2 * lwi)), 1.0 / sum(exp(2 * lwfi))) * r_eff_i, lwi = lwi, i = i) } #' Update the importance weights, Pareto diagnostic and log-likelihood #' for observation `i` based on model `x`. #' #' @noRd #' @param x A fitted model object. #' @param upars A matrix representing a sample of vector-valued parameters in #' the unconstrained space. #' @param i observation number. #' @param orig_log_prob log probability densities of the original draws from #' the model `x`. #' @param log_prob_upars A function that takes arguments `x` and #' `upars` and returns a matrix of log-posterior density values of the #' unconstrained posterior draws passed via `upars`. #' @param log_lik_i_upars A function that takes arguments `x`, `upars`, #' and `i` and returns a vector of log-likeliood draws of the `i`th #' observation based on the unconstrained posterior draws passed via #' `upars`. #' @param r_eff_i MCMC effective sample size divided by the total sample size #' for 1/exp(log_ratios) for observation i. #' @template is_method #' @return List with the updated importance weights, Pareto diagnostics and #' log-likelihood values. #' update_quantities_i <- function(x, upars, i, orig_log_prob, log_prob_upars, log_lik_i_upars, r_eff_i, is_method, ...) { log_prob_new <- log_prob_upars(x, upars = upars, ...) log_liki_new <- log_lik_i_upars(x, upars = upars, i = i, ...) # compute new log importance weights is_obj_new <- suppressWarnings(importance_sampling.default(-log_liki_new + log_prob_new - orig_log_prob, method = is_method, r_eff = r_eff_i, cores = 1)) lwi_new <- as.vector(weights(is_obj_new)) ki_new <- is_obj_new$diagnostics$pareto_k is_obj_f_new <- suppressWarnings(importance_sampling.default(log_prob_new - orig_log_prob, method = is_method, r_eff = r_eff_i, cores = 1)) lwfi_new <- as.vector(weights(is_obj_f_new)) kfi_new <- is_obj_f_new$diagnostics$pareto_k # gather results list( lwi = lwi_new, lwfi = lwfi_new, ki = ki_new, kfi = kfi_new, log_liki = log_liki_new ) } #' Shift a matrix of parameters to their weighted mean. #' Also calls update_quantities_i which updates the importance weights based on #' the supplied model object. #' #' @noRd #' @param x A fitted model object. #' @param upars A matrix representing a sample of vector-valued parameters in #' the unconstrained space #' @param lwi A vector representing the log-weight of each parameter #' @return List with the shift that was performed, and the new parameter matrix. #' shift <- function(x, upars, lwi) { # compute moments using log weights mean_original <- colMeans(upars) mean_weighted <- colSums(exp(lwi) * upars) shift <- mean_weighted - mean_original # transform posterior draws upars_new <- sweep(upars, 2, shift, "+") list( upars = upars_new, shift = shift ) } #' Shift a matrix of parameters to their weighted mean and scale the marginal #' variances to match the weighted marginal variances. Also calls #' update_quantities_i which updates the importance weights based on #' the supplied model object. #' #' @noRd #' @param x A fitted model object. #' @param upars A matrix representing a sample of vector-valued parameters in #' the unconstrained space #' @param lwi A vector representing the log-weight of each parameter #' @return List with the shift and scaling that were performed, and the new #' parameter matrix. #' #' shift_and_scale <- function(x, upars, lwi) { # compute moments using log weights S <- dim(upars)[1] mean_original <- colMeans(upars) mean_weighted <- colSums(exp(lwi) * upars) shift <- mean_weighted - mean_original mii <- exp(lwi)* upars^2 mii <- colSums(mii) - mean_weighted^2 mii <- mii*S/(S-1) scaling <- sqrt(mii / matrixStats::colVars(upars)) # transform posterior draws upars_new <- sweep(upars, 2, mean_original, "-") upars_new <- sweep(upars_new, 2, scaling, "*") upars_new <- sweep(upars_new, 2, mean_weighted, "+") list( upars = upars_new, shift = shift, scaling = scaling ) } #' Shift a matrix of parameters to their weighted mean and scale the covariance #' to match the weighted covariance. #' Also calls update_quantities_i which updates the importance weights based on #' the supplied model object. #' #' @noRd #' @param x A fitted model object. #' @param upars A matrix representing a sample of vector-valued parameters in #' the unconstrained space #' @param lwi A vector representing the log-weight of each parameter #' @return List with the shift and mapping that were performed, and the new #' parameter matrix. #' shift_and_cov <- function(x, upars, lwi, ...) { # compute moments using log weights mean_original <- colMeans(upars) mean_weighted <- colSums(exp(lwi) * upars) shift <- mean_weighted - mean_original covv <- stats::cov(upars) wcovv <- stats::cov.wt(upars, wt = exp(lwi))$cov chol1 <- tryCatch( { chol(wcovv) }, error = function(cond) { return(NULL) } ) if (is.null(chol1)) { mapping <- diag(length(mean_original)) } else { chol2 <- chol(covv) mapping <- t(chol1) %*% solve(t(chol2)) } # transform posterior draws upars_new <- sweep(upars, 2, mean_original, "-") upars_new <- tcrossprod(upars_new, mapping) upars_new <- sweep(upars_new, 2, mean_weighted, "+") colnames(upars_new) <- colnames(upars) list( upars = upars_new, shift = shift, mapping = mapping ) } #' Warning message if max_iters is reached #' @noRd throw_moment_match_max_iters_warning <- function() { warning( "The maximum number of moment matching iterations ('max_iters' argument) was reached.\n", "Increasing the value may improve accuracy.", call. = FALSE ) } #' Warning message if not using split transformation and accuracy is compromised #' @noRd throw_large_kf_warning <- function(kf, k_threshold = 0.5) { if (any(kf > k_threshold)) { warning( "The accuracy of self-normalized importance sampling may be bad.\n", "Setting the argument 'split' to 'TRUE' will likely improve accuracy.", call. = FALSE ) } } #' warnings about pareto k values ------------------------------------------ #' @noRd psislw_warnings <- function(k) { if (any(k > 0.7)) { .warn( "Some Pareto k diagnostic values are too high. ", .k_help() ) } else if (any(k > 0.5)) { .warn( "Some Pareto k diagnostic values are slightly high. ", .k_help() ) } } loo/R/print.R0000644000176200001440000001112513575772017012541 0ustar liggesusers#' Print methods #' #' @export #' @param x An object returned by [loo()], [psis()], or [waic()]. #' @param digits An integer passed to [base::round()]. #' @param plot_k Logical. If `TRUE` the estimates of the Pareto shape #' parameter \eqn{k} are plotted. Ignored if `x` was generated by #' [waic()]. To just plot \eqn{k} without printing use the #' [plot()][pareto-k-diagnostic] method for 'loo' objects. #' @param ... Arguments passed to [plot.psis_loo()] if `plot_k` is #' `TRUE`. #' #' @return `x`, invisibly. #' #' @seealso [pareto-k-diagnostic] #' print.loo <- function(x, digits = 1, ...) { cat("\n") print_dims(x) if (!("estimates" %in% names(x))) { x <- convert_old_object(x) } cat("\n") print(.fr(as.data.frame(x$estimates), digits), quote = FALSE) return(invisible(x)) } #' @export #' @rdname print.loo print.waic <- function(x, digits = 1, ...) { print.loo(x, digits = digits, ...) throw_pwaic_warnings(x$pointwise[, "p_waic"], digits = digits, warn = FALSE) invisible(x) } #' @export #' @rdname print.loo print.psis_loo <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") print_mcse_summary(x, digits = digits) if (length(pareto_k_ids(x, threshold = 0.5))) { cat("\n") } print(pareto_k_table(x), digits = digits) cat(.k_help()) if (plot_k) { graphics::plot(x, ...) } invisible(x) } #' @export #' @rdname print.loo print.importance_sampling_loo <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") invisible(x) } #' @export #' @rdname print.loo print.psis_loo_ap <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") cat("Posterior approximation correction used.\n") print_mcse_summary(x, digits = digits) if (length(pareto_k_ids(x, threshold = 0.5))) { cat("\n") } print(pareto_k_table(x), digits = digits) cat(.k_help()) if (plot_k) { graphics::plot(x, ...) } invisible(x) } #' @export #' @rdname print.loo print.psis <- function(x, digits = 1, plot_k = FALSE, ...) { print_dims(x) print(pareto_k_table(x), digits = digits) cat(.k_help()) if (plot_k) { graphics::plot(x, ...) } invisible(x) } #' @export #' @rdname print.loo print.importance_sampling <- function(x, digits = 1, plot_k = FALSE, ...) { print_dims(x) if (plot_k) { graphics::plot(x, ...) } invisible(x) } # internal ---------------------------------------------------------------- #' Print dimensions of log-likelihood or log-weights matrix #' #' @export #' @keywords internal #' #' @param x The object returned by [psis()], [loo()], or [waic()]. #' @param ... Ignored. print_dims <- function(x, ...) UseMethod("print_dims") #' @rdname print_dims #' @export print_dims.importance_sampling <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), "log-weights matrix\n" ) } #' @rdname print_dims #' @export print_dims.psis_loo <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), "log-likelihood matrix\n" ) } #' @rdname print_dims #' @export print_dims.importance_sampling_loo <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), "log-likelihood matrix using", class(x)[1], "\n" ) } #' @rdname print_dims #' @export print_dims.waic <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), "log-likelihood matrix\n" ) } #' @rdname print_dims #' @export print_dims.kfold <- function(x, ...) { K <- attr(x, "K", exact = TRUE) if (!is.null(K)) { cat("Based on", paste0(K, "-fold"), "cross-validation\n") } } #' @rdname print_dims #' @export print_dims.psis_loo_ss <- function(x, ...) { cat( "Computed from", paste(c(dim(x)[1], nobs(x)) , collapse = " by "), "subsampled log-likelihood\nvalues from", length(x$loo_subsampling$elpd_loo_approx), "total observations.\n" ) } print_mcse_summary <- function(x, digits) { mcse_val <- mcse_loo(x) cat( "Monte Carlo SE of elpd_loo is", paste0(.fr(mcse_val, digits), ".\n") ) } # print and warning helpers .fr <- function(x, digits) format(round(x, digits), nsmall = digits) .warn <- function(..., call. = FALSE) warning(..., call. = call.) .k_help <- function() "See help('pareto-k-diagnostic') for details.\n" # compatibility with old loo objects convert_old_object <- function(x, digits = 1, ...) { z <- x[-grep("pointwise|pareto_k|n_eff", names(x))] uz <- unlist(z) nms <- names(uz) ses <- grepl("se", nms) list(estimates = data.frame(Estimate = uz[!ses], SE = uz[ses])) } loo/R/E_loo.R0000644000176200001440000002102514407123455012431 0ustar liggesusers#' Compute weighted expectations #' #' The `E_loo()` function computes weighted expectations (means, variances, #' quantiles) using the importance weights obtained from the #' [PSIS][psis()] smoothing procedure. The expectations estimated by the #' `E_loo()` function assume that the PSIS approximation is working well. #' **A small [Pareto k][pareto-k-diagnostic] estimate is necessary, #' but not sufficient, for `E_loo()` to give reliable estimates.** Additional #' diagnostic checks for gauging the reliability of the estimates are in #' development and will be added in a future release. #' #' @export #' @param x A numeric vector or matrix. #' @param psis_object An object returned by [psis()]. #' @param log_ratios Optionally, a vector or matrix (the same dimensions as `x`) #' of raw (not smoothed) log ratios. If working with log-likelihood values, #' the log ratios are the **negative** of those values. If `log_ratios` is #' specified we are able to compute [Pareto k][pareto-k-diagnostic] #' diagnostics specific to `E_loo()`. #' @param type The type of expectation to compute. The options are #' `"mean"`, `"variance"`, and `"quantile"`. #' @param probs For computing quantiles, a vector of probabilities. #' @param ... Arguments passed to individual methods. #' #' @return A named list with the following components: #' \describe{ #' \item{`value`}{ #' The result of the computation. #' #' For the matrix method, `value` is a vector with `ncol(x)` #' elements, with one exception: when `type="quantile"` and #' multiple values are specified in `probs` the `value` component of #' the returned object is a `length(probs)` by `ncol(x)` matrix. #' #' For the default/vector method the `value` component is scalar, with #' one exception: when `type` is `"quantile"` and multiple values #' are specified in `probs` the `value` component is a vector with #' `length(probs)` elements. #' } #' \item{`pareto_k`}{ #' Function-specific diagnostic. #' #' If `log_ratios` is not specified when calling `E_loo()`, #' `pareto_k` will be `NULL`. Otherwise, for the matrix method it #' will be a vector of length `ncol(x)` containing estimates of the shape #' parameter \eqn{k} of the generalized Pareto distribution. For the #' default/vector method, the estimate is a scalar. #' } #' } #' #' #' @examples #' \donttest{ #' if (requireNamespace("rstanarm", quietly = TRUE)) { #' # Use rstanarm package to quickly fit a model and get both a log-likelihood #' # matrix and draws from the posterior predictive distribution #' library("rstanarm") #' #' # data from help("lm") #' ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14) #' trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69) #' d <- data.frame( #' weight = c(ctl, trt), #' group = gl(2, 10, 20, labels = c("Ctl","Trt")) #' ) #' fit <- stan_glm(weight ~ group, data = d, refresh = 0) #' yrep <- posterior_predict(fit) #' dim(yrep) #' #' log_ratios <- -1 * log_lik(fit) #' dim(log_ratios) #' #' r_eff <- relative_eff(exp(-log_ratios), chain_id = rep(1:4, each = 1000)) #' psis_object <- psis(log_ratios, r_eff = r_eff, cores = 2) #' #' E_loo(yrep, psis_object, type = "mean") #' E_loo(yrep, psis_object, type = "var") #' E_loo(yrep, psis_object, type = "quantile", probs = 0.5) # median #' E_loo(yrep, psis_object, type = "quantile", probs = c(0.1, 0.9)) #' #' # To get Pareto k diagnostic with E_loo we also need to provide the negative #' # log-likelihood values using the log_ratios argument. #' E_loo(yrep, psis_object, type = "mean", log_ratios = log_ratios) #' } #' } #' E_loo <- function(x, psis_object, ...) { UseMethod("E_loo") } #' @rdname E_loo #' @export E_loo.default <- function(x, psis_object, ..., type = c("mean", "variance", "quantile"), probs = NULL, log_ratios = NULL) { stopifnot( is.numeric(x), is.psis(psis_object), length(x) == dim(psis_object)[1], is.null(log_ratios) || (length(x) == length(log_ratios)) ) type <- match.arg(type) E_fun <- .E_fun(type) r_eff <- NULL if (type == "variance") { r_eff <- relative_eff(psis_object) } w <- as.vector(weights(psis_object, log = FALSE)) x <- as.vector(x) out <- E_fun(x, w, probs, r_eff) if (is.null(log_ratios)) { warning("'log_ratios' not specified. Can't compute k-hat diagnostic.", call. = FALSE) khat <- NULL } else { khat <- E_loo_khat.default(x, psis_object, log_ratios) } list(value = out, pareto_k = khat) } #' @rdname E_loo #' @export E_loo.matrix <- function(x, psis_object, ..., type = c("mean", "variance", "quantile"), probs = NULL, log_ratios = NULL) { stopifnot( is.numeric(x), is.psis(psis_object), identical(dim(x), dim(psis_object)), is.null(log_ratios) || identical(dim(x), dim(log_ratios)) ) type <- match.arg(type) E_fun <- .E_fun(type) fun_val <- numeric(1) r_eff <- NULL if (type == "variance") { r_eff <- relative_eff(psis_object) } else if (type == "quantile") { stopifnot( is.numeric(probs), length(probs) >= 1, all(probs > 0 & probs < 1) ) fun_val <- numeric(length(probs)) } w <- weights(psis_object, log = FALSE) out <- vapply(seq_len(ncol(x)), function(i) { E_fun(x[, i], w[, i], probs = probs, r_eff = r_eff[i]) }, FUN.VALUE = fun_val) if (is.null(log_ratios)) { warning("'log_ratios' not specified. Can't compute k-hat diagnostic.", call. = FALSE) khat <- NULL } else { khat <- E_loo_khat.matrix(x, psis_object, log_ratios) } list(value = out, pareto_k = khat) } #' Select the function to use based on user's 'type' argument #' #' @noRd #' @param type User's `type` argument. #' @return The function for computing the weighted expectation specified by #' `type`. #' .E_fun <- function(type = c("mean", "variance", "quantile")) { switch( type, "mean" = .wmean, "variance" = .wvar, "quantile" = .wquant ) } #' loo-weighted mean, variance, and quantiles #' #' @noRd #' @param x,w Vectors of the same length. This should be checked inside #' `E_loo()` before calling these functions. #' @param probs Vector of probabilities. #' @param ... ignored. Having ... allows `probs` to be passed to `.wmean()` and #' `.wvar()` in `E_loo()` without resulting in an error. #' .wmean <- function(x, w, ...) { sum(w * x) } .wvar <- function(x, w, r_eff = NULL, ...) { if (is.null(r_eff)) { r_eff <- 1 } r <- (x - .wmean(x, w))^2 sum(w^2 * r) / r_eff } .wquant <- function(x, w, probs, ...) { if (all(w == w[1])) { return(quantile(x, probs = probs, names = FALSE)) } ord <- order(x) x <- x[ord] w <- w[ord] ww <- cumsum(w) ww <- ww / ww[length(ww)] qq <- numeric(length(probs)) for (j in seq_along(probs)) { ids <- which(ww >= probs[j]) wi <- min(ids) if (wi == 1) { qq[j] <- x[1] } else { w1 <- ww[wi - 1] x1 <- x[wi - 1] qq[j] <- x1 + (x[wi] - x1) * (probs[j] - w1) / (ww[wi] - w1) } } return(qq) } #' Compute function-specific k-hat diagnostics #' #' @noRd #' @param log_ratios Vector or matrix of raw (not smoothed) log ratios with the #' same dimensions as `x`. If working with log-likelihood values, the log #' ratios are the negative of those values. #' @return Vector (of length `NCOL(x)`) of k-hat estimates. #' E_loo_khat <- function(x, psis_object, log_ratios, ...) { UseMethod("E_loo_khat") } E_loo_khat.default <- function(x, psis_object, log_ratios, ...) { .E_loo_khat_i(x, log_ratios, attr(psis_object, "tail_len")) } E_loo_khat.matrix <- function(x, psis_object, log_ratios, ...) { tail_lengths <- attr(psis_object, "tail_len") sapply(seq_len(ncol(x)), function(i) { .E_loo_khat_i(x[, i], log_ratios[, i], tail_lengths[i]) }) } #' Compute function-specific khat estimates #' #' @noRd #' @param x_i Vector of values of function h(theta) #' @param log_ratios_i S-vector of log_ratios, log(r(theta)), for a single #' observation. #' @param tail_len_i Integer tail length used for fitting GPD. #' @return Scalar h-specific k-hat estimate. #' .E_loo_khat_i <- function(x_i, log_ratios_i, tail_len_i) { h_theta <- x_i r_theta <- exp(log_ratios_i - max(log_ratios_i)) a <- sqrt(1 + h_theta^2) * r_theta log_a <- sort(log(a)) S <- length(log_a) tail_ids <- seq(S - tail_len_i + 1, S) tail_sample <- log_a[tail_ids] cutoff <- log_a[min(tail_ids) - 1] smoothed <- psis_smooth_tail(tail_sample, cutoff) smoothed$k } loo/R/sysdata.rda0000644000176200001440000071000713261751552013420 0ustar liggesusersBZh91AY&SYXYYw{;>o@a;w}y_yn{Ν<ݾ{4} mk{ͱ[v׺}w>λ]8ڼn{{nr{sv6fj#֩뵧w ;;ۭʽЦoGʼ]ws;{{T[g<)ǭub.v{jKmi^޶{RqUo/{֫ݽڻ-;H2 mZZ:+D+#Y,H`A\`Jg}G^4'8"k1-yklG˃شr-3tLH8.s3 b^KA`b[fbϥ/M-/թi6- QJblHub/@ ]OR;q%$㱲?NI]L[U_ClGU]/@_h-ۅ-n? 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However, we recommend LOO-CV using PSIS (as implemented by #' the [loo()] function) because PSIS provides useful diagnostics as well as #' effective sample size and Monte Carlo estimates. #' #' @export waic waic.array waic.matrix waic.function #' @inheritParams loo #' #' @return A named list (of class `c("waic", "loo")`) with components: #' #' \describe{ #' \item{`estimates`}{ #' A matrix with two columns (`"Estimate"`, `"SE"`) and three #' rows (`"elpd_waic"`, `"p_waic"`, `"waic"`). This contains #' point estimates and standard errors of the expected log pointwise predictive #' density (`elpd_waic`), the effective number of parameters #' (`p_waic`) and the information criterion `waic` (which is just #' `-2 * elpd_waic`, i.e., converted to deviance scale). #' } #' \item{`pointwise`}{ #' A matrix with three columns (and number of rows equal to the number of #' observations) containing the pointwise contributions of each of the above #' measures (`elpd_waic`, `p_waic`, `waic`). #' } #' } #' #' @seealso #' * The __loo__ package [vignettes](https://mc-stan.org/loo/articles/) and #' Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, #' and Gabry (2019) for more details on why we prefer `loo()` to `waic()`. #' * [loo_compare()] for comparing models on approximate LOO-CV or WAIC. #' #' @references #' Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and #' widely application information criterion in singular learning theory. #' *Journal of Machine Learning Research* **11**, 3571-3594. #' #' @template loo-and-psis-references #' #' @examples #' ### Array and matrix methods #' LLarr <- example_loglik_array() #' dim(LLarr) #' #' LLmat <- example_loglik_matrix() #' dim(LLmat) #' #' waic_arr <- waic(LLarr) #' waic_mat <- waic(LLmat) #' identical(waic_arr, waic_mat) #' #' #' \dontrun{ #' log_lik1 <- extract_log_lik(stanfit1) #' log_lik2 <- extract_log_lik(stanfit2) #' (waic1 <- waic(log_lik1)) #' (waic2 <- waic(log_lik2)) #' print(compare(waic1, waic2), digits = 2) #' } #' waic <- function(x, ...) { UseMethod("waic") } #' @export #' @templateVar fn waic #' @template array #' waic.array <- function(x, ...) { waic.matrix(llarray_to_matrix(x), ...) } #' @export #' @templateVar fn waic #' @template matrix #' waic.matrix <- function(x, ...) { ll <- validate_ll(x) lldim <- dim(ll) lpd <- matrixStats::colLogSumExps(ll) - log(nrow(ll)) # colLogMeanExps p_waic <- matrixStats::colVars(ll) elpd_waic <- lpd - p_waic waic <- -2 * elpd_waic pointwise <- cbind(elpd_waic, p_waic, waic) throw_pwaic_warnings(pointwise[, "p_waic"], digits = 1) return(waic_object(pointwise, dims = lldim)) } #' @export #' @templateVar fn waic #' @template function #' @param draws,data,... For the function method only. See the #' **Methods (by class)** section below for details on these arguments. #' waic.function <- function(x, ..., data = NULL, draws = NULL) { stopifnot(is.data.frame(data) || is.matrix(data), !is.null(draws)) .llfun <- validate_llfun(x) N <- dim(data)[1] S <- length(as.vector(.llfun(data_i = data[1,, drop=FALSE], draws = draws, ...))) waic_list <- lapply(seq_len(N), FUN = function(i) { ll_i <- .llfun(data_i = data[i,, drop=FALSE], draws = draws, ...) ll_i <- as.vector(ll_i) lpd_i <- logMeanExp(ll_i) p_waic_i <- var(ll_i) elpd_waic_i <- lpd_i - p_waic_i c(elpd_waic = elpd_waic_i, p_waic = p_waic_i) }) pointwise <- do.call(rbind, waic_list) pointwise <- cbind(pointwise, waic = -2 * pointwise[, "elpd_waic"]) throw_pwaic_warnings(pointwise[, "p_waic"], digits = 1) waic_object(pointwise, dims = c(S, N)) } #' @export dim.waic <- function(x) { attr(x, "dims") } #' @rdname waic #' @export is.waic <- function(x) { inherits(x, "waic") && is.loo(x) } # internal ---------------------------------------------------------------- # structure the object returned by the waic methods waic_object <- function(pointwise, dims) { estimates <- table_of_estimates(pointwise) out <- nlist(estimates, pointwise) # maintain backwards compatibility old_nms <- c("elpd_waic", "p_waic", "waic", "se_elpd_waic", "se_p_waic", "se_waic") out <- c(out, setNames(as.list(estimates), old_nms)) structure( out, dims = dims, class = c("waic", "loo") ) } # waic warnings # @param p 'p_waic' estimates throw_pwaic_warnings <- function(p, digits = 1, warn = TRUE) { badp <- p > 0.4 if (any(badp)) { count <- sum(badp) prop <- count / length(badp) msg <- paste0("\n", count, " (", .fr(100 * prop, digits), "%) p_waic estimates greater than 0.4. ", "We recommend trying loo instead.") if (warn) .warn(msg) else cat(msg, "\n") } invisible(NULL) } loo/R/loo.R0000644000176200001440000005753614214417101012173 0ustar liggesusers#' Efficient approximate leave-one-out cross-validation (LOO) #' #' The `loo()` methods for arrays, matrices, and functions compute PSIS-LOO #' CV, efficient approximate leave-one-out (LOO) cross-validation for Bayesian #' models using Pareto smoothed importance sampling ([PSIS][psis()]). This is #' an implementation of the methods described in Vehtari, Gelman, and Gabry #' (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019). #' #' @export loo loo.array loo.matrix loo.function #' @param x A log-likelihood array, matrix, or function. The **Methods (by class)** #' section, below, has detailed descriptions of how to specify the inputs for #' each method. #' @param r_eff Vector of relative effective sample size estimates for the #' likelihood (`exp(log_lik)`) of each observation. This is related to #' the relative efficiency of estimating the normalizing term in #' self-normalizing importance sampling when using posterior draws obtained #' with MCMC. If MCMC draws are used and `r_eff` is not provided then #' the reported PSIS effective sample sizes and Monte Carlo error estimates #' will be over-optimistic. If the posterior draws are independent then #' `r_eff=1` and can be omitted. The warning message thrown when `r_eff` is #' not specified can be disabled by setting `r_eff` to `NA`. See the #' [relative_eff()] helper functions for computing `r_eff`. #' @param save_psis Should the `"psis"` object created internally by `loo()` be #' saved in the returned object? The `loo()` function calls [psis()] #' internally but by default discards the (potentially large) `"psis"` object #' after using it to compute the LOO-CV summaries. Setting `save_psis=TRUE` #' will add a `psis_object` component to the list returned by `loo`. #' Currently this is only needed if you plan to use the [E_loo()] function to #' compute weighted expectations after running `loo`. #' @template cores #' @template is_method #' #' @details The `loo()` function is an S3 generic and methods are provided for #' 3-D pointwise log-likelihood arrays, pointwise log-likelihood matrices, and #' log-likelihood functions. The array and matrix methods are the most #' convenient, but for models fit to very large datasets the `loo.function()` #' method is more memory efficient and may be preferable. #' #' @section Defining `loo()` methods in a package: Package developers can define #' `loo()` methods for fitted models objects. See the example `loo.stanfit()` #' method in the **Examples** section below for an example of defining a #' method that calls `loo.array()`. The `loo.stanreg()` method in the #' **rstanarm** package is an example of defining a method that calls #' `loo.function()`. #' #' @return The `loo()` methods return a named list with class #' `c("psis_loo", "loo")` and components: #' \describe{ #' \item{`estimates`}{ #' A matrix with two columns (`Estimate`, `SE`) and three rows (`elpd_loo`, #' `p_loo`, `looic`). This contains point estimates and standard errors of the #' expected log pointwise predictive density ([`elpd_loo`][loo-glossary]), the #' effective number of parameters ([`p_loo`][loo-glossary]) and the LOO #' information criterion `looic` (which is just `-2 * elpd_loo`, i.e., #' converted to deviance scale). #' } #' #' \item{`pointwise`}{ #' A matrix with five columns (and number of rows equal to the number of #' observations) containing the pointwise contributions of the measures #' (`elpd_loo`, `mcse_elpd_loo`, `p_loo`, `looic`, `influence_pareto_k`). #' in addition to the three measures in `estimates`, we also report #' pointwise values of the Monte Carlo standard error of [`elpd_loo`][loo-glossary] #' ([`mcse_elpd_loo`][loo-glossary]), and statistics describing the influence of #' each observation on the posterior distribution (`influence_pareto_k`). #' These are the estimates of the shape parameter \eqn{k} of the #' generalized Pareto fit to the importance ratios for each leave-one-out #' distribution (see the [pareto-k-diagnostic] page for details). #' } #' #' \item{`diagnostics`}{ #' A named list containing two vectors: #' * `pareto_k`: Importance sampling reliability diagnostics. By default, #' these are equal to the `influence_pareto_k` in `pointwise`. #' Some algorithms can improve importance sampling reliability and #' modify these diagnostics. See the [pareto-k-diagnostic] page for details. #' * `n_eff`: PSIS effective sample size estimates. #' } #' #' \item{`psis_object`}{ #' This component will be `NULL` unless the `save_psis` argument is set to #' `TRUE` when calling `loo()`. In that case `psis_object` will be the object #' of class `"psis"` that is created when the `loo()` function calls [psis()] #' internally to do the PSIS procedure. #' } #' } #' #' @seealso #' * The __loo__ package [vignettes](https://mc-stan.org/loo/articles/index.html) #' for demonstrations. #' * The [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. #' * [psis()] for the underlying Pareto Smoothed Importance Sampling (PSIS) #' procedure used in the LOO-CV approximation. #' * [pareto-k-diagnostic] for convenience functions for looking at diagnostics. #' * [loo_compare()] for model comparison. #' #' @template loo-and-psis-references #' #' @examples #' ### Array and matrix methods (using example objects included with loo package) #' # Array method #' LLarr <- example_loglik_array() #' rel_n_eff <- relative_eff(exp(LLarr)) #' loo(LLarr, r_eff = rel_n_eff, cores = 2) #' #' # Matrix method #' LLmat <- example_loglik_matrix() #' rel_n_eff <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = 500)) #' loo(LLmat, r_eff = rel_n_eff, cores = 2) #' #' #' ### Using log-likelihood function instead of array or matrix #' set.seed(124) #' #' # Simulate data and draw from posterior #' N <- 50; K <- 10; S <- 100; a0 <- 3; b0 <- 2 #' p <- rbeta(1, a0, b0) #' y <- rbinom(N, size = K, prob = p) #' a <- a0 + sum(y); b <- b0 + N * K - sum(y) #' fake_posterior <- as.matrix(rbeta(S, a, b)) #' dim(fake_posterior) # S x 1 #' fake_data <- data.frame(y,K) #' dim(fake_data) # N x 2 #' #' llfun <- function(data_i, draws) { #' # each time called internally within loo the arguments will be equal to: #' # data_i: ith row of fake_data (fake_data[i,, drop=FALSE]) #' # draws: entire fake_posterior matrix #' dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE) #' } #' #' # Use the loo_i function to check that llfun works on a single observation #' # before running on all obs. For example, using the 3rd obs in the data: #' loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) #' print(loo_3$pointwise[, "elpd_loo"]) #' #' # Use loo.function method (setting r_eff=NA since this posterior not obtained via MCMC) #' loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data, r_eff = NA) #' #' # If we look at the elpd_loo contribution from the 3rd obs it should be the #' # same as what we got above with the loo_i function and i=3: #' print(loo_with_fn$pointwise[3, "elpd_loo"]) #' print(loo_3$pointwise[, "elpd_loo"]) #' #' # Check that the loo.matrix method gives same answer as loo.function method #' log_lik_matrix <- sapply(1:N, function(i) { #' llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior) #' }) #' loo_with_mat <- loo(log_lik_matrix, r_eff = NA) #' all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE! #' #' #' \dontrun{ #' ### For package developers: defining loo methods #' #' # An example of a possible loo method for 'stanfit' objects (rstan package). #' # A similar method is included in the rstan package. #' # In order for users to be able to call loo(stanfit) instead of #' # loo.stanfit(stanfit) the NAMESPACE needs to be handled appropriately #' # (roxygen2 and devtools packages are good for that). #' # #' loo.stanfit <- #' function(x, #' pars = "log_lik", #' ..., #' save_psis = FALSE, #' cores = getOption("mc.cores", 1)) { #' stopifnot(length(pars) == 1L) #' LLarray <- loo::extract_log_lik(stanfit = x, #' parameter_name = pars, #' merge_chains = FALSE) #' r_eff <- loo::relative_eff(x = exp(LLarray), cores = cores) #' loo::loo.array(LLarray, #' r_eff = r_eff, #' cores = cores, #' save_psis = save_psis) #' } #' } #' #' loo <- function(x, ...) { UseMethod("loo") } #' @export #' @templateVar fn loo #' @template array #' loo.array <- function(x, ..., r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { if (is.null(r_eff)) throw_loo_r_eff_warning() is_method <- match.arg(is_method) psis_out <- importance_sampling.array(log_ratios = -x, r_eff = r_eff, cores = cores, method = is_method) ll <- llarray_to_matrix(x) pointwise <- pointwise_loo_calcs(ll, psis_out) importance_sampling_loo_object( pointwise = pointwise, diagnostics = psis_out$diagnostics, dims = dim(psis_out), is_method = is_method, is_object = if (save_psis) psis_out else NULL ) } #' @export #' @templateVar fn loo #' @template matrix #' loo.matrix <- function(x, ..., r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { is_method <- match.arg(is_method) if (is.null(r_eff)) { throw_loo_r_eff_warning() } psis_out <- importance_sampling.matrix( log_ratios = -x, r_eff = r_eff, cores = cores, method = is_method ) pointwise <- pointwise_loo_calcs(x, psis_out) importance_sampling_loo_object( pointwise = pointwise, diagnostics = psis_out$diagnostics, dims = dim(psis_out), is_method = is_method, is_object = if (save_psis) psis_out else NULL ) } #' @export #' @templateVar fn loo #' @template function #' @param data,draws,... For the `loo.function()` method and the `loo_i()` #' function, these are the data, posterior draws, and other arguments to pass #' to the log-likelihood function. See the **Methods (by class)** section #' below for details on how to specify these arguments. #' loo.function <- function(x, ..., data = NULL, draws = NULL, r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { is_method <- match.arg(is_method) cores <- loo_cores(cores) stopifnot(is.data.frame(data) || is.matrix(data), !is.null(draws)) assert_importance_sampling_method_is_implemented(is_method) .llfun <- validate_llfun(x) N <- dim(data)[1] if (is.null(r_eff)) { throw_loo_r_eff_warning() } else { r_eff <- prepare_psis_r_eff(r_eff, len = N) } psis_list <- parallel_importance_sampling_list( N = N, .loo_i = .loo_i, .llfun = .llfun, data = data, draws = draws, r_eff = r_eff, save_psis = save_psis, cores = cores, method = is_method, ... ) pointwise <- lapply(psis_list, "[[", "pointwise") if (save_psis) { psis_object_list <- lapply(psis_list, "[[", "psis_object") psis_out <- list2importance_sampling(psis_object_list) diagnostics <- psis_out$diagnostics } else { diagnostics_list <- lapply(psis_list, "[[", "diagnostics") diagnostics <- list( pareto_k = psis_apply(diagnostics_list, "pareto_k"), n_eff = psis_apply(diagnostics_list, "n_eff") ) } importance_sampling_loo_object( pointwise = do.call(rbind, pointwise), diagnostics = diagnostics, dims = c(attr(psis_list[[1]], "S"), N), is_method = is_method, is_object = if (save_psis) psis_out else NULL ) } #' @description The `loo_i()` function enables testing log-likelihood #' functions for use with the `loo.function()` method. #' #' @rdname loo #' @export #' #' @param i For `loo_i()`, an integer in `1:N`. #' @param llfun For `loo_i()`, the same as `x` for the #' `loo.function()` method. A log-likelihood function as described in the #' **Methods (by class)** section. #' #' @return The `loo_i()` function returns a named list with components #' `pointwise` and `diagnostics`. These components have the same #' structure as the `pointwise` and `diagnostics` components of the #' object returned by `loo()` except they contain results for only a single #' observation. #' loo_i <- function(i, llfun, ..., data = NULL, draws = NULL, r_eff = NULL, is_method = "psis" ) { stopifnot( i == as.integer(i), is.function(llfun) || is.character(llfun), is.data.frame(data) || is.matrix(data), i <= dim(data)[1], !is.null(draws), is_method %in% implemented_is_methods() ) .loo_i( i = as.integer(i), llfun = match.fun(llfun), data = data, draws = draws, r_eff = r_eff[i], save_psis = FALSE, is_method = is_method, ... ) } # Function that is passed to the FUN argument of lapply, mclapply, or parLapply # for the loo.function method. The arguments and return value are the same as # the ones documented above for the user-facing loo_i function. .loo_i <- function(i, llfun, ..., data, draws, r_eff = NULL, save_psis = FALSE, is_method) { if (!is.null(r_eff)) { r_eff <- r_eff[i] } d_i <- data[i, , drop = FALSE] ll_i <- llfun(data_i = d_i, draws = draws, ...) if (!is.matrix(ll_i)) { ll_i <- as.matrix(ll_i) } psis_out <- importance_sampling.matrix( log_ratios = -ll_i, r_eff = r_eff, cores = 1, method = is_method ) structure( list( pointwise = pointwise_loo_calcs(ll_i, psis_out), diagnostics = psis_out$diagnostics, psis_object = if (save_psis) psis_out else NULL ), S = dim(psis_out)[1], N = 1 ) } #' @export dim.loo <- function(x) { attr(x, "dims") } #' @rdname loo #' @export is.loo <- function(x) { inherits(x, "loo") } #' @export dim.psis_loo <- function(x) { attr(x, "dims") } #' @rdname loo #' @export is.psis_loo <- function(x) { inherits(x, "psis_loo") && is.loo(x) } # internal ---------------------------------------------------------------- #' Compute pointwise elpd_loo, p_loo, looic from log lik matrix and #' psis log weights #' #' @noRd #' @param ll Log-likelihood matrix. #' @param psis_object The object returned by `psis()`. #' @return Named list with pointwise elpd_loo, mcse_elpd_loo, p_loo, looic, #' and influence_pareto_k. #' pointwise_loo_calcs <- function(ll, psis_object) { if (!is.matrix(ll)) { ll <- as.matrix(ll) } lw <- weights(psis_object, normalize = TRUE, log = TRUE) elpd_loo <- matrixStats::colLogSumExps(ll + lw) lpd <- matrixStats::colLogSumExps(ll) - log(nrow(ll)) # colLogMeanExps p_loo <- lpd - elpd_loo mcse_elpd_loo <- mcse_elpd(ll, lw, E_elpd = elpd_loo, r_eff = relative_eff(psis_object)) looic <- -2 * elpd_loo influence_pareto_k <- psis_object$diagnostics$pareto_k cbind(elpd_loo, mcse_elpd_loo, p_loo, looic, influence_pareto_k) } #' Structure the object returned by the loo methods #' #' @noRd #' @param pointwise Matrix containing columns elpd_loo, mcse_elpd_loo, p_loo, #' looic, influence_pareto_k. #' @param diagnostics Named list containing vector `pareto_k` and vector `n_eff`. #' @param dims Log likelihood matrix dimensions (attribute of `"psis"` object). #' @template is_method #' @param is_object An object of class `"psis"/"tis"/"sis"`, as returned by the [psis()/tis()/sis()] function. #' @return A `'importance_sampling_loo'` object as described in the Value section of the [loo()] #' function documentation. #' importance_sampling_loo_object <- function(pointwise, diagnostics, dims, is_method, is_object = NULL) { if (!is.matrix(pointwise)) stop("Internal error ('pointwise' must be a matrix)") if (!is.list(diagnostics)) stop("Internal error ('diagnostics' must be a list)") assert_importance_sampling_method_is_implemented(is_method) cols_to_summarize <- !(colnames(pointwise) %in% c("mcse_elpd_loo", "influence_pareto_k")) estimates <- table_of_estimates(pointwise[, cols_to_summarize, drop=FALSE]) out <- nlist(estimates, pointwise, diagnostics) if (is.null(is_object)) { out[paste0(is_method, "_object")] <- list(NULL) } else { out[[paste0(is_method, "_object")]] <- is_object } # maintain backwards compatibility old_nms <- c("elpd_loo", "p_loo", "looic", "se_elpd_loo", "se_p_loo", "se_looic") out <- c(out, setNames(as.list(estimates), old_nms)) structure( out, dims = dims, class = c(paste0(is_method, "_loo"), "importance_sampling_loo", "loo") ) } #' Compute Monte Carlo standard error for ELPD #' #' @noRd #' @param ll Log-likelihood matrix. #' @param E_elpd elpd_loo column of pointwise matrix. #' @param psis_object Object returned by [psis()]. #' @param n_samples Number of draws to take from `Normal(E[epd_i], SD[epd_i])`. #' @return Vector of standard error estimates. #' mcse_elpd <- function(ll, lw, E_elpd, r_eff, n_samples = 1000) { lik <- exp(ll) w2 <- exp(lw)^2 E_epd <- exp(E_elpd) # zn is approximate ordered statistics of unit normal distribution with offset # recommended by Blom (1958) S <- n_samples c <- 3/8 r <- 1:n_samples zn <- qnorm((r - c) / (S - 2 * c + 1)) var_elpd <- vapply( seq_len(ncol(w2)), FUN.VALUE = numeric(1), FUN = function(i) { var_epd_i <- sum(w2[, i] * (lik[, i] - E_epd[i]) ^ 2) sd_epd_i <- sqrt(var_epd_i) z <- E_epd[i] + sd_epd_i * zn var(log(z[z > 0])) } ) sqrt(var_elpd / r_eff) } #' Warning message if r_eff not specified #' @noRd throw_loo_r_eff_warning <- function() { warning( "Relative effective sample sizes ('r_eff' argument) not specified.\n", "For models fit with MCMC, the reported PSIS effective sample sizes and \n", "MCSE estimates will be over-optimistic.", call. = FALSE ) } #' Combine many psis objects into a single psis object #' #' @noRd #' @param objects List of `"psis"` objects, each for a single observation. #' @return A single `"psis"` object. #' list2importance_sampling <- function(objects) { log_weights <- sapply(objects, "[[", "log_weights") diagnostics <- lapply(objects, "[[", "diagnostics") method <- psis_apply(objects, "method", fun = "attr", fun_val = character(1)) methods <- unique(method) if (length(methods) == 1) { method <- methods classes <- c(methods, "importance_sampling", "list") } else { classes <- c("importance_sampling", "list") } structure( list( log_weights = log_weights, diagnostics = list( pareto_k = psis_apply(diagnostics, item = "pareto_k"), n_eff = psis_apply(diagnostics, item = "n_eff") ) ), norm_const_log = psis_apply(objects, "norm_const_log", fun = "attr"), tail_len = psis_apply(objects, "tail_len", fun = "attr"), r_eff = psis_apply(objects, "r_eff", fun = "attr"), dims = dim(log_weights), method = method, class = classes ) } #' Extractor methods #' #' These are only defined in order to deprecate with a warning (rather than #' remove and break backwards compatibility) the old way of accessing the point #' estimates in a `"psis_loo"` or `"psis"` object. The new way as of #' v2.0.0 is to get them from the `"estimates"` component of the object. #' #' @name old-extractors #' @keywords internal #' @param x,i,exact,name See \link{Extract}. #' NULL #' @rdname old-extractors #' @keywords internal #' @export `[.loo` <- function(x, i) { flags <- c("elpd_loo", "se_elpd_loo", "p_loo", "se_p_loo", "looic", "se_looic", "elpd_waic", "se_elpd_waic", "p_waic", "se_p_waic", "waic", "se_waic") if (is.character(i)) { needs_warning <- which(flags == i) if (length(needs_warning)) { warning( "Accessing ", flags[needs_warning], " using '[' is deprecated ", "and will be removed in a future release. ", "Please extract the ", flags[needs_warning], " estimate from the 'estimates' component instead.", call. = FALSE ) } } NextMethod() } #' @rdname old-extractors #' @keywords internal #' @export `[[.loo` <- function(x, i, exact=TRUE) { flags <- c("elpd_loo", "se_elpd_loo", "p_loo", "se_p_loo", "looic", "se_looic", "elpd_waic", "se_elpd_waic", "p_waic", "se_p_waic", "waic", "se_waic") if (is.character(i)) { needs_warning <- which(flags == i) if (length(needs_warning)) { warning( "Accessing ", flags[needs_warning], " using '[[' is deprecated ", "and will be removed in a future release. ", "Please extract the ", flags[needs_warning], " estimate from the 'estimates' component instead.", call. = FALSE ) } } NextMethod() } #' @rdname old-extractors #' @keywords internal #' @export #' `$.loo` <- function(x, name) { flags <- c("elpd_loo", "se_elpd_loo", "p_loo", "se_p_loo", "looic", "se_looic", "elpd_waic", "se_elpd_waic", "p_waic", "se_p_waic", "waic", "se_waic") needs_warning <- which(flags == name) if (length(needs_warning)) { warning( "Accessing ", flags[needs_warning], " using '$' is deprecated ", "and will be removed in a future release. ", "Please extract the ", flags[needs_warning], " estimate from the 'estimates' component instead.", call. = FALSE ) } NextMethod() } #' Parallel psis list computations #' #' @details Refactored function to handle parallel computations #' for psis_list #' #' @keywords internal #' @inheritParams loo.function #' @param .loo_i The function used to compute individual loo contributions. #' @param .llfun See `llfun` in [loo.function()]. #' @param N The total number of observations (i.e. `nrow(data)`). #' @param method See `is_method` for [loo()] #' parallel_psis_list <- function(N, .loo_i, .llfun, data, draws, r_eff, save_psis, cores, ...){ parallel_importance_sampling_list(N, .loo_i, .llfun, data, draws, r_eff, save_psis, cores, method = "psis", ...) } #' @rdname parallel_psis_list parallel_importance_sampling_list <- function(N, .loo_i, .llfun, data, draws, r_eff, save_psis, cores, method, ...){ if (cores == 1) { psis_list <- lapply( X = seq_len(N), FUN = .loo_i, llfun = .llfun, data = data, draws = draws, r_eff = r_eff, save_psis = save_psis, is_method = method, ... ) } else { if (!os_is_windows()) { # On Mac or Linux use mclapply() for multiple cores psis_list <- parallel::mclapply( mc.cores = cores, X = seq_len(N), FUN = .loo_i, llfun = .llfun, data = data, draws = draws, r_eff = r_eff, save_psis = save_psis, is_method = method, ... ) } else { # On Windows use makePSOCKcluster() and parLapply() for multiple cores cl <- parallel::makePSOCKcluster(cores) on.exit(parallel::stopCluster(cl)) psis_list <- parallel::parLapply( cl = cl, X = seq_len(N), fun = .loo_i, llfun = .llfun, data = data, draws = draws, r_eff = r_eff, save_psis = save_psis, is_method = method, ... ) } } } loo/R/zzz.R0000644000176200001440000000122213575772017012237 0ustar liggesusers.onAttach <- function(...) { ver <- utils::packageVersion("loo") packageStartupMessage("This is loo version ", ver) packageStartupMessage( "- Online documentation and vignettes at mc-stan.org/loo" ) packageStartupMessage( "- As of v2.0.0 loo defaults to 1 core ", "but we recommend using as many as possible. ", "Use the 'cores' argument or set options(mc.cores = NUM_CORES) ", "for an entire session. " ) if (os_is_windows()) { packageStartupMessage( "- Windows 10 users: loo may be very slow if 'mc.cores' ", "is set in your .Rprofile file (see https://github.com/stan-dev/loo/issues/94)." ) } } loo/R/loo_compare.psis_loo_ss_list.R0000644000176200001440000001752613575772017017305 0ustar liggesusers#' Compare `psis_loo_ss` objects #' @noRd #' @param x A list with `psis_loo` objects. #' @param ... Currently ignored. #' @return A `compare.loo_ss` object. #' @author Mans Magnusson loo_compare.psis_loo_ss_list <- function(x, ...) { checkmate::assert_list(x, any.missing = FALSE, min.len = 1) for(i in seq_along(x)){ if (!inherits(x[[i]], "psis_loo_ss")) x[[i]] <- as.psis_loo_ss.psis_loo(x[[i]]) } loo_compare_checks.psis_loo_ss_list(x) comp <- loo_compare_matrix.psis_loo_ss_list(x) ord <- loo_compare_order(x) names(x) <- rownames(comp)[ord] rnms <- rownames(comp) elpd_diff_mat <- matrix(0, nrow = nrow(comp), ncol = 3, dimnames = list(rnms, c("elpd_diff", "se_diff", "subsampling_se_diff"))) for(i in 2:length(ord)){ elpd_diff_mat[i,] <- loo_compare_ss(ref_loo = x[ord[1]], compare_loo = x[ord[i]]) } comp <- cbind(elpd_diff_mat, comp) rownames(comp) <- rnms class(comp) <- c("compare.loo_ss", "compare.loo", class(comp)) return(comp) } #' Compare a reference loo object with a comaprison loo object #' @noRd #' @param ref_loo A named list with a `psis_loo_ss` object. #' @param compare_loo A named list with a `psis_loo_ss` object. #' @return A 1 by 3 elpd_diff estimation. loo_compare_ss <- function(ref_loo, compare_loo){ checkmate::assert_list(ref_loo, names = "named") checkmate::assert_list(compare_loo, names = "named") checkmate::assert_class(ref_loo[[1]], "psis_loo_ss") checkmate::assert_class(compare_loo[[1]], "psis_loo_ss") ref_idx <- obs_idx(ref_loo[[1]]) compare_idx <- obs_idx(compare_loo[[1]]) intersect_idx <- base::intersect(ref_idx, compare_idx) ref_subset_of_compare <- base::setequal(intersect_idx, ref_idx) compare_subset_of_ref <- base::setequal(intersect_idx, compare_idx) # Using HH estimation if (ref_loo[[1]]$loo_subsampling$estimator == "hh_pps" | compare_loo[[1]]$loo_subsampling$estimator == "hh_pps"){ warning("Hansen-Hurwitz estimator used. Naive diff SE is used.", call. = FALSE) return(loo_compare_ss_naive(ref_loo, compare_loo)) } # Same observations in both if (compare_subset_of_ref & ref_subset_of_compare){ return(loo_compare_ss_diff(ref_loo, compare_loo)) } # Use subset if (compare_subset_of_ref | ref_subset_of_compare){ if (compare_subset_of_ref) ref_loo[[1]] <- update(object = ref_loo[[1]], observations = compare_loo[[1]]) if (ref_subset_of_compare) compare_loo[[1]] <- update(compare_loo[[1]], observations = ref_loo[[1]]) message("Estimated elpd_diff using observations included in loo calculations for all models.") return(loo_compare_ss_diff(ref_loo, compare_loo)) } # If different samples if (!compare_subset_of_ref & !ref_subset_of_compare){ warning("Different subsamples in '", names(ref_loo), "' and '", names(compare_loo), "'. Naive diff SE is used.", call. = FALSE) return(loo_compare_ss_naive(ref_loo, compare_loo)) } } #' Compute a naive diff SE #' @noRd #' @inheritParams loo_compare_ss #' @return a 1 by 3 elpd_diff estimation loo_compare_ss_naive <- function(ref_loo, compare_loo){ checkmate::assert_list(ref_loo, names = "named") checkmate::assert_list(compare_loo, names = "named") checkmate::assert_class(ref_loo[[1]], "psis_loo_ss") checkmate::assert_class(compare_loo[[1]], "psis_loo_ss") elpd_loo_diff <- ref_loo[[1]]$estimates["elpd_loo","Estimate"] - compare_loo[[1]]$estimates["elpd_loo","Estimate"] elpd_loo_diff_se <- sqrt( (ref_loo[[1]]$estimates["elpd_loo","SE"])^2 + (compare_loo[[1]]$estimates["elpd_loo","SE"])^2) elpd_loo_diff_subsampling_se <- sqrt( (ref_loo[[1]]$estimates["elpd_loo","subsampling SE"])^2 + (compare_loo[[1]]$estimates["elpd_loo","subsampling SE"])^2) c(elpd_loo_diff, elpd_loo_diff_se, elpd_loo_diff_subsampling_se) } #' Compare a effective diff SE #' @noRd #' @inheritParams loo_compare_ss #' @return a 1 by 3 elpd_diff estimation loo_compare_ss_diff <- function(ref_loo, compare_loo){ checkmate::assert_list(ref_loo, names = "named") checkmate::assert_list(compare_loo, names = "named") checkmate::assert_class(ref_loo[[1]], "psis_loo_ss") checkmate::assert_class(compare_loo[[1]], "psis_loo_ss") checkmate::assert_true(identical(obs_idx(ref_loo[[1]]), obs_idx(compare_loo[[1]]))) # Assert not none as loo approximation checkmate::assert_true(ref_loo[[1]]$loo_subsampling$loo_approximation != "none") checkmate::assert_true(compare_loo[[1]]$loo_subsampling$loo_approximation != "none") diff_approx <- ref_loo[[1]]$loo_subsampling$elpd_loo_approx - compare_loo[[1]]$loo_subsampling$elpd_loo_approx diff_sample <- ref_loo[[1]]$pointwise[,"elpd_loo"] - compare_loo[[1]]$pointwise[,"elpd_loo"] est <- srs_diff_est(diff_approx, y = diff_sample, y_idx = ref_loo[[1]]$pointwise[,"idx"]) elpd_loo_diff <- est$y_hat elpd_loo_diff_se <- sqrt(est$hat_v_y) elpd_loo_diff_subsampling_se <- sqrt(est$v_y_hat) c(elpd_loo_diff, elpd_loo_diff_se, elpd_loo_diff_subsampling_se) } #' Check list of `psis_loo` objects #' @details Similar to `loo_compare_checks()` but checks dim size rather than #' pointwise dim since different pointwise sizes of `psis_loo_ss` will work. #' Can probably be removed by refactoring `loo_compare_checks()`. #' @noRd #' @inheritParams loo_compare_ss #' @return A 1 by 3 elpd_diff estimation. loo_compare_checks.psis_loo_ss_list <- function(loos) { ## errors if (length(loos) <= 1L) { stop("'loo_compare' requires at least two models.", call.=FALSE) } if (!all(sapply(loos, is.loo))) { stop("All inputs should have class 'loo'.", call.=FALSE) } Ns <- sapply(loos, function(x) x$loo_subsampling$data_dim[1]) if (!all(Ns == Ns[1L])) { stop("Not all models have the same number of data points.", call.=FALSE) } ## warnings yhash <- lapply(loos, attr, which = "yhash") yhash_ok <- sapply(yhash, function(x) { # ok only if all yhash are same (all NULL is ok) isTRUE(all.equal(x, yhash[[1]])) }) if (!all(yhash_ok)) { warning("Not all models have the same y variable. ('yhash' attributes do not match)", call. = FALSE) } if (all(sapply(loos, is.kfold))) { Ks <- unlist(lapply(loos, attr, which = "K")) if (!all(Ks == Ks[1])) { warning("Not all kfold objects have the same K value. ", "For a more accurate comparison use the same number of folds. ", call. = FALSE) } } else if (any(sapply(loos, is.kfold)) && any(sapply(loos, is.psis_loo))) { warning("Comparing LOO-CV to K-fold-CV. ", "For a more accurate comparison use the same number of folds ", "or loo for all models compared.", call. = FALSE) } } #' @rdname loo_compare #' @export print.compare.loo_ss <- function(x, ..., digits = 1, simplify = TRUE) { xcopy <- x if (inherits(xcopy, "old_compare.loo")) { if (NCOL(xcopy) >= 2 && simplify) { patts <- "^elpd_|^se_diff|^p_|^waic$|^looic$" xcopy <- xcopy[, grepl(patts, colnames(xcopy))] } } else if (NCOL(xcopy) >= 2 && simplify) { xcopy <- xcopy[, c("elpd_diff", "se_diff", "subsampling_se_diff")] } print(.fr(xcopy, digits), quote = FALSE) invisible(x) } #' Compute comparison matrix for `psis_loo_ss` objects #' @noRd #' @keywords internal #' @param loos List of `psis_loo_ss` objects. #' @return A `compare.loo_ss` matrix. loo_compare_matrix.psis_loo_ss_list <- function(loos){ tmp <- sapply(loos, function(x) { est <- x$estimates setNames(c(est), nm = c(rownames(est), paste0("se_", rownames(est)), paste0("subsampling_se_", rownames(est)))) }) colnames(tmp) <- find_model_names(loos) rnms <- rownames(tmp) comp <- tmp ord <- loo_compare_order(loos) comp <- t(comp)[ord, ] patts <- c("elpd", "p_", "^waic$|^looic$", "se_waic$|se_looic$") col_ord <- unlist(sapply(patts, function(p) grep(p, colnames(comp))), use.names = FALSE) comp <- comp[, col_ord] comp } loo/R/effective_sample_sizes.R0000644000176200001440000002174114411421542016110 0ustar liggesusers#' Convenience function for computing relative efficiencies #' #' `relative_eff()` computes the the MCMC effective sample size divided by #' the total sample size. #' #' @export #' @param x A vector, matrix, 3-D array, or function. See the **Methods (by #' class)** section below for details on specifying `x`, but where #' "log-likelihood" is mentioned replace it with one of the following #' depending on the use case: #' * For use with the [loo()] function, the values in `x` (or generated by #' `x`, if a function) should be **likelihood** values #' (i.e., `exp(log_lik)`), not on the log scale. #' * For generic use with [psis()], the values in `x` should be the reciprocal #' of the importance ratios (i.e., `exp(-log_ratios)`). #' @param chain_id A vector of length `NROW(x)` containing MCMC chain #' indexes for each each row of `x` (if a matrix) or each value in #' `x` (if a vector). No `chain_id` is needed if `x` is a 3-D #' array. If there are `C` chains then valid chain indexes are values #' in `1:C`. #' @param cores The number of cores to use for parallelization. #' @return A vector of relative effective sample sizes. #' #' @examples #' LLarr <- example_loglik_array() #' LLmat <- example_loglik_matrix() #' dim(LLarr) #' dim(LLmat) #' #' rel_n_eff_1 <- relative_eff(exp(LLarr)) #' rel_n_eff_2 <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = 500)) #' all.equal(rel_n_eff_1, rel_n_eff_2) #' relative_eff <- function(x, ...) { UseMethod("relative_eff") } #' @export #' @templateVar fn relative_eff #' @template vector #' relative_eff.default <- function(x, chain_id, ...) { dim(x) <- c(length(x), 1) class(x) <- "matrix" relative_eff.matrix(x, chain_id) } #' @export #' @templateVar fn relative_eff #' @template matrix #' relative_eff.matrix <- function(x, chain_id, ..., cores = getOption("mc.cores", 1)) { x <- llmatrix_to_array(x, chain_id) relative_eff.array(x, cores = cores) } #' @export #' @templateVar fn relative_eff #' @template array #' relative_eff.array <- function(x, ..., cores = getOption("mc.cores", 1)) { stopifnot(length(dim(x)) == 3) S <- prod(dim(x)[1:2]) # posterior sample size = iter * chains if (cores == 1) { n_eff_vec <- apply(x, 3, ess_rfun) } else { if (!os_is_windows()) { n_eff_list <- parallel::mclapply( mc.cores = cores, X = seq_len(dim(x)[3]), FUN = function(i) ess_rfun(x[, , i, drop = TRUE]) ) } else { cl <- parallel::makePSOCKcluster(cores) on.exit(parallel::stopCluster(cl)) n_eff_list <- parallel::parLapply( cl = cl, X = seq_len(dim(x)[3]), fun = function(i) ess_rfun(x[, , i, drop = TRUE]) ) } n_eff_vec <- unlist(n_eff_list, use.names = FALSE) } return(n_eff_vec / S) } #' @export #' @templateVar fn relative_eff #' @template function #' @param data,draws,... Same as for the [loo()] function method. #' relative_eff.function <- function(x, chain_id, ..., cores = getOption("mc.cores", 1), data = NULL, draws = NULL) { f_i <- validate_llfun(x) # not really an llfun, should return exp(ll) or exp(-ll) N <- dim(data)[1] if (cores == 1) { n_eff_list <- lapply( X = seq_len(N), FUN = function(i) { val_i <- f_i(data_i = data[i, , drop = FALSE], draws = draws, ...) relative_eff.default(as.vector(val_i), chain_id = chain_id, cores = 1) } ) } else { if (!os_is_windows()) { n_eff_list <- parallel::mclapply( X = seq_len(N), FUN = function(i) { val_i <- f_i(data_i = data[i, , drop = FALSE], draws = draws, ...) relative_eff.default(as.vector(val_i), chain_id = chain_id, cores = 1) }, mc.cores = cores ) } else { cl <- parallel::makePSOCKcluster(cores) parallel::clusterExport(cl=cl, varlist=c("draws", "chain_id", "data"), envir=environment()) on.exit(parallel::stopCluster(cl)) n_eff_list <- parallel::parLapply( cl = cl, X = seq_len(N), fun = function(i) { val_i <- f_i(data_i = data[i, , drop = FALSE], draws = draws, ...) relative_eff.default(as.vector(val_i), chain_id = chain_id, cores = 1) } ) } } n_eff_vec <- unlist(n_eff_list, use.names = FALSE) return(n_eff_vec) } #' @export #' @describeIn relative_eff #' If `x` is an object of class `"psis"`, `relative_eff()` simply returns #' the `r_eff` attribute of `x`. relative_eff.importance_sampling <- function(x, ...) { attr(x, "r_eff") } # internal ---------------------------------------------------------------- #' Effective sample size for PSIS #' #' @noRd #' @param w A vector or matrix (one column per observation) of normalized Pareto #' smoothed weights (not log weights). #' @param r_eff Relative effective sample size of `exp(log_lik)` or #' `exp(-log_ratios)`. `r_eff` should be a scalar if `w` is a #' vector and a vector of length `ncol(w)` if `w` is a matrix. #' @return A scalar if `w` is a vector. A vector of length `ncol(w)` #' if `w` is matrix. #' psis_n_eff <- function(w, ...) { UseMethod("psis_n_eff") } psis_n_eff.default <- function(w, r_eff = NULL, ...) { ss <- sum(w^2) if (is.null(r_eff)) { warning("PSIS n_eff not adjusted based on MCMC n_eff.", call. = FALSE) return(1 / ss) } stopifnot(length(r_eff) == 1) 1 / ss * r_eff } psis_n_eff.matrix <- function(w, r_eff = NULL, ...) { ss <- colSums(w^2) if (is.null(r_eff)) { warning("PSIS n_eff not adjusted based on MCMC n_eff.", call. = FALSE) return(1 / ss) } if (length(r_eff) != length(ss)) stop("r_eff must have length ncol(w).", call. = FALSE) 1 / ss * r_eff } #' MCMC effective sample size calculation #' #' @noRd #' @param sims An iterations by chains matrix of draws for a single parameter. #' In the case of the **loo** package, this will be the **exponentiated** #' log-likelihood values for the ith observation. #' @return MCMC effective sample size based on RStan's calculation. #' ess_rfun <- function(sims) { if (is.vector(sims)) dim(sims) <- c(length(sims), 1) chains <- ncol(sims) n_samples <- nrow(sims) if (requireNamespace("posterior", quietly = TRUE)) { acov <- lapply(1:chains, FUN = function(i) posterior::autocovariance(sims[,i])) } else { acov <- lapply(1:chains, FUN = function(i) autocovariance(sims[,i])) } acov <- do.call(cbind, acov) chain_mean <- colMeans(sims) mean_var <- mean(acov[1,]) * n_samples / (n_samples - 1) var_plus <- mean_var * (n_samples - 1) / n_samples if (chains > 1) var_plus <- var_plus + var(chain_mean) # Geyer's initial positive sequence rho_hat_t <- rep.int(0, n_samples) t <- 0 rho_hat_even <- 1 rho_hat_t[t + 1] <- rho_hat_even rho_hat_odd <- 1 - (mean_var - mean(acov[t + 2, ])) / var_plus rho_hat_t[t + 2] <- rho_hat_odd while (t < nrow(acov) - 5 && !is.nan(rho_hat_even + rho_hat_odd) && (rho_hat_even + rho_hat_odd > 0)) { t <- t + 2 rho_hat_even = 1 - (mean_var - mean(acov[t + 1, ])) / var_plus rho_hat_odd = 1 - (mean_var - mean(acov[t + 2, ])) / var_plus if ((rho_hat_even + rho_hat_odd) >= 0) { rho_hat_t[t + 1] <- rho_hat_even rho_hat_t[t + 2] <- rho_hat_odd } } max_t <- t # this is used in the improved estimate if (rho_hat_even>0) rho_hat_t[max_t + 1] <- rho_hat_even # Geyer's initial monotone sequence t <- 0 while (t <= max_t - 4) { t <- t + 2 if (rho_hat_t[t + 1] + rho_hat_t[t + 2] > rho_hat_t[t - 1] + rho_hat_t[t]) { rho_hat_t[t + 1] = (rho_hat_t[t - 1] + rho_hat_t[t]) / 2; rho_hat_t[t + 2] = rho_hat_t[t + 1]; } } ess <- chains * n_samples # Geyer's truncated estimate # tau_hat <- -1 + 2 * sum(rho_hat_t[1:max_t]) # Improved estimate reduces variance in antithetic case tau_hat <- -1 + 2 * sum(rho_hat_t[1:max_t]) + rho_hat_t[max_t+1] # Safety check for negative values and with max ess equal to ess*log10(ess) tau_hat <- max(tau_hat, 1/log10(ess)) ess <- ess / tau_hat ess } fft_next_good_size <- function(N) { # Find the optimal next size for the FFT so that # a minimum number of zeros are padded. if (N <= 2) return(2) while (TRUE) { m = N while ((m %% 2) == 0) m = m / 2 while ((m %% 3) == 0) m = m / 3 while ((m %% 5) == 0) m = m / 5 if (m <= 1) return(N) N = N + 1 } } # autocovariance function to use if posterior::autocovariance is not available autocovariance <- function(y) { # Compute autocovariance estimates for every lag for the specified # input sequence using a fast Fourier transform approach. N <- length(y) M <- fft_next_good_size(N) Mt2 <- 2 * M yc <- y - mean(y) yc <- c(yc, rep.int(0, Mt2-N)) transform <- stats::fft(yc) ac <- stats::fft(Conj(transform) * transform, inverse = TRUE) # use "biased" estimate as recommended by Geyer (1992) ac <- Re(ac)[1:N] / (N^2 * 2) ac } loo/R/helpers.R0000644000176200001440000001161714372466116013051 0ustar liggesusers#' Detect if OS is Windows #' @noRd os_is_windows <- function() { checkmate::test_os("windows") } #' More stable version of `log(mean(exp(x)))` #' #' @noRd #' @param x A numeric vector. #' @return A scalar equal to `log(mean(exp(x)))`. #' logMeanExp <- function(x) { logS <- log(length(x)) matrixStats::logSumExp(x) - logS } #' More stable version of `log(colMeans(exp(x)))` #' #' @noRd #' @param x A matrix. #' @return A vector where each element is `logMeanExp()` of a column of `x`. #' colLogMeanExps <- function(x) { logS <- log(nrow(x)) matrixStats::colLogSumExps(x) - logS } #' Compute point estimates and standard errors from pointwise vectors #' #' @noRd #' @param x A matrix. #' @return An `ncol(x)` by 2 matrix with columns `"Estimate"` and `"SE"` #' and rownames equal to `colnames(x)`. #' table_of_estimates <- function(x) { out <- cbind( Estimate = matrixStats::colSums2(x), SE = sqrt(nrow(x) * matrixStats::colVars(x)) ) rownames(out) <- colnames(x) return(out) } # validating and reshaping arrays/matrices ------------------------------- #' Check for `NA` and non-finite values in log-lik (or log-ratios) #' array/matrix/vector #' #' @noRd #' @param x Array/matrix/vector of log-likelihood or log-ratio values. #' @return `x`, invisibly, if no error is thrown. #' validate_ll <- function(x) { if (is.list(x)) { stop("List not allowed as input.") } else if (anyNA(x)) { stop("NAs not allowed in input.") } else if (!all(is.finite(x))) { stop("All input values must be finite.") } invisible(x) } #' Convert iter by chain by obs array to (iter * chain) by obs matrix #' #' @noRd #' @param x Array to convert. #' @return An (iter * chain) by obs matrix. #' llarray_to_matrix <- function(x) { stopifnot(is.array(x), length(dim(x)) == 3) xdim <- dim(x) dim(x) <- c(prod(xdim[1:2]), xdim[3]) unname(x) } #' Convert (iter * chain) by obs matrix to iter by chain by obs array #' #' @noRd #' @param x matrix to convert. #' @param chain_id vector of chain ids. #' @return iter by chain by obs array #' llmatrix_to_array <- function(x, chain_id) { stopifnot(is.matrix(x), all(chain_id == as.integer(chain_id))) lldim <- dim(x) n_chain <- length(unique(chain_id)) chain_id <- as.integer(chain_id) chain_counts <- as.numeric(table(chain_id)) if (length(chain_id) != lldim[1]) { stop("Number of rows in matrix not equal to length(chain_id).", call. = FALSE) } else if (any(chain_counts != chain_counts[1])) { stop("Not all chains have same number of iterations.", call. = FALSE) } else if (max(chain_id) != n_chain) { stop("max(chain_id) not equal to the number of chains.", call. = FALSE) } n_iter <- lldim[1] / n_chain n_obs <- lldim[2] a <- array(data = NA, dim = c(n_iter, n_chain, n_obs)) for (c in seq_len(n_chain)) { a[, c, ] <- x[chain_id == c, , drop = FALSE] } return(a) } #' Validate that log-lik function exists and has correct arg names #' #' @noRd #' @param x A function with arguments `data_i` and `draws`. #' @return Either returns `x` or throws an error. #' validate_llfun <- function(x) { f <- match.fun(x) must_have <- c("data_i", "draws") arg_names <- names(formals(f)) if (!all(must_have %in% arg_names)) { stop( "Log-likelihood function must have at least the arguments ", "'data_i' and 'draws'", call. = FALSE ) } return(f) } #' Named lists #' #' Create a named list using specified names or, if names are omitted, using the #' names of the objects in the list. The code `list(a = a, b = b)` becomes #' `nlist(a,b)` and `list(a = a, b = 2)` becomes `nlist(a, b = 2)`, etc. #' #' @export #' @keywords internal #' @param ... Objects to include in the list. #' @return A named list. #' @examples #' #' # All variables already defined #' a <- rnorm(100) #' b <- mat.or.vec(10, 3) #' nlist(a,b) #' #' # Define some variables in the call and take the rest from the environment #' nlist(a, b, veggies = c("lettuce", "spinach"), fruits = c("banana", "papaya")) #' nlist <- function(...) { m <- match.call() out <- list(...) no_names <- is.null(names(out)) has_name <- if (no_names) FALSE else nzchar(names(out)) if (all(has_name)) return(out) nms <- as.character(m)[-1L] if (no_names) { names(out) <- nms } else { names(out)[!has_name] <- nms[!has_name] } return(out) } # Check how many cores to use and throw deprecation warning if loo.cores is used loo_cores <- function(cores) { loo_cores_op <- getOption("loo.cores", NA) if (!is.na(loo_cores_op) && (loo_cores_op != cores)) { cores <- loo_cores_op warning("'loo.cores' is deprecated, please use 'mc.cores' or pass 'cores' explicitly.", call. = FALSE) } return(cores) } # nocov start # release reminders (for devtools) release_questions <- function() { c( "Have you updated references?", "Have you updated inst/CITATION?", "Have you updated the vignettes?" ) } # nocov end loo/R/gpdfit.R0000644000176200001440000000624614407123455012661 0ustar liggesusers#' Estimate parameters of the Generalized Pareto distribution #' #' Given a sample \eqn{x}, Estimate the parameters \eqn{k} and \eqn{\sigma} of #' the generalized Pareto distribution (GPD), assuming the location parameter is #' 0. By default the fit uses a prior for \eqn{k}, which will stabilize #' estimates for very small sample sizes (and low effective sample sizes in the #' case of MCMC samples). The weakly informative prior is a Gaussian prior #' centered at 0.5. #' #' @export #' @param x A numeric vector. The sample from which to estimate the parameters. #' @param wip Logical indicating whether to adjust \eqn{k} based on a weakly #' informative Gaussian prior centered on 0.5. Defaults to `TRUE`. #' @param min_grid_pts The minimum number of grid points used in the fitting #' algorithm. The actual number used is `min_grid_pts + floor(sqrt(length(x)))`. #' @param sort_x If `TRUE` (the default), the first step in the fitting #' algorithm is to sort the elements of `x`. If `x` is already #' sorted in ascending order then `sort_x` can be set to `FALSE` to #' skip the initial sorting step. #' @return A named list with components `k` and `sigma`. #' #' @details Here the parameter \eqn{k} is the negative of \eqn{k} in Zhang & #' Stephens (2009). #' #' @seealso [psis()], [pareto-k-diagnostic] #' #' @references #' Zhang, J., and Stephens, M. A. (2009). A new and efficient estimation method #' for the generalized Pareto distribution. *Technometrics* **51**, 316-325. #' gpdfit <- function(x, wip = TRUE, min_grid_pts = 30, sort_x = TRUE) { # See section 4 of Zhang and Stephens (2009) if (sort_x) { x <- sort.int(x) } N <- length(x) prior <- 3 M <- min_grid_pts + floor(sqrt(N)) jj <- seq_len(M) xstar <- x[floor(N / 4 + 0.5)] # first quartile of sample theta <- 1 / x[N] + (1 - sqrt(M / (jj - 0.5))) / prior / xstar l_theta <- N * lx(theta, x) # profile log-lik w_theta <- exp(l_theta - matrixStats::logSumExp(l_theta)) # normalize theta_hat <- sum(theta * w_theta) k <- mean.default(log1p(-theta_hat * x)) sigma <- -k / theta_hat if (wip) { k <- adjust_k_wip(k, n = N) } if (is.nan(k)) { k <- Inf } nlist(k, sigma) } # internal ---------------------------------------------------------------- lx <- function(a,x) { a <- -a k <- vapply(a, FUN = function(a_i) mean(log1p(a_i * x)), FUN.VALUE = numeric(1)) log(a / k) - k - 1 } #' Adjust k based on weakly informative prior, Gaussian centered on 0.5. This #' will stabilize estimates for very small Monte Carlo sample sizes and low neff #' cases. #' #' @noRd #' @param k Scalar khat estimate. #' @param n Integer number of tail samples used to fit GPD. #' @return Scalar adjusted khat estimate. #' adjust_k_wip <- function(k, n) { a <- 10 n_plus_a <- n + a k * n / n_plus_a + a * 0.5 / n_plus_a } #' Inverse CDF of generalized pareto distribution #' (assuming location parameter is 0) #' #' @noRd #' @param p Vector of probabilities. #' @param k Scalar shape parameter. #' @param sigma Scalar scale parameter. #' @return Vector of quantiles. #' qgpd <- function(p, k, sigma) { if (is.nan(sigma) || sigma <= 0) { return(rep(NaN, length(p))) } sigma * expm1(-k * log1p(-p)) / k } loo/R/loo-package.R0000644000176200001440000001027114407123455013557 0ustar liggesusers#' Efficient LOO-CV and WAIC for Bayesian models #' #' @docType package #' @name loo-package #' #' @importFrom stats sd var quantile setNames weights rnorm qnorm #' @importFrom matrixStats logSumExp colLogSumExps colSums2 colVars colMaxs #' #' @description #' \if{html}{ #' \figure{stanlogo.png}{options: width="50" alt="mc-stan.org"} #' } #' *Stan Development Team* #' #' This package implements the methods described in Vehtari, Gelman, and #' Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2019), and #' Yao et al. (2018). To get started see the **loo** package #' [vignettes](https://mc-stan.org/loo/articles/index.html), the #' [loo()] function for efficient approximate leave-one-out #' cross-validation (LOO-CV), the [psis()] function for the Pareto #' smoothed importance sampling (PSIS) algorithm, or #' [loo_model_weights()] for an implementation of Bayesian stacking of #' predictive distributions from multiple models. #' #' #' @details Leave-one-out cross-validation (LOO-CV) and the widely applicable #' information criterion (WAIC) are methods for estimating pointwise #' out-of-sample prediction accuracy from a fitted Bayesian model using the #' log-likelihood evaluated at the posterior simulations of the parameter #' values. LOO-CV and WAIC have various advantages over simpler estimates of #' predictive error such as AIC and DIC but are less used in practice because #' they involve additional computational steps. This package implements the #' fast and stable computations for approximate LOO-CV laid out in Vehtari, #' Gelman, and Gabry (2017). From existing posterior simulation draws, we #' compute LOO-CV using Pareto smoothed importance sampling (PSIS; Vehtari, #' Simpson, Gelman, Yao, and Gabry, 2019), a new procedure for stabilizing #' and diagnosing importance weights. As a byproduct of our calculations, #' we also obtain approximate standard errors for estimated predictive #' errors and for comparing of predictive errors between two models. #' #' We recommend PSIS-LOO-CV instead of WAIC, because PSIS provides useful #' diagnostics and effective sample size and Monte Carlo standard error #' estimates. #' #' #' @template loo-and-psis-references #' @template stacking-references #' @template loo-large-data-references #' #' @references #' Epifani, I., MacEachern, S. N., and Peruggia, M. (2008). Case-deletion #' importance sampling estimators: Central limit theorems and related results. #' *Electronic Journal of Statistics* **2**, 774-806. #' #' Gelfand, A. E. (1996). Model determination using sampling-based methods. In #' *Markov Chain Monte Carlo in Practice*, ed. W. R. Gilks, S. Richardson, #' D. J. Spiegelhalter, 145-162. London: Chapman and Hall. #' #' Gelfand, A. E., Dey, D. K., and Chang, H. (1992). Model determination using #' predictive distributions with implementation via sampling-based methods. In #' *Bayesian Statistics 4*, ed. J. M. Bernardo, J. O. Berger, A. P. Dawid, #' and A. F. M. Smith, 147-167. Oxford University Press. #' #' Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive #' information criteria for Bayesian models. *Statistics and Computing* #' **24**, 997-1016. #' #' Ionides, E. L. (2008). Truncated importance sampling. *Journal of #' Computational and Graphical Statistics* **17**, 295-311. #' #' Koopman, S. J., Shephard, N., and Creal, D. (2009). Testing the assumptions #' behind importance sampling. *Journal of Econometrics* **149**, 2-11. #' #' Peruggia, M. (1997). On the variability of case-deletion importance sampling #' weights in the Bayesian linear model. *Journal of the American #' Statistical Association* **92**, 199-207. #' #' Stan Development Team (2017). The Stan C++ Library, Version 2.17.0. #' . #' #' Stan Development Team (2018). RStan: the R interface to Stan, Version 2.17.3. #' . #' #' Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and #' widely application information criterion in singular learning theory. #' *Journal of Machine Learning Research* **11**, 3571-3594. #' #' Zhang, J., and Stephens, M. A. (2009). A new and efficient estimation method #' for the generalized Pareto distribution. *Technometrics* **51**, #' 316-325. #' NULL loo/R/loo-glossary.R0000644000176200001440000001430614214417101014020 0ustar liggesusers#' LOO package glossary #' #' @name loo-glossary #' #' @template loo-and-psis-references #' @template bayesvis-reference #' #' @description #' The pages provides definitions to key terms. Also see the #' [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. #' #' Note: VGG2017 refers to Vehtari, Gelman, and Gabry (2017). See #' **References**, below. #' #' @section ELPD and `elpd_loo`: #' #' The ELPD is the theoretical expected log pointwise predictive density for a new #' dataset (Eq 1 in VGG2017), which can be estimated, e.g., using #' cross-validation. `elpd_loo` is the Bayesian LOO estimate of the #' expected log pointwise predictive density (Eq 4 in VGG2017) and #' is a sum of N individual pointwise log predictive densities. Probability #' densities can be smaller or larger than 1, and thus log predictive densities #' can be negative or positive. For simplicity the ELPD acronym is used also for #' expected log pointwise predictive probabilities for discrete models. #' Probabilities are always equal or less than 1, and thus log predictive #' probabilities are 0 or negative. #' #' @section Standard error of `elpd_loo`: #' #' As `elpd_loo` is defined as the sum of N independent components (Eq 4 in #' VGG2017), we can compute the standard error by using the standard deviation #' of the N components and multiplying by `sqrt(N)` (Eq 23 in VGG2017). #' This standard error is a coarse description of our uncertainty about the #' predictive performance for unknown future data. When N is small or there is #' severe model misspecification, the current SE estimate is overoptimistic and #' the actual SE can even be twice as large. Even for moderate N, when the SE #' estimate is an accurate estimate for the scale, it ignores the skewness. When #' making model comparisons, the SE of the component-wise (pairwise) differences #' should be used instead (see the `se_diff` section below and Eq 24 in #' VGG2017). #' #' @section Monte Carlo SE of elpd_loo: #' #' The Monte Carlo standard error is the estimate for the computational accuracy #' of MCMC and importance sampling used to compute `elpd_loo`. Usually this #' is negligible compared to the standard describing the uncertainty due to #' finite number of observations (Eq 23 in VGG2017). #' #' @section `p_loo` (effective number of parameters): #' #' `p_loo` is the difference between `elpd_loo` and the non-cross-validated #' log posterior predictive density. It describes how much more difficult it #' is to predict future data than the observed data. Asymptotically under #' certain regularity conditions, `p_loo` can be interpreted as the #' *effective number of parameters*. In well behaving cases `p_loo < N` and #' `p_loo < p`, where `p` is the total number of parameters in the #' model. `p_loo > N` or `p_loo > p` indicates that the model has very #' weak predictive capability and may indicate a severe model misspecification. #' See below for more on interpreting `p_loo` when there are warnings #' about high Pareto k diagnostic values. #' #' @section Pareto k estimates: #' #' The Pareto `k` estimate is a diagnostic for Pareto smoothed importance #' sampling (PSIS), which is used to compute components of `elpd_loo`. In #' importance-sampling LOO (the full posterior distribution is used as the #' proposal distribution). The Pareto k diagnostic estimates how far an #' individual leave-one-out distribution is from the full distribution. If #' leaving out an observation changes the posterior too much then importance #' sampling is not able to give reliable estimate. If `k<0.5`, then the #' corresponding component of `elpd_loo` is estimated with high accuracy. #' If `0.50.7`, #' then importance sampling is not able to provide useful estimate for that #' component/observation. Pareto k is also useful as a measure of influence of #' an observation. Highly influential observations have high k values. Very high #' k values often indicate model misspecification, outliers or mistakes in data #' processing. See Section 6 of Gabry et al. (2019) for an example. #' #' \subsection{Interpreting `p_loo` when Pareto `k` is large}{ #' If `k > 0.7` then we can also look at the `p_loo` estimate for #' some additional information about the problem: #' #' \itemize{ #' \item If `p_loo << p` (the total number of parameters in the model), #' then the model is likely to be misspecified. Posterior predictive checks #' (PPCs) are then likely to also detect the problem. Try using an overdispersed #' model, or add more structural information (nonlinearity, mixture model, #' etc.). #' #' \item If `p_loo < p` and the number of parameters `p` is relatively #' large compared to the number of observations (e.g., `p>N/5`), it is #' likely that the model is so flexible or the population prior so weak that it’s #' difficult to predict the left out observation (even for the true model). #' This happens, for example, in the simulated 8 schools (in VGG2017), random #' effect models with a few observations per random effect, and Gaussian #' processes and spatial models with short correlation lengths. #' #' \item If `p_loo > p`, then the model is likely to be badly misspecified. #' If the number of parameters `p< for an example. #' If `p` is relatively large compared to the number of #' observations, say `p>N/5` (more accurately we should count number of #' observations influencing each parameter as in hierarchical models some groups #' may have few observations and other groups many), it is possible that PPCs won't #' detect the problem. #' } #' } #' #' @section elpd_diff: #' `elpd_diff` is the difference in `elpd_loo` for two models. If more #' than two models are compared, the difference is computed relative to the #' model with highest `elpd_loo`. #' #' @section se_diff: #' #' The standard error of component-wise differences of elpd_loo (Eq 24 in #' VGG2017) between two models. This SE is *smaller* than the SE for #' individual models due to correlation (i.e., if some observations are easier #' and some more difficult to predict for all models). #' NULL loo/R/psis_approximate_posterior.R0000644000176200001440000001066213575772017017107 0ustar liggesusers#' Diagnostics for Laplace and ADVI approximations and Laplace-loo and ADVI-loo #' #' @param log_p The log-posterior (target) evaluated at S samples from the #' proposal distribution (g). A vector of length S. #' @param log_g The log-density (proposal) evaluated at S samples from the #' proposal distribution (g). A vector of length S. #' @param log_q Deprecated argument name (the same as log_g). #' @param log_liks A log-likelihood matrix of size S * N, where N is the number #' of observations and S is the number of samples from q. See #' [loo.matrix()] for details. Default is `NULL`. Then only the #' posterior is evaluated using the k_hat diagnostic. #' @inheritParams loo #' #' @return #' If log likelihoods are supplied, the function returns a `"loo"` object, #' otherwise the function returns a `"psis"` object. #' #' @seealso [loo()] and [psis()] #' #' @template loo-and-psis-references #' #' @keywords internal #' psis_approximate_posterior <- function(log_p = NULL, log_g = NULL, log_liks = NULL, cores, save_psis, ..., log_q = NULL) { if (!is.null(log_q)) { .Deprecated(msg = "psis_approximate_posterior() argument log_q has been changed to log_g") log_g <- log_q } checkmate::assert_numeric(log_p, any.missing = FALSE, len = length(log_g), null.ok = FALSE) checkmate::assert_numeric(log_g, any.missing = FALSE, len = length(log_p), null.ok = FALSE) checkmate::assert_matrix(log_liks, null.ok = TRUE, nrows = length(log_p)) checkmate::assert_integerish(cores) checkmate::assert_flag(save_psis) if (is.null(log_liks)) { approx_correction <- log_p - log_g # Handle underflow/overflow approx_correction <- approx_correction - max(approx_correction) log_ratios <- matrix(approx_correction, ncol = 1) } else { log_ratios <- correct_log_ratios(log_ratios = -log_liks, log_p = log_p, log_g = log_g) } psis_out <- psis.matrix(log_ratios, cores = cores, r_eff = rep(1, ncol(log_ratios))) if (is.null(log_liks)) { return(psis_out) } pointwise <- pointwise_loo_calcs(log_liks, psis_out) importance_sampling_loo_object( pointwise = pointwise, diagnostics = psis_out$diagnostics, dims = dim(psis_out), is_method = "psis", is_object = if (save_psis) psis_out else NULL ) } #' Correct log ratios for posterior approximations #' #' @inheritParams psis_approximate_posterior #' @inheritParams ap_psis #' @noRd #' @keywords internal correct_log_ratios <- function(log_ratios, log_p, log_g) { approx_correction <- log_p - log_g log_ratios <- log_ratios + approx_correction # Handle underflow/overflow log_ratio_max <- apply(log_ratios, 2, max) log_ratios <- sweep(log_ratios, MARGIN = 2, STATS = log_ratio_max) log_ratios } #' Pareto smoothed importance sampling (PSIS) #' using approximate posteriors #' @inheritParams psis_approximate_posterior #' @param log_ratios The log-likelihood ratios (ie -log_liks) #' @param ... Currently not in use. ap_psis <- function(log_ratios, log_p, log_g, ...) { UseMethod("ap_psis") } #' @export #' @templateVar fn ap_psis #' @template array #' ap_psis.array <- function(log_ratios, log_p, log_g, ..., cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) stopifnot(length(dim(log_ratios)) == 3) log_ratios <- validate_ll(log_ratios) log_ratios <- llarray_to_matrix(log_ratios) r_eff <- prepare_psis_r_eff(r_eff, len = ncol(log_ratios)) ap_psis.matrix(log_ratios = log_ratios, log_p = log_p, log_g = log_g, cores = 1) } #' @export #' @templateVar fn ap_psis #' @template matrix #' ap_psis.matrix <- function(log_ratios, log_p, log_g, ..., cores = getOption("mc.cores", 1)) { checkmate::assert_numeric(log_p, len = nrow(log_ratios)) checkmate::assert_numeric(log_g, len = nrow(log_ratios)) cores <- loo_cores(cores) log_ratios <- validate_ll(log_ratios) log_ratios <- correct_log_ratios(log_ratios, log_p = log_p, log_g = log_g) do_psis(log_ratios, r_eff = rep(1, ncol(log_ratios)), cores = cores) } #' @export #' @templateVar fn ap_psis #' @template vector #' ap_psis.default <- function(log_ratios, log_p, log_g, ...) { stopifnot(is.null(dim(log_ratios)) || length(dim(log_ratios)) == 1) dim(log_ratios) <- c(length(log_ratios), 1) warning("llfun values do not return a matrix, coerce to matrix") ap_psis.matrix(as.matrix(log_ratios), log_p, log_g, cores = 1) } loo/R/psis.R0000644000176200001440000003171714407123455012363 0ustar liggesusers#' Pareto smoothed importance sampling (PSIS) #' #' Implementation of Pareto smoothed importance sampling (PSIS), a method for #' stabilizing importance ratios. The version of PSIS implemented here #' corresponds to the algorithm presented in Vehtari, Simpson, Gelman, Yao, #' and Gabry (2019). #' For PSIS diagnostics see the [pareto-k-diagnostic] page. #' #' @export #' @param log_ratios An array, matrix, or vector of importance ratios on the log #' scale (for PSIS-LOO these are *negative* log-likelihood values). See the #' **Methods (by class)** section below for a detailed description of how #' to specify the inputs for each method. #' @param ... Arguments passed on to the various methods. #' @template cores #' @param r_eff Vector of relative effective sample size estimates containing #' one element per observation. The values provided should be the relative #' effective sample sizes of `1/exp(log_ratios)` (i.e., `1/ratios`). #' This is related to the relative efficiency of estimating the normalizing #' term in self-normalizing importance sampling. If `r_eff` is not #' provided then the reported PSIS effective sample sizes and Monte Carlo #' error estimates will be over-optimistic. See the [relative_eff()] #' helper function for computing `r_eff`. If using `psis` with #' draws of the `log_ratios` not obtained from MCMC then the warning #' message thrown when not specifying `r_eff` can be disabled by #' setting `r_eff` to `NA`. #' #' @return The `psis()` methods return an object of class `"psis"`, #' which is a named list with the following components: #' #' \describe{ #' \item{`log_weights`}{ #' Vector or matrix of smoothed (and truncated) but *unnormalized* log #' weights. To get normalized weights use the #' [`weights()`][weights.importance_sampling] method provided for objects of #' class `"psis"`. #' } #' \item{`diagnostics`}{ #' A named list containing two vectors: #' * `pareto_k`: Estimates of the shape parameter \eqn{k} of the #' generalized Pareto distribution. See the [pareto-k-diagnostic] #' page for details. #' * `n_eff`: PSIS effective sample size estimates. #' } #' } #' #' Objects of class `"psis"` also have the following [attributes][attributes()]: #' \describe{ #' \item{`norm_const_log`}{ #' Vector of precomputed values of `colLogSumExps(log_weights)` that are #' used internally by the `weights` method to normalize the log weights. #' } #' \item{`tail_len`}{ #' Vector of tail lengths used for fitting the generalized Pareto distribution. #' } #' \item{`r_eff`}{ #' If specified, the user's `r_eff` argument. #' } #' \item{`dims`}{ #' Integer vector of length 2 containing `S` (posterior sample size) #' and `N` (number of observations). #' } #' \item{`method`}{ #' Method used for importance sampling, here `psis`. #' } #' } #' #' @seealso #' * [loo()] for approximate LOO-CV using PSIS. #' * [pareto-k-diagnostic] for PSIS diagnostics. #' * The __loo__ package [vignettes](https://mc-stan.org/loo/articles/index.html) #' for demonstrations. #' * The [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. #' #' @template loo-and-psis-references #' #' @examples #' log_ratios <- -1 * example_loglik_array() #' r_eff <- relative_eff(exp(-log_ratios)) #' psis_result <- psis(log_ratios, r_eff = r_eff) #' str(psis_result) #' plot(psis_result) #' #' # extract smoothed weights #' lw <- weights(psis_result) # default args are log=TRUE, normalize=TRUE #' ulw <- weights(psis_result, normalize=FALSE) # unnormalized log-weights #' #' w <- weights(psis_result, log=FALSE) # normalized weights (not log-weights) #' uw <- weights(psis_result, log=FALSE, normalize = FALSE) # unnormalized weights #' #' #' psis <- function(log_ratios, ...) UseMethod("psis") #' @export #' @templateVar fn psis #' @template array #' psis.array <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.array(log_ratios = log_ratios, ..., r_eff = r_eff, cores = cores, method = "psis") } #' @export #' @templateVar fn psis #' @template matrix #' psis.matrix <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.matrix(log_ratios, ..., r_eff = r_eff, cores = cores, method = "psis") } #' @export #' @templateVar fn psis #' @template vector #' psis.default <- function(log_ratios, ..., r_eff = NULL) { importance_sampling.default(log_ratios = log_ratios, ..., r_eff = r_eff, method = "psis") } #' @rdname psis #' @export #' @param x For `is.psis()`, an object to check. is.psis <- function(x) { inherits(x, "psis") && is.list(x) } # internal ---------------------------------------------------------------- #' @noRd #' @seealso importance_sampling_object psis_object <- function(unnormalized_log_weights, pareto_k, tail_len, r_eff) { importance_sampling_object(unnormalized_log_weights = unnormalized_log_weights, pareto_k = pareto_k, tail_len = tail_len, r_eff = r_eff, method = "psis") } #' @noRd #' @seealso do_importance_sampling do_psis <- function(log_ratios, r_eff, cores, method){ do_importance_sampling(log_ratios = log_ratios, r_eff = r_eff, cores = cores, method = "psis") } #' Extract named components from each list in the list of lists obtained by #' parallelizing `do_psis_i()` #' #' @noRd #' @param x List of lists. #' @param item String naming the component or attribute to pull out of each list #' (or list-like object). #' @param fun,fun.val passed to `vapply()`'s `FUN` and `FUN.VALUE` arguments. #' @return Numeric vector or matrix. #' psis_apply <- function(x, item, fun = c("[[", "attr"), fun_val = numeric(1)) { if (!is.list(x)) stop("Internal error ('x' must be a list for psis_apply)") vapply(x, FUN = match.arg(fun), FUN.VALUE = fun_val, item) } #' PSIS on a single vector #' #' @noRd #' @param log_ratios_i A vector of log importance ratios (for `loo()`, negative #' log likelihoods). #' @param tail_len_i An integer tail length. #' @param ... Not used. Included to conform to API for differen IS methods. #' #' @details #' * If there are enough tail samples then the tail is smoothed with PSIS #' * The log weights (or log ratios if no smoothing) larger than the largest raw #' ratio are set to the largest raw ratio #' #' @return A named list containing: #' * `lw`: vector of unnormalized log weights #' * `pareto_k`: scalar Pareto k estimate. #' do_psis_i <- function(log_ratios_i, tail_len_i, ...) { S <- length(log_ratios_i) # shift log ratios for safer exponentation lw_i <- log_ratios_i - max(log_ratios_i) khat <- Inf if (enough_tail_samples(tail_len_i)) { ord <- sort.int(lw_i, index.return = TRUE) tail_ids <- seq(S - tail_len_i + 1, S) lw_tail <- ord$x[tail_ids] if (abs(max(lw_tail) - min(lw_tail)) < .Machine$double.eps/100) { warning( "Can't fit generalized Pareto distribution ", "because all tail values are the same.", call. = FALSE ) } else { cutoff <- ord$x[min(tail_ids) - 1] # largest value smaller than tail values smoothed <- psis_smooth_tail(lw_tail, cutoff) khat <- smoothed$k lw_i[ord$ix[tail_ids]] <- smoothed$tail } } # truncate at max of raw wts (i.e., 0 since max has been subtracted) lw_i[lw_i > 0] <- 0 # shift log weights back so that the smallest log weights remain unchanged lw_i <- lw_i + max(log_ratios_i) list(log_weights = lw_i, pareto_k = khat) } #' PSIS tail smoothing for a single vector #' #' @noRd #' @param x Vector of tail elements already sorted in ascending order. #' @return A named list containing: #' * `tail`: vector same size as `x` containing the logs of the #' order statistics of the generalized pareto distribution. #' * `k`: scalar shape parameter estimate. #' psis_smooth_tail <- function(x, cutoff) { len <- length(x) exp_cutoff <- exp(cutoff) # save time not sorting since x already sorted fit <- gpdfit(exp(x) - exp_cutoff, sort_x = FALSE) k <- fit$k sigma <- fit$sigma if (is.finite(k)) { p <- (seq_len(len) - 0.5) / len qq <- qgpd(p, k, sigma) + exp_cutoff tail <- log(qq) } else { tail <- x } list(tail = tail, k = k) } #' Calculate tail lengths to use for fitting the GPD #' #' The number of weights (i.e., tail length) used to fit the generalized Pareto #' distribution is now decreasing with the number of posterior draws S, and is #' also adjusted based on the relative MCMC neff for `exp(log_lik)`. This will #' answer the questions about the asymptotic properties, works better for thick #' tailed proposal distributions, and is adjusted based on dependent Markov chain #' samples. Specifically, the tail length is now `3*sqrt(S)/r_eff` but capped at #' 20% of the total number of weights. #' #' @noRd #' @param r_eff A N-vector of relative MCMC effective sample sizes of `exp(log-lik matrix)`. #' @param S The (integer) size of posterior sample. #' @return An N-vector of tail lengths. #' n_pareto <- function(r_eff, S) { ceiling(pmin(0.2 * S, 3 * sqrt(S / r_eff))) } #' Check for enough tail samples to fit GPD #' #' @noRd #' @param tail_len Integer tail length. #' @param min_len The minimum allowed tail length. #' @return `TRUE` or `FALSE` #' enough_tail_samples <- function(tail_len, min_len = 5) { tail_len >= min_len } #' Throw warnings about pareto k estimates #' #' @noRd #' @param k A vector of pareto k estimates. #' @param high The value at which to warn about slighly high estimates. #' @param too_high The value at which to warn about very high estimates. #' @return Nothing, just possibly throws warnings. #' throw_pareto_warnings <- function(k, high = 0.5, too_high = 0.7) { if (isTRUE(any(k > too_high))) { .warn("Some Pareto k diagnostic values are too high. ", .k_help()) } else if (isTRUE(any(k > high))) { .warn("Some Pareto k diagnostic values are slightly high. ", .k_help()) } } #' Warn if not enough tail samples to fit GPD #' #' @noRd #' @param tail_lengths Vector of tail lengths. #' @return `tail_lengths`, invisibly. #' throw_tail_length_warnings <- function(tail_lengths) { tail_len_bad <- !sapply(tail_lengths, enough_tail_samples) if (any(tail_len_bad)) { if (length(tail_lengths) == 1) { warning( "Not enough tail samples to fit the generalized Pareto distribution.", call. = FALSE, immediate. = TRUE ) } else { bad <- which(tail_len_bad) Nbad <- length(bad) warning( "Not enough tail samples to fit the generalized Pareto distribution ", "in some or all columns of matrix of log importance ratios. ", "Skipping the following columns: ", paste(if (Nbad <= 10) bad else bad[1:10], collapse = ", "), if (Nbad > 10) paste0(", ... [", Nbad - 10, " more not printed].\n") else "\n", call. = FALSE, immediate. = TRUE ) } } invisible(tail_lengths) } #' Prepare `r_eff` to pass to `psis()` and throw warnings/errors if necessary #' #' @noRd #' @param r_eff User's `r_eff` argument. #' @param len The length `r_eff` should have if not `NULL` or `NA`. #' @return #' * If `r_eff` has length `len` then `r_eff` is returned. #' * If `r_eff` is `NULL` then a warning is thrown and `rep(1, len)` is returned. #' * If `r_eff` is `NA` then the warning is skipped and #' `rep(1, len)` is returned. #' * If `r_eff` has length `len` but has `NA`s then an error is thrown. #' prepare_psis_r_eff <- function(r_eff, len) { if (isTRUE(is.null(r_eff) || all(is.na(r_eff)))) { if (!called_from_loo() && is.null(r_eff)) { throw_psis_r_eff_warning() } r_eff <- rep(1, len) } else if (length(r_eff) != len) { stop("'r_eff' must have one value per observation.", call. = FALSE) } else if (anyNA(r_eff)) { stop("Can't mix NA and not NA values in 'r_eff'.", call. = FALSE) } r_eff } #' Check if `psis()` was called from one of the loo methods #' #' @noRd #' @return `TRUE` if the `loo()` array, matrix, or function method is found in #' the active call list, `FALSE` otherwise. #' called_from_loo <- function() { calls <- sys.calls() txt <- unlist(lapply(calls, deparse)) patts <- "loo.array\\(|loo.matrix\\(|loo.function\\(" check <- sapply(txt, function(x) grepl(patts, x)) isTRUE(any(check)) } #' Warning message about missing `r_eff` argument #' @noRd throw_psis_r_eff_warning <- function() { warning( "Relative effective sample sizes ('r_eff' argument) not specified. ", "PSIS n_eff will not be adjusted based on MCMC n_eff.", call. = FALSE ) } loo/R/extract_log_lik.R0000644000176200001440000000545313762013700014547 0ustar liggesusers#' Extract pointwise log-likelihood from a Stan model #' #' Convenience function for extracting the pointwise log-likelihood #' matrix or array from a `stanfit` object from the \pkg{rstan} package. #' Note: recent versions of \pkg{rstan} now include a `loo()` method for #' `stanfit` objects that handles this internally. #' #' @export #' @param stanfit A `stanfit` object (\pkg{rstan} package). #' @param parameter_name A character string naming the parameter (or generated #' quantity) in the Stan model corresponding to the log-likelihood. #' @param merge_chains If `TRUE` (the default), all Markov chains are #' merged together (i.e., stacked) and a matrix is returned. If `FALSE` #' they are kept separate and an array is returned. #' @return If `merge_chains=TRUE`, an \eqn{S} by \eqn{N} matrix of #' (post-warmup) extracted draws, where \eqn{S} is the size of the posterior #' sample and \eqn{N} is the number of data points. If #' `merge_chains=FALSE`, an \eqn{I} by \eqn{C} by \eqn{N} array, where #' \eqn{I \times C = S}{I * C = S}. #' #' #' @details Stan does not automatically compute and store the log-likelihood. It #' is up to the user to incorporate it into the Stan program if it is to be #' extracted after fitting the model. In a Stan model, the pointwise log #' likelihood can be coded as a vector in the transformed parameters block #' (and then summed up in the model block) or it can be coded entirely in the #' generated quantities block. We recommend using the generated quantities #' block so that the computations are carried out only once per iteration #' rather than once per HMC leapfrog step. #' #' For example, the following is the `generated quantities` block for #' computing and saving the log-likelihood for a linear regression model with #' `N` data points, outcome `y`, predictor matrix `X`, #' coefficients `beta`, and standard deviation `sigma`: #' #' `vector[N] log_lik;` #' #' `for (n in 1:N) log_lik[n] = normal_lpdf(y[n] | X[n, ] * beta, sigma);` #' #' @references #' Stan Development Team (2017). The Stan C++ Library, Version 2.16.0. #' #' #' Stan Development Team (2017). RStan: the R interface to Stan, Version 2.16.1. #' #' extract_log_lik <- function(stanfit, parameter_name = "log_lik", merge_chains = TRUE) { if (!inherits(stanfit, "stanfit")) stop("Not a stanfit object.", call. = FALSE) if (stanfit@mode != 0) stop("Stan model does not contain posterior draws.", call. = FALSE) if (!requireNamespace("rstan", quietly = TRUE)) stop("Please load the 'rstan' package.", call. = FALSE) if (merge_chains) { log_lik <- as.matrix(stanfit, pars = parameter_name) } else { log_lik <- as.array(stanfit, pars = parameter_name) } unname(log_lik) } loo/R/loo_approximate_posterior.R0000644000176200001440000001624313575772017016723 0ustar liggesusers#' Efficient approximate leave-one-out cross-validation (LOO) for posterior #' approximations #' #' @param x A log-likelihood array, matrix, or function. #' The **Methods (by class)** section, below, has detailed descriptions of how #' to specify the inputs for each method. #' @param save_psis Should the `"psis"` object created internally by #' `loo_approximate_posterior()` be saved in the returned object? See #' [loo()] for details. #' @template cores #' @inheritParams psis_approximate_posterior #' #' @details The `loo_approximate_posterior()` function is an S3 generic and #' methods are provided for 3-D pointwise log-likelihood arrays, pointwise #' log-likelihood matrices, and log-likelihood functions. The implementation #' works for posterior approximations where it is possible to compute the log #' density for the posterior approximation. #' #' @return The `loo_approximate_posterior()` methods return a named list with #' class `c("psis_loo_ap", "psis_loo", "loo")`. It has the same structure #' as the objects returned by [loo()] but with the additional slot: #' \describe{ #' \item{`posterior_approximation`}{ #' A list with two vectors, `log_p` and `log_g` of the same length #' containing the posterior density and the approximation density #' for the individual draws. #' } #' } #' #' @seealso [loo()], [psis()], [loo_compare()] #' @template loo-large-data-references #' #' @export loo_approximate_posterior #' @export loo_approximate_posterior.array loo_approximate_posterior.matrix loo_approximate_posterior.function #' loo_approximate_posterior <- function(x, log_p, log_g, ...) { UseMethod("loo_approximate_posterior") } #' @export #' @templateVar fn loo_approximate_posterior #' @template array loo_approximate_posterior.array <- function(x, log_p, log_g, ..., save_psis = FALSE, cores = getOption("mc.cores", 1)) { checkmate::assert_flag(save_psis) checkmate::assert_int(cores) checkmate::assert_matrix(log_p, mode = "numeric", nrows = dim(x)[1], ncols = dim(x)[2]) checkmate::assert_matrix(log_g, mode = "numeric", nrows = nrow(log_p), ncols = ncol(log_p)) ll <- llarray_to_matrix(x) log_p <- as.vector(log_p) log_g <- as.vector(log_g) loo_approximate_posterior.matrix( ll, log_p = log_p, log_g = log_g, ..., save_psis = save_psis, cores = cores ) } #' @export #' @templateVar fn loo_approximate_posterior #' @template matrix loo_approximate_posterior.matrix <- function(x, log_p, log_g, ..., save_psis = FALSE, cores = getOption("mc.cores", 1)) { checkmate::assert_flag(save_psis) checkmate::assert_int(cores) checkmate::assert_numeric(log_p, len = nrow(x)) checkmate::assert_null(dim(log_p)) checkmate::assert_numeric(log_g, len = length(log_p)) checkmate::assert_null(dim(log_g)) ap_psis <- psis_approximate_posterior( log_p = log_p, log_g = log_g, log_liks = x, ..., cores = cores, save_psis = save_psis ) ap_psis$approximate_posterior <- list(log_p = log_p, log_g = log_g) class(ap_psis) <- c("psis_loo_ap", class(ap_psis)) assert_psis_loo_ap(ap_psis) ap_psis } #' @export #' @templateVar fn loo_approximate_posterior #' @template function #' @param data,draws,... For the `loo_approximate_posterior.function()` method, #' these are the data, posterior draws, and other arguments to pass to the #' log-likelihood function. See the **Methods (by class)** section below for #' details on how to specify these arguments. #' loo_approximate_posterior.function <- function(x, ..., data = NULL, draws = NULL, log_p = NULL, log_g = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1)) { checkmate::assert_numeric(log_p, len = length(log_g)) checkmate::assert_numeric(log_g, len = length(log_p)) cores <- loo_cores(cores) stopifnot(is.data.frame(data) || is.matrix(data), !is.null(draws)) .llfun <- validate_llfun(x) N <- dim(data)[1] psis_list <- parallel_psis_list(N = N, .loo_i = .loo_ap_i, .llfun = .llfun, data = data, draws = draws, r_eff = NULL, # r_eff is ignored save_psis = save_psis, log_p = log_p, log_g = log_g, cores = cores, ...) pointwise <- lapply(psis_list, "[[", "pointwise") if (save_psis) { psis_object_list <- lapply(psis_list, "[[", "psis_object") psis_out <- list2importance_sampling(psis_object_list) diagnostics <- psis_out$diagnostics } else { diagnostics_list <- lapply(psis_list, "[[", "diagnostics") diagnostics <- list( pareto_k = psis_apply(diagnostics_list, "pareto_k"), n_eff = psis_apply(diagnostics_list, "n_eff") ) } ap_psis <- importance_sampling_loo_object( pointwise = do.call(rbind, pointwise), diagnostics = diagnostics, dims = c(attr(psis_list[[1]], "S"), N), is_method = "psis", is_object = if (save_psis) psis_out else NULL ) ap_psis$approximate_posterior <- list(log_p = log_p, log_g = log_g) class(ap_psis) <- c("psis_loo_ap", class(ap_psis)) assert_psis_loo_ap(ap_psis) ap_psis } # Function that is passed to the FUN argument of lapply, mclapply, or parLapply # for the loo_approximate_posterior.function method. The arguments and return # value are the same as the ones documented for the user-facing loo_i function. .loo_ap_i <- function(i, llfun, ..., data, draws, log_p, log_g, r_eff = NULL, save_psis = FALSE, is_method) { if (!is.null(r_eff)) warning("r_eff not implemented for aploo.") if (is_method != "psis") stop(is_method, " not implemented for aploo.") d_i <- data[i, , drop = FALSE] ll_i <- llfun(data_i = d_i, draws = draws, ...) if (!is.matrix(ll_i)) { ll_i <- as.matrix(ll_i) } psis_out <- ap_psis(log_ratios = -ll_i, log_p = log_p, log_g = log_g, cores = 1) structure( list( pointwise = pointwise_loo_calcs(ll_i, psis_out), diagnostics = psis_out$diagnostics, psis_object = if (save_psis) psis_out else NULL ), S = dim(psis_out)[1], N = 1 ) } assert_psis_loo_ap <- function(x) { checkmate::assert_class(x, "psis_loo_ap") checkmate::assert_names(names(x), must.include = c("estimates", "pointwise", "diagnostics", "psis_object", "approximate_posterior")) checkmate::assert_names(names(x$approximate_posterior), must.include = c("log_p", "log_g")) checkmate::assert_numeric(x$approximate_posterior$log_p, len = length(x$approximate_posterior$log_g), any.missing = FALSE) checkmate::assert_numeric(x$approximate_posterior$log_g, len = length(x$approximate_posterior$log_p), any.missing = FALSE) } loo/R/diagnostics.R0000644000176200001440000003206714407123455013713 0ustar liggesusers#' Diagnostics for Pareto smoothed importance sampling (PSIS) #' #' Print a diagnostic table summarizing the estimated Pareto shape parameters #' and PSIS effective sample sizes, find the indexes of observations for which #' the estimated Pareto shape parameter \eqn{k} is larger than some #' `threshold` value, or plot observation indexes vs. diagnostic estimates. #' The **Details** section below provides a brief overview of the #' diagnostics, but we recommend consulting Vehtari, Gelman, and Gabry (2017) #' and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for full details. #' #' @name pareto-k-diagnostic #' @param x An object created by [loo()] or [psis()]. #' @param threshold For `pareto_k_ids()`, `threshold` is the minimum \eqn{k} #' value to flag (default is `0.5`). For `mcse_loo()`, if any \eqn{k} #' estimates are greater than `threshold` the MCSE estimate is returned as #' `NA` (default is `0.7`). See **Details** for the motivation behind these #' defaults. #' #' @details #' The reliability and approximate convergence rate of the PSIS-based estimates #' can be assessed using the estimates for the shape parameter \eqn{k} of the #' generalized Pareto distribution: #' * If \eqn{k < 0.5} then the distribution of raw importance ratios has #' finite variance and the central limit theorem holds. However, as \eqn{k} #' approaches \eqn{0.5} the RMSE of plain importance sampling (IS) increases #' significantly while PSIS has lower RMSE. #' #' * If \eqn{0.5 \leq k < 1}{0.5 <= k < 1} then the variance of the raw #' importance ratios is infinite, but the mean exists. TIS and PSIS estimates #' have finite variance by accepting some bias. The convergence of the #' estimate is slower with increasing \eqn{k}. #' If \eqn{k} is between 0.5 and approximately 0.7 then we observe practically #' useful convergence rates and Monte Carlo error estimates with PSIS (the #' bias of TIS increases faster than the bias of PSIS). If \eqn{k > 0.7} we #' observe impractical convergence rates and unreliable Monte Carlo error #' estimates. #' #' * If \eqn{k \geq 1}{k >= 1} then neither the variance nor the mean of #' the raw importance ratios exists. The convergence rate is close to #' zero and bias can be large with practical sample sizes. #' #' \subsection{What if the estimated tail shape parameter \eqn{k} exceeds #' \eqn{0.5}}{ Importance sampling is likely to work less well if the #' marginal posterior \eqn{p(\theta^s | y)} and LOO posterior #' \eqn{p(\theta^s | y_{-i})} are very different, which is more likely to #' happen with a non-robust model and highly influential observations. #' If the estimated tail shape parameter \eqn{k} exceeds \eqn{0.5}, the user #' should be warned. (Note: If \eqn{k} is greater than \eqn{0.5} #' then WAIC is also likely to fail, but WAIC lacks its own diagnostic.) #' In practice, we have observed good performance for values of \eqn{k} up to 0.7. #' When using PSIS in the context of approximate LOO-CV, we recommend one of #' the following actions when \eqn{k > 0.7}: #' #' * With some additional computations, it is possible to transform the MCMC #' draws from the posterior distribution to obtain more reliable importance #' sampling estimates. This results in a smaller shape parameter \eqn{k}. #' See [loo_moment_match()] for an example of this. #' #' * Sampling directly from \eqn{p(\theta^s | y_{-i})} for the problematic #' observations \eqn{i}, or using \eqn{k}-fold cross-validation will generally #' be more stable. #' #' * Using a model that is more robust to anomalous observations will #' generally make approximate LOO-CV more stable. #' #' } #' #' \subsection{Observation influence statistics}{ The estimated shape parameter #' \eqn{k} for each observation can be used as a measure of the observation's #' influence on posterior distribution of the model. These can be obtained with #' `pareto_k_influence_values()`. #' } #' #' \subsection{Effective sample size and error estimates}{ In the case that we #' obtain the samples from the proposal distribution via MCMC the **loo** #' package also computes estimates for the Monte Carlo error and the effective #' sample size for importance sampling, which are more accurate for PSIS than #' for IS and TIS (see Vehtari et al (2019) for details). However, the PSIS #' effective sample size estimate will be #' **over-optimistic when the estimate of \eqn{k} is greater than 0.7**. #' } #' #' @seealso #' * [psis()] for the implementation of the PSIS algorithm. #' * The [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. #' #' @template loo-and-psis-references #' NULL #' @rdname pareto-k-diagnostic #' @export #' @return `pareto_k_table()` returns an object of class #' `"pareto_k_table"`, which is a matrix with columns `"Count"`, #' `"Proportion"`, and `"Min. n_eff"`, and has its own print method. #' pareto_k_table <- function(x) { k <- pareto_k_values(x) n_eff <- try(psis_n_eff_values(x), silent = TRUE) if (inherits(n_eff, "try-error")) { n_eff <- rep(NA, length(k)) } kcut <- k_cut(k) min_n_eff <- min_n_eff_by_k(n_eff, kcut) count <- table(kcut) out <- cbind( Count = count, Proportion = prop.table(count), "Min. n_eff" = min_n_eff ) structure(out, class = c("pareto_k_table", class(out))) } #' @export print.pareto_k_table <- function(x, digits = 1, ...) { count <- x[, "Count"] if (sum(count[2:4]) == 0) { cat("\nAll Pareto k estimates are good (k < 0.5).\n") } else { tab <- cbind( " " = rep("", 4), " " = c("(good)", "(ok)", "(bad)", "(very bad)"), "Count" = .fr(count, 0), "Pct. " = paste0(.fr(100 * x[, "Proportion"], digits), "%"), "Min. n_eff" = round(x[, "Min. n_eff"]) ) tab2 <- rbind(tab) cat("Pareto k diagnostic values:\n") rownames(tab2) <- format(rownames(tab2), justify = "right") print(tab2, quote = FALSE) if (sum(count[3:4]) == 0) cat("\nAll Pareto k estimates are ok (k < 0.7).\n") invisible(x) } } #' @rdname pareto-k-diagnostic #' @export #' @return `pareto_k_ids()` returns an integer vector indicating which #' observations have Pareto \eqn{k} estimates above `threshold`. #' pareto_k_ids <- function(x, threshold = 0.5) { k <- pareto_k_values(x) which(k > threshold) } #' @rdname pareto-k-diagnostic #' @export #' @return `pareto_k_values()` returns a vector of the estimated Pareto #' \eqn{k} parameters. These represent the reliability of sampling. pareto_k_values <- function(x) { k <- x$diagnostics[["pareto_k"]] if (is.null(k)) { # for compatibility with objects from loo < 2.0.0 k <- x[["pareto_k"]] } if (is.null(k)) { stop("No Pareto k estimates found.", call. = FALSE) } return(k) } #' @rdname pareto-k-diagnostic #' @export #' @return `pareto_k_influence_values()` returns a vector of the estimated Pareto #' \eqn{k} parameters. These represent influence of the observations on the #' model posterior distribution. pareto_k_influence_values <- function(x) { if ("influence_pareto_k" %in% colnames(x$pointwise)) { k <- x$pointwise[,"influence_pareto_k"] } else { stop("No Pareto k influence estimates found.", call. = FALSE) } return(k) } #' @rdname pareto-k-diagnostic #' @export #' @return `psis_n_eff_values()` returns a vector of the estimated PSIS #' effective sample sizes. psis_n_eff_values <- function(x) { n_eff <- x$diagnostics[["n_eff"]] if (is.null(n_eff)) { stop("No PSIS n_eff estimates found.", call. = FALSE) } return(n_eff) } #' @rdname pareto-k-diagnostic #' @export #' @return `mcse_loo()` returns the Monte Carlo standard error (MCSE) #' estimate for PSIS-LOO. MCSE will be NA if any Pareto \eqn{k} values are #' above `threshold`. #' mcse_loo <- function(x, threshold = 0.7) { stopifnot(is.psis_loo(x)) if (any(pareto_k_values(x) > 0.7, na.rm = TRUE)) { return(NA) } mc_var <- x$pointwise[, "mcse_elpd_loo"]^2 sqrt(sum(mc_var)) } #' @rdname pareto-k-diagnostic #' @aliases plot.loo #' @export #' @param label_points,... For the `plot()` method, if `label_points` is #' `TRUE` the observation numbers corresponding to any values of \eqn{k} #' greater than 0.5 will be displayed in the plot. Any arguments specified in #' `...` will be passed to [graphics::text()] and can be used #' to control the appearance of the labels. #' @param diagnostic For the `plot` method, which diagnostic should be #' plotted? The options are `"k"` for Pareto \eqn{k} estimates (the #' default) or `"n_eff"` for PSIS effective sample size estimates. #' @param main For the `plot()` method, a title for the plot. #' #' @return The `plot()` method is called for its side effect and does not #' return anything. If `x` is the result of a call to [loo()] #' or [psis()] then `plot(x, diagnostic)` produces a plot of #' the estimates of the Pareto shape parameters (`diagnostic = "k"`) or #' estimates of the PSIS effective sample sizes (`diagnostic = "n_eff"`). #' plot.psis_loo <- function(x, diagnostic = c("k", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot") { diagnostic <- match.arg(diagnostic) k <- pareto_k_values(x) k[is.na(k)] <- 0 # FIXME when reloo is changed to make NA k values -Inf k_inf <- !is.finite(k) if (any(k_inf)) { warning(signif(100 * mean(k_inf), 2), "% of Pareto k estimates are Inf/NA/NaN and not plotted.") } if (diagnostic == "n_eff") { n_eff <- psis_n_eff_values(x) } else { n_eff <- NULL } plot_diagnostic( k = k, n_eff = n_eff, ..., label_points = label_points, main = main ) } #' @export #' @noRd #' @rdname pareto-k-diagnostic plot.loo <- plot.psis_loo #' @export #' @rdname pareto-k-diagnostic plot.psis <- function(x, diagnostic = c("k", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot") { plot.psis_loo(x, diagnostic = diagnostic, ..., label_points = label_points, main = main) } # internal ---------------------------------------------------------------- plot_diagnostic <- function(k, n_eff = NULL, ..., label_points = FALSE, main = "PSIS diagnostic plot") { use_n_eff <- !is.null(n_eff) graphics::plot( x = if (use_n_eff) n_eff else k, xlab = "Data point", ylab = if (use_n_eff) "PSIS n_eff" else "Pareto shape k", type = "n", bty = "l", yaxt = "n", main = main ) graphics::axis(side = 2, las = 1) in_range <- function(x, lb_ub) { x >= lb_ub[1L] & x <= lb_ub[2L] } if (!use_n_eff) { krange <- range(k, na.rm = TRUE) breaks <- c(0, 0.5, 0.7, 1) hex_clrs <- c("#C79999", "#A25050", "#7C0000") ltys <- c(3, 4, 2, 1) for (j in seq_along(breaks)) { val <- breaks[j] if (in_range(val, krange)) graphics::abline( h = val, col = ifelse(val == 0, "darkgray", hex_clrs[j - 1]), lty = ltys[j], lwd = 1 ) } } breaks <- c(-Inf, 0.5, 1) hex_clrs <- c("#6497b1", "#005b96", "#03396c") clrs <- ifelse( in_range(k, breaks[1:2]), hex_clrs[1], ifelse(in_range(k, breaks[2:3]), hex_clrs[2], hex_clrs[3]) ) if (all(k < 0.5) || !label_points) { graphics::points(x = if (use_n_eff) n_eff else k, col = clrs, pch = 3, cex = .6) return(invisible()) } else { graphics::points(x = if (use_n_eff) n_eff[k < 0.5] else k[k < 0.5], col = clrs[k < 0.5], pch = 3, cex = .6) sel <- !in_range(k, breaks[1:2]) dots <- list(...) txt_args <- c( list( x = seq_along(k)[sel], y = if (use_n_eff) n_eff[sel] else k[sel], labels = seq_along(k)[sel] ), if (length(dots)) dots ) if (!("adj" %in% names(txt_args))) txt_args$adj <- 2 / 3 if (!("cex" %in% names(txt_args))) txt_args$cex <- 0.75 if (!("col" %in% names(txt_args))) txt_args$col <- clrs[sel] do.call(graphics::text, txt_args) } } #' Convert numeric Pareto k values to a factor variable. #' #' @noRd #' @param k Vector of Pareto k estimates. #' @return A factor variable (the same length as k) with 4 levels. #' k_cut <- function(k) { cut( k, breaks = c(-Inf, 0.5, 0.7, 1, Inf), labels = c("(-Inf, 0.5]", "(0.5, 0.7]", "(0.7, 1]", "(1, Inf)") ) } #' Calculate the minimum PSIS n_eff within groups defined by Pareto k values #' #' @noRd #' @param n_eff Vector of PSIS n_eff estimates. #' @param kcut Factor returned by the k_cut() function. #' @return Vector of length `nlevels(kcut)` containing the minimum n_eff within #' each k group. If there are no k values in a group the corresponding element #' of the returned vector is NA. min_n_eff_by_k <- function(n_eff, kcut) { n_eff_split <- split(n_eff, f = kcut) n_eff_split <- sapply(n_eff_split, function(x) { # some k groups might be empty. # split gives numeric(0) but replace with NA if (!length(x)) NA else x }) sapply(n_eff_split, min) } loo/R/kfold-helpers.R0000644000176200001440000000700113575772017014142 0ustar liggesusers#' Helper functions for K-fold cross-validation #' #' @description These functions can be used to generate indexes for use with #' K-fold cross-validation. See the **Details** section for explanations. #' #' @name kfold-helpers #' @param K The number of folds to use. #' @param N The number of observations in the data. #' @param x A discrete variable of length `N` with at least `K` levels (unique #' values). Will be coerced to a [factor][factor()]. #' #' @return An integer vector of length `N` where each element is an index in `1:K`. #' #' @details #' `kfold_split_random()` splits the data into `K` groups #' of equal size (or roughly equal size). #' #' For a categorical variable `x` `kfold_split_stratified()` #' splits the observations into `K` groups ensuring that relative #' category frequencies are approximately preserved. #' #' For a grouping variable `x`, `kfold_split_grouped()` places #' all observations in `x` from the same group/level together in #' the same fold. The selection of which groups/levels go into which #' fold (relevant when when there are more groups than folds) is #' randomized. #' #' @examples #' ids <- kfold_split_random(K = 5, N = 20) #' print(ids) #' table(ids) #' #' #' x <- sample(c(0, 1), size = 200, replace = TRUE, prob = c(0.05, 0.95)) #' table(x) #' ids <- kfold_split_stratified(K = 5, x = x) #' print(ids) #' table(ids, x) #' #' grp <- gl(n = 50, k = 15, labels = state.name) #' length(grp) #' head(table(grp)) #' #' ids_10 <- kfold_split_grouped(K = 10, x = grp) #' (tab_10 <- table(grp, ids_10)) #' colSums(tab_10) #' #' ids_9 <- kfold_split_grouped(K = 9, x = grp) #' (tab_9 <- table(grp, ids_9)) #' colSums(tab_9) #' NULL #' @rdname kfold-helpers #' @export kfold_split_random <- function(K = 10, N = NULL) { stopifnot( !is.null(N), K == as.integer(K), N == as.integer(N), length(K) == 1, length(N) == 1, K > 1, K <= N ) perm <- sample.int(N) idx <- ceiling(seq(from = 1, to = N, length.out = K + 1)) bins <- .bincode(perm, breaks = idx, right = FALSE, include.lowest = TRUE) return(bins) } #' @rdname kfold-helpers #' @export kfold_split_stratified <- function(K = 10, x = NULL) { stopifnot( !is.null(x), K == as.integer(K), length(K) == 1, K > 1, K <= length(x) ) x <- as.integer(as.factor(x)) Nlev <- length(unique(x)) N <- length(x) xids <- numeric() for (l in 1:Nlev) { xids <- c(xids, sample(which(x==l))) } bins <- rep(NA, N) bins[xids] <- rep(1:K, ceiling(N/K))[1:N] return(bins) } #' @rdname kfold-helpers #' @export kfold_split_grouped <- function(K = 10, x = NULL) { stopifnot( !is.null(x), K == as.integer(K), length(K) == 1, K > 1, K <= length(x) ) Nlev <- length(unique(x)) if (Nlev < K) { stop("'K' must not be bigger than the number of levels/groups in 'x'.") } x <- as.integer(as.factor(x)) if (Nlev == K) { return(x) } # Otherwise we have Nlev > K S1 <- ceiling(Nlev / K) # number of levels in largest groups of levels N_S2 <- S1 * K - Nlev # number of groups of levels of size S1 - 1 N_S1 <- K - N_S2 # number of groups of levels of size S1 perm <- sample.int(Nlev) # permute group levels brks <- seq(from = S1 + 0.5, by = S1, length.out = N_S1) if (N_S2 > 0) { brks2 <- seq(from = brks[N_S1] + S1 - 1, by = S1 - 1, length.out = N_S2 - 1) brks <- c(brks, brks2) } grps <- findInterval(perm, vec = brks) + 1 # +1 so min is 1 not 0 bins <- rep(NA, length(x)) for (j in perm) { bins[x == j] <- grps[j] } return(bins) } loo/R/split_moment_matching.R0000644000176200001440000001350413753273364015773 0ustar liggesusers#' Split moment matching for efficient approximate leave-one-out cross-validation (LOO) #' #' A function that computes the split moment matching importance sampling loo. #' Takes in the moment matching total transformation, transforms only half #' of the draws, and computes a single elpd using multiple importance sampling. #' #' @param x A fitted model object. #' @param upars A matrix containing the model parameters in unconstrained space #' where they can have any real value. #' @param cov Logical; Indicate whether to match the covariance matrix of the #' samples or not. If `FALSE`, only the mean and marginal variances are #' matched. #' @param total_shift A vector representing the total shift made by the moment #' matching algorithm. #' @param total_scaling A vector representing the total scaling of marginal #' variance made by the moment matching algorithm. #' @param total_mapping A vector representing the total covariance #' transformation made by the moment matching algorithm. #' @param i Observation index. #' @param log_prob_upars A function that takes arguments `x` and `upars` and #' returns a matrix of log-posterior density values of the unconstrained #' posterior draws passed via `upars`. #' @param log_lik_i_upars A function that takes arguments `x`, `upars`, and `i` #' and returns a vector of log-likeliood draws of the `i`th observation based #' on the unconstrained posterior draws passed via `upars`. #' @param r_eff_i MCMC relative effective sample size of the `i`'th log #' likelihood draws. #' @template cores #' @template is_method #' @param ... Further arguments passed to the custom functions documented above. #' #' @return A list containing the updated log-importance weights and #' log-likelihood values. Also returns the updated MCMC effective sample size #' and the integrand-specific log-importance weights. #' #' #' @seealso [loo()], [loo_moment_match()] #' @template moment-matching-references #' #' loo_moment_match_split <- function(x, upars, cov, total_shift, total_scaling, total_mapping, i, log_prob_upars, log_lik_i_upars, r_eff_i, cores, is_method, ...) { S <- dim(upars)[1] S_half <- as.integer(0.5 * S) mean_original <- colMeans(upars) # accumulated affine transformation upars_trans <- sweep(upars, 2, mean_original, "-") upars_trans <- sweep(upars_trans, 2, total_scaling, "*") if (cov) { upars_trans <- tcrossprod(upars_trans, total_mapping) } upars_trans <- sweep(upars_trans, 2, total_shift + mean_original, "+") attributes(upars_trans) <- attributes(upars) # inverse accumulated affine transformation upars_trans_inv <- sweep(upars, 2, mean_original, "-") if (cov) { upars_trans_inv <- tcrossprod(upars_trans_inv, solve(total_mapping)) } upars_trans_inv <- sweep(upars_trans_inv, 2, total_scaling, "/") upars_trans_inv <- sweep(upars_trans_inv, 2, mean_original - total_shift, "+") attributes(upars_trans_inv) <- attributes(upars) # first half of upars_trans_half are T(theta) # second half are theta upars_trans_half <- upars take <- seq_len(S_half) upars_trans_half[take, ] <- upars_trans[take, , drop = FALSE] # first half of upars_half_inv are theta # second half are T^-1 (theta) upars_trans_half_inv <- upars take <- seq_len(S)[-seq_len(S_half)] upars_trans_half_inv[take, ] <- upars_trans_inv[take, , drop = FALSE] # compute log likelihoods and log probabilities log_prob_half_trans <- log_prob_upars(x, upars = upars_trans_half, ...) log_prob_half_trans_inv <- log_prob_upars(x, upars = upars_trans_half_inv, ...) log_liki_half <- log_lik_i_upars(x, upars = upars_trans_half, i = i, ...) # compute weights log_prob_half_trans_inv <- (log_prob_half_trans_inv - log(prod(total_scaling)) - log(det(total_mapping))) stable_S <- log_prob_half_trans > log_prob_half_trans_inv lwi_half <- -log_liki_half + log_prob_half_trans lwi_half[stable_S] <- lwi_half[stable_S] - (log_prob_half_trans[stable_S] + log1p(exp(log_prob_half_trans_inv[stable_S] - log_prob_half_trans[stable_S]))) lwi_half[!stable_S] <- lwi_half[!stable_S] - (log_prob_half_trans_inv[!stable_S] + log1p(exp(log_prob_half_trans[!stable_S] - log_prob_half_trans_inv[!stable_S]))) is_obj_half <- suppressWarnings(importance_sampling.default(lwi_half, method = is_method, r_eff = r_eff_i, cores = cores)) lwi_half <- as.vector(weights(is_obj_half)) is_obj_f_half <- suppressWarnings(importance_sampling.default(lwi_half + log_liki_half, method = is_method, r_eff = r_eff_i, cores = cores)) lwfi_half <- as.vector(weights(is_obj_f_half)) # relative_eff recomputation # currently ignores chain information # since we have two proposal distributions # compute S_eff separately from both and take the smaller take <- seq_len(S)[-seq_len(S_half)] log_liki_half_1 <- log_liki_half[take, drop = FALSE] dim(log_liki_half_1) <- c(length(take), 1, 1) take <- seq_len(S)[-seq_len(S_half)] log_liki_half_2 <- log_liki_half[take, drop = FALSE] dim(log_liki_half_2) <- c(length(take), 1, 1) r_eff_i1 <- loo::relative_eff(exp(log_liki_half_1), cores = cores) r_eff_i2 <- loo::relative_eff(exp(log_liki_half_2), cores = cores) r_eff_i <- min(r_eff_i1,r_eff_i2) list( lwi = lwi_half, lwfi = lwfi_half, log_liki = log_liki_half, r_eff_i = r_eff_i ) } loo/R/crps.R0000644000176200001440000001543414411130104012332 0ustar liggesusers#' Continuously ranked probability score #' #' The `crps()` and `scrps()` functions and their `loo_*()` counterparts can be #' used to compute the continuously ranked probability score (CRPS) and scaled #' CRPS (SCRPS) (see Bolin and Wallin, 2022). CRPS is a proper scoring rule, and #' strictly proper when the first moment of the predictive distribution is #' finite. Both can be expressed in terms of samples form the predictive #' distribution. See e.g. Gneiting and Raftery (2007) for a comprehensive #' discussion on CRPS. #' #' To compute (S)CRPS, the user needs to provide two sets of draws, `x` and #' `x2`, from the predictive distribution. This is due to the fact that formulas #' used to compute CRPS involve an expectation of the absolute difference of `x` #' and `x2`, both having the same distribution. See the `permutations` argument, #' as well as Gneiting and Raftery (2007) for details. #' #' @export #' @param x A `S` by `N` matrix (draws by observations), or a vector of length #' `S` when only single observation is provided in `y`. #' @param x2 Independent draws from the same distribution as draws in `x`. #' Should be of the identical dimension. #' @param y A vector of observations or a single value. #' @param permutations An integer, with default value of 1, specifying how many #' times the expected value of |X - X'| (`|x - x2|`) is computed. The row #' order of `x2` is shuffled as elements `x` and `x2` are typically drawn #' given the same values of parameters. This happens, e.g., when one calls #' `posterior_predict()` twice for a fitted \pkg{rstanarm} or \pkg{brms} #' model. Generating more permutations is expected to decrease the variance of #' the computed expected value. #' @param ... Passed on to [E_loo()] in the `loo_*()` version of these #' functions. #' #' @return A list containing two elements: `estimates` and `pointwise`. #' The former reports estimator and standard error and latter the pointwise #' values. #' #' @examples #' \dontrun{ #' # An example using rstanarm #' library(rstanarm) #' data("kidiq") #' fit <- stan_glm(kid_score ~ mom_hs + mom_iq, data = kidiq) #' ypred1 <- posterior_predict(fit) #' ypred2 <- posterior_predict(fit) #' crps(ypred1, ypred2, y = fit$y) #' loo_crps(ypred1, ypred2, y = fit$y, log_lik = log_lik(fit)) #' } #' #' @references #' Bolin, D., & Wallin, J. (2022). Local scale invariance and robustness of #' proper scoring rules. arXiv. \doi{10.48550/arXiv.1912.05642} #' #' Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, #' Prediction, and Estimation. Journal of the American Statistical Association, #' 102(477), 359–378. crps <- function(x, ...) { UseMethod("crps") } #' @rdname crps #' @export scrps <- function(x, ...) { UseMethod("scrps") } #' @rdname crps #' @export loo_crps <- function(x, ...) { UseMethod("loo_crps") } #' @rdname crps #' @export loo_scrps <- function(x, ...) { UseMethod("loo_scrps") } #' @rdname crps #' @export crps.matrix <- function(x, x2, y, ..., permutations = 1) { validate_crps_input(x, x2, y) repeats <- replicate(permutations, EXX_compute(x, x2), simplify = F) EXX <- Reduce(`+`, repeats) / permutations EXy <- colMeans(abs(sweep(x, 2, y))) crps_output(.crps_fun(EXX, EXy)) } #' Method for a single data point #' @rdname crps #' @export crps.numeric <- function(x, x2, y, ..., permutations = 1) { stopifnot(length(x) == length(x2), length(y) == 1) crps.matrix(as.matrix(x), as.matrix(x2), y, permutations) } #' @rdname crps #' @export #' @param log_lik A log-likelihood matrix the same size as `x`. #' @param r_eff An optional vector of relative effective sample size estimates #' containing one element per observation. See [psis()] for details. #' @param cores The number of cores to use for parallelization of `[psis()]`. #' See [psis()] for details. loo_crps.matrix <- function(x, x2, y, log_lik, ..., permutations = 1, r_eff = NULL, cores = getOption("mc.cores", 1)) { validate_crps_input(x, x2, y, log_lik) repeats <- replicate(permutations, EXX_loo_compute(x, x2, log_lik, r_eff = r_eff, ...), simplify = F) EXX <- Reduce(`+`, repeats) / permutations psis_obj <- psis(-log_lik, r_eff = r_eff, cores = cores) EXy <- E_loo(abs(sweep(x, 2, y)), psis_obj, log_ratios = -log_lik, ...)$value crps_output(.crps_fun(EXX, EXy)) } #' @rdname crps #' @export scrps.matrix <- function(x, x2, y, ..., permutations = 1) { validate_crps_input(x, x2, y) repeats <- replicate(permutations, EXX_compute(x, x2), simplify = F) EXX <- Reduce(`+`, repeats) / permutations EXy <- colMeans(abs(sweep(x, 2, y))) crps_output(.crps_fun(EXX, EXy, scale = TRUE)) } #' @rdname crps #' @export scrps.numeric <- function(x, x2, y, ..., permutations = 1) { stopifnot(length(x) == length(x2), length(y) == 1) scrps.matrix(as.matrix(x), as.matrix(x2), y, permutations) } #' @rdname crps #' @export loo_scrps.matrix <- function( x, x2, y, log_lik, ..., permutations = 1, r_eff = NULL, cores = getOption("mc.cores", 1)) { validate_crps_input(x, x2, y, log_lik) repeats <- replicate(permutations, EXX_loo_compute(x, x2, log_lik, r_eff = r_eff, ...), simplify = F) EXX <- Reduce(`+`, repeats) / permutations psis_obj <- psis(-log_lik, r_eff = r_eff, cores = cores) EXy <- E_loo(abs(sweep(x, 2, y)), psis_obj, log_ratios = -log_lik, ...)$value crps_output(.crps_fun(EXX, EXy, scale = TRUE)) } # ------------ Internals ---------------- EXX_compute <- function(x, x2) { S <- nrow(x) colMeans(abs(x - x2[sample(1:S),])) } EXX_loo_compute <- function(x, x2, log_lik, r_eff = NULL, ...) { S <- nrow(x) shuffle <- sample (1:S) x2 <- x2[shuffle,] log_lik2 <- log_lik[shuffle,] psis_obj_joint <- psis(-log_lik - log_lik2 , r_eff = r_eff) E_loo(abs(x - x2), psis_obj_joint, log_ratios = -log_lik - log_lik2, ...)$value } #' Function to compute crps and scrps #' @noRd .crps_fun <- function(EXX, EXy, scale = FALSE) { if (scale) return(-EXy/EXX - 0.5 * log(EXX)) 0.5 * EXX - EXy } #' Compute output data for crps functions #' @noRd crps_output <- function(crps_pw) { n <- length(crps_pw) out <- list() out$estimates <- c(mean(crps_pw), sd(crps_pw) / sqrt(n)) names(out$estimates) <- c('Estimate', 'SE') out$pointwise <- crps_pw out } #' Validate input of CRPS functions #' #' Check that predictive draws and observed data are of compatible shape #' @noRd validate_crps_input <- function(x, x2, y, log_lik = NULL) { stopifnot(is.numeric(x), is.numeric(x2), is.numeric(y), identical(dim(x), dim(x2)), ncol(x) == length(y), ifelse(is.null(log_lik), TRUE, identical(dim(log_lik), dim(x))) ) } loo/R/loo_predictive_metric.R0000644000176200001440000001464214411130104015735 0ustar liggesusers#' Estimate leave-one-out predictive performance.. #' #' The `loo_predictive_metric()` function computes estimates of leave-one-out #' predictive metrics given a set of predictions and observations. Currently #' supported metrics are mean absolute error, mean squared error and root mean #' squared error for continuous predictions and accuracy and balanced accuracy #' for binary classification. Predictions are passed on to the [E_loo()] #' function, so this function assumes that the PSIS approximation is working #' well. #' #' @param x A numeric matrix of predictions. #' @param y A numeric vector of observations. Length should be equal to the #' number of rows in `x`. #' @param log_lik A matrix of pointwise log-likelihoods. Should be of same #' dimension as `x`. #' @param metric The type of predictive metric to be used. Currently #' supported options are `"mae"`, `"rmse"` and `"mse"` for regression and #' for binary classification `"acc"` and `"balanced_acc"`. #' \describe{ #' \item{`"mae"`}{ #' Mean absolute error. #' } #' \item{`"mse"`}{ #' Mean squared error. #' } #' \item{`"rmse"`}{ #' Root mean squared error, given by as the square root of `MSE`. #' } #' \item{`"acc"`}{ #' The proportion of predictions indicating the correct outcome. #' } #' \item{`"balanced_acc"`}{ #' Balanced accuracy is given by the average of true positive and true #' negative rates. #' } #' } #' @param r_eff A Vector of relative effective sample size estimates containing #' one element per observation. See [psis()] for more details. #' @param cores The number of cores to use for parallelization of `[psis()]`. #' See [psis()] for details. #' @param ... Additional arguments passed on to [E_loo()] #' #' @return A list with the following components: #' \describe{ #' \item{`estimate`}{ #' Estimate of the given metric. #' } #' \item{`se`}{ #' Standard error of the estimate. #' } #' } #' @export #' #' @examples #' \donttest{ #' if (requireNamespace("rstanarm", quietly = TRUE)) { #' # Use rstanarm package to quickly fit a model and get both a log-likelihood #' # matrix and draws from the posterior predictive distribution #' library("rstanarm") #' #' # data from help("lm") #' ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14) #' trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69) #' d <- data.frame( #' weight = c(ctl, trt), #' group = gl(2, 10, 20, labels = c("Ctl","Trt")) #' ) #' fit <- stan_glm(weight ~ group, data = d, refresh = 0) #' ll <- log_lik(fit) #' r_eff <- relative_eff(exp(-ll), chain_id = rep(1:4, each = 1000)) #' #' mu_pred <- posterior_epred(fit) #' # Leave-one-out mean absolute error of predictions #' mae <- loo_predictive_metric(x = mu_pred, y = d$weight, log_lik = ll, #' pred_error = 'mae', r_eff = r_eff) #' # Leave-one-out 90%-quantile of mean absolute error #' mae_90q <- loo_predictive_metric(x = mu_pred, y = d$weight, log_lik = ll, #' pred_error = 'mae', r_eff = r_eff, #' type = 'quantile', probs = 0.9) #' } #' } loo_predictive_metric <- function(x, ...) { UseMethod("loo_predictive_metric") } #' @rdname loo_predictive_metric #' @export loo_predictive_metric.matrix <- function(x, y, log_lik, ..., metric = c("mae", "rmse", "mse", "acc", "balanced_acc"), r_eff = NULL, cores = getOption("mc.cores", 1)) { stopifnot( is.numeric(x), is.numeric(y), identical(ncol(x), length(y)), identical(dim(x), dim(log_lik)) ) metric <- match.arg(metric) psis_object <- psis(-log_lik, r_eff = r_eff, cores = cores) pred_loo <- E_loo(x, psis_object = psis_object, log_ratios = -log_lik, ...)$value predictive_metric_fun <- .loo_predictive_metric_fun(metric) predictive_metric_fun(y, pred_loo) } # ----------------------------- Internals ----------------------------- #' Select predictive metric function based on user's `metric` argument #' #' @noRd #' @param metric The metric used. #' @return The function used to compute predictive error or accuracy specified #' by the argument `metric`. .loo_predictive_metric_fun <- function(metric) { switch( metric, 'mae' = .mae, 'rmse' = .rmse, 'mse' = .mse, 'acc' = .accuracy, 'balanced_acc' = .balanced_accuracy ) } #' Mean absolute error #' #' @noRd #' @param y A vector of observed values #' @param yhat A vector of predictions .mae <-function(y, yhat) { stopifnot(length(y) == length(yhat)) n <- length(y) e <- abs(y - yhat) list(estimate = mean(e), se = sd(e) / sqrt(n)) } #' Mean squared error #' #' @noRd #' @param y A vector of observed values #' @param yhat A vector of predictions .mse <-function(y, yhat) { stopifnot(length(y) == length(yhat)) n <- length(y) e <- (y - yhat)^2 list(estimate = mean(e), se = sd(e) / sqrt(n)) } #' Root mean squared error #' #' @noRd #' @param y A vector of observed values #' @param yhat A vector of predictions .rmse <-function(y, yhat) { est <- .mse(y, yhat) mean_mse <- est$estimate var_mse <- est$se^2 var_rmse <- var_mse / mean_mse / 4 # Comes from the first order Taylor approx. return(list(estimate = sqrt(mean_mse), se = sqrt(var_rmse))) } #' Classification accuracy #' #' @noRd #' @param y A vector of observed values #' @param yhat A vector of predictions .accuracy <- function(y, yhat) { stopifnot(length(y) == length(yhat), all(y <= 1 & y >= 0), all(yhat <= 1 & yhat >= 0)) n <- length(y) yhat <- as.integer(yhat > 0.5) acc <- as.integer(yhat == y) est <- mean(acc) list(estimate = est, se = sqrt(est * (1-est) / n) ) } #' Balanced classification accuracy #' #' @noRd #' @param y A vector of observed values #' @param yhat A vector of predictions .balanced_accuracy <- function(y, yhat) { stopifnot(length(y) == length(yhat), all(y <= 1 & y >= 0), all(yhat <= 1 & yhat >= 0)) n <- length(y) yhat <- as.integer(yhat > 0.5) mask <- y == 0 tn <- mean(yhat[mask] == y[mask]) # True negatives tp <- mean(yhat[!mask] == y[!mask]) # True positives bls_acc <- (tp + tn) / 2 # This approximation has quite large bias for small samples bls_acc_var <- (tp * (1 - tp) + tn * (1 - tn)) / 4 list(estimate = bls_acc, se = sqrt(bls_acc_var / n)) } loo/R/loo_subsample.R0000644000176200001440000014172514407123455014252 0ustar liggesusers#' Efficient approximate leave-one-out cross-validation (LOO) using subsampling #' #' @param x A function. The **Methods (by class)** section, below, has detailed #' descriptions of how to specify the inputs. #' #' @inheritParams loo #' @param save_psis Should the `"psis"` object created internally by #' `loo_subsample()` be saved in the returned object? See [loo()] for details. #' @template cores #' #' @details The `loo_subsample()` function is an S3 generic and a methods is #' currently provided for log-likelihood functions. The implementation works #' for both MCMC and for posterior approximations where it is possible to #' compute the log density for the approximation. #' #' @return `loo_subsample()` returns a named list with class `c("psis_loo_ss", #' "psis_loo", "loo")`. This has the same structure as objects returned by #' [loo()] but with the additional slot: #' * `loo_subsampling`: A list with two vectors, `log_p` and `log_g`, of the #' same length containing the posterior density and the approximation density #' for the individual draws. #' #' @seealso [loo()], [psis()], [loo_compare()] #' @template loo-large-data-references #' #' @export loo_subsample loo_subsample.function #' loo_subsample <- function(x, ...) { UseMethod("loo_subsample") } #' @export #' @templateVar fn loo_subsample #' @template function #' @param data,draws,... For `loo_subsample.function()`, these are the data, #' posterior draws, and other arguments to pass to the log-likelihood #' function. Note that for some `loo_approximation`s, the draws will be replaced #' by the posteriors summary statistics to compute loo approximations. See #' argument `loo_approximation` for details. #' @param observations The subsample observations to use. The argument can take #' four (4) types of arguments: #' * `NULL` to use all observations. The algorithm then just uses #' standard `loo()` or `loo_approximate_posterior()`. #' * A single integer to specify the number of observations to be subsampled. #' * A vector of integers to provide the indices used to subset the data. #' _These observations need to be subsampled with the same scheme as given by #' the `estimator` argument_. #' * A `psis_loo_ss` object to use the same observations that were used in a #' previous call to `loo_subsample()`. #' #' @param log_p,log_g Should be supplied only if approximate posterior draws are #' used. The default (`NULL`) indicates draws are from "true" posterior (i.e. #' using MCMC). If not `NULL` then they should be specified as described in #' [loo_approximate_posterior()]. #' #' @param loo_approximation What type of approximation of the loo_i's should be used? #' The default is `"plpd"` (the log predictive density using the posterior expectation). #' There are six different methods implemented to approximate loo_i's #' (see the references for more details): #' * `"plpd"`: uses the lpd based on point estimates (i.e., \eqn{p(y_i|\hat{\theta})}). #' * `"lpd"`: uses the lpds (i,e., \eqn{p(y_i|y)}). #' * `"tis"`: uses truncated importance sampling to approximate PSIS-LOO. #' * `"waic"`: uses waic (i.e., \eqn{p(y_i|y) - p_{waic}}). #' * `"waic_grad_marginal"`: uses waic approximation using first order delta #' method and posterior marginal variances to approximate \eqn{p_{waic}} (ie. #' \eqn{p(y_i|\hat{\theta})}-p_waic_grad_marginal). Requires gradient of #' likelihood function. #' * `"waic_grad"`: uses waic approximation using first order delta method and #' posterior covariance to approximate \eqn{p_{waic}} (ie. #' \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient of likelihood #' function. #' * `"waic_hess"`: uses waic approximation using second order delta method and #' posterior covariance to approximate \eqn{p_{waic}} (ie. #' \eqn{p(y_i|\hat{\theta})}-p_waic_grad). Requires gradient and Hessian of #' likelihood function. #' #' As point estimates of \eqn{\hat{\theta}}, the posterior expectations #' of the parameters are used. #' #' @param loo_approximation_draws The number of posterior draws used when #' integrating over the posterior. This is used if `loo_approximation` is set #' to `"lpd"`, `"waic"`, or `"tis"`. #' #' @param estimator How should `elpd_loo`, `p_loo` and `looic` be estimated? #' The default is `"diff_srs"`. #' * `"diff_srs"`: uses the difference estimator with simple random sampling #' (srs). `p_loo` is estimated using standard srs. #' * `"hh"`: uses the Hansen-Hurwitz estimator with sampling proportional to #' size, where `abs` of loo_approximation is used as size. #' * `"srs"`: uses simple random sampling and ordinary estimation. #' #' @param llgrad The gradient of the log-likelihood. This #' is only used when `loo_approximation` is `"waic_grad"`, #' `"waic_grad_marginal"`, or `"waic_hess"`. The default is `NULL`. #' @param llhess The hessian of the log-likelihood. This is only used #' with `loo_approximation = "waic_hess"`. The default is `NULL`. #' loo_subsample.function <- function(x, ..., data = NULL, draws = NULL, observations = 400, log_p = NULL, log_g = NULL, r_eff = NULL, save_psis = FALSE, cores = getOption("mc.cores", 1), loo_approximation = "plpd", loo_approximation_draws = NULL, estimator = "diff_srs", llgrad = NULL, llhess = NULL) { cores <- loo_cores(cores) # Asserting inputs .llfun <- validate_llfun(x) stopifnot(is.data.frame(data) || is.matrix(data), !is.null(draws)) observations <- assert_observations(observations, N = dim(data)[1], estimator) checkmate::assert_numeric(log_p, len = length(log_g), null.ok = TRUE) checkmate::assert_null(dim(log_p)) checkmate::assert_numeric(log_g, len = length(log_p), null.ok = TRUE) checkmate::assert_null(dim(log_g)) if (is.null(log_p) && is.null(log_g)) { if (is.null(r_eff)) { throw_loo_r_eff_warning() } else { r_eff <- prepare_psis_r_eff(r_eff, len = dim(data)[1]) } } checkmate::assert_flag(save_psis) cores <- loo_cores(cores) checkmate::assert_choice(loo_approximation, choices = loo_approximation_choices(), null.ok = FALSE) checkmate::assert_int(loo_approximation_draws, lower = 1, upper = .ndraws(draws), null.ok = TRUE) checkmate::assert_choice(estimator, choices = estimator_choices()) .llgrad <- .llhess <- NULL if (!is.null(llgrad)) .llgrad <- validate_llfun(llgrad) if (!is.null(llhess)) .llhess <- validate_llfun(llhess) # Fallbacks if (is.null(observations)) { if (is.null(log_p) && is.null(log_g)) { loo_obj <- loo.function( .llfun, ..., data = data, draws = draws, r_eff = r_eff, save_psis = save_psis, cores = cores ) } else { loo_obj <- loo_approximate_posterior.function( .llfun, ..., log_p = log_p, log_g = log_g, data = data, draws = draws, save_psis = save_psis, cores = cores ) } return(loo_obj) } # Compute loo approximation elpd_loo_approx <- elpd_loo_approximation( .llfun = .llfun, data = data, draws = draws, cores = cores, loo_approximation = loo_approximation, loo_approximation_draws = loo_approximation_draws, .llgrad = .llgrad, .llhess = .llhess ) # Draw subsample of observations if (length(observations) == 1) { # Compute idxs idxs <- subsample_idxs( estimator = estimator, elpd_loo_approximation = elpd_loo_approx, observations = observations ) } else { # Compute idxs idxs <- compute_idxs(observations) } data_subsample <- data[idxs$idx,, drop = FALSE] if (length(r_eff) > 1) { r_eff <- r_eff[idxs$idx] } # Compute elpd_loo if (!is.null(log_p) && !is.null(log_g)) { loo_obj <- loo_approximate_posterior.function( x = .llfun, data = data_subsample, draws = draws, log_p = log_p, log_g = log_g, save_psis = save_psis, cores = cores ) } else { loo_obj <- loo.function( x = .llfun, data = data_subsample, draws = draws, r_eff = r_eff, save_psis = save_psis, cores = cores ) } # Construct ss object and estimate loo_ss <- psis_loo_ss_object(x = loo_obj, idxs = idxs, elpd_loo_approx = elpd_loo_approx, loo_approximation = loo_approximation, loo_approximation_draws = loo_approximation_draws, estimator = estimator, .llfun = .llfun, .llgrad = .llgrad, .llhess = .llhess, data_dim = dim(data), ndraws = .ndraws(draws)) loo_ss } #' Update `psis_loo_ss` objects #' #' @details #' If `observations` is updated then if a vector of indices or a `psis_loo_ss` #' object is supplied the updated object will have exactly the observations #' indicated by the vector or `psis_loo_ss` object. If a single integer is #' supplied, new observations will be sampled to reach the supplied sample size. #' #' @export #' @inheritParams loo_subsample.function #' @param data,draws See [loo_subsample.function()]. #' @param object A `psis_loo_ss` object to update. #' @param ... Currently not used. #' @return A `psis_loo_ss` object. #' @importFrom stats update update.psis_loo_ss <- function(object, ..., data = NULL, draws = NULL, observations = NULL, r_eff = NULL, cores = getOption("mc.cores", 1), loo_approximation = NULL, loo_approximation_draws = NULL, llgrad = NULL, llhess = NULL) { # Fallback if (is.null(observations) & is.null(loo_approximation) & is.null(loo_approximation_draws) & is.null(llgrad) & is.null(llhess)) return(object) if (!is.null(data)) { stopifnot(is.data.frame(data) || is.matrix(data)) checkmate::assert_true(all(dim(data) == object$loo_subsampling$data_dim)) } if (!is.null(draws)) { # No current checks } cores <- loo_cores(cores) # Update elpd approximations if (!is.null(loo_approximation) | !is.null(loo_approximation_draws)) { stopifnot(is.data.frame(data) || is.matrix(data) & !is.null(draws)) if (object$loo_subsampling$estimator %in% "hh_pps") { # HH estimation uses elpd_loo approx to sample, # so updating it will lead to incorrect results stop("Can not update loo_approximation when using PPS sampling.", call. = FALSE) } if (is.null(loo_approximation)) loo_approximation <- object$loo_subsampling$loo_approximation if (is.null(loo_approximation_draws)) loo_approximation_draws <- object$loo_subsampling$loo_approximation_draws if (is.null(llgrad)) .llgrad <- object$loo_subsampling$.llgrad else .llgrad <- validate_llfun(llgrad) if (is.null(llhess)) .llhess <- object$loo_subsampling$.llhess else .llhess <- validate_llfun(llhess) # Compute loo approximation elpd_loo_approx <- elpd_loo_approximation(.llfun = object$loo_subsampling$.llfun, data = data, draws = draws, cores = cores, loo_approximation = loo_approximation, loo_approximation_draws = loo_approximation_draws, .llgrad = .llgrad, .llhess = .llhess) # Update object object$loo_subsampling$elpd_loo_approx <- elpd_loo_approx object$loo_subsampling$loo_approximation <- loo_approximation object$loo_subsampling["loo_approximation_draws"] <- list(loo_approximation_draws) object$loo_subsampling$.llgrad <- .llgrad object$loo_subsampling$.llhess <- .llhess object$pointwise[, "elpd_loo_approx"] <- object$loo_subsampling$elpd_loo_approx[object$pointwise[, "idx"]] } # Update observations if (!is.null(observations)) { observations <- assert_observations(observations, N = object$loo_subsampling$data_dim[1], object$loo_subsampling$estimator) if (length(observations) == 1) { checkmate::assert_int(observations, lower = nobs(object) + 1) stopifnot(is.data.frame(data) || is.matrix(data) & !is.null(draws)) } # Compute subsample indecies if (length(observations) > 1) { idxs <- compute_idxs(observations) } else { current_obs <- nobs(object) # If sampling with replacement if (object$loo_subsampling$estimator %in% c("hh_pps")) { idxs <- subsample_idxs(estimator = object$loo_subsampling$estimator, elpd_loo_approximation = object$loo_subsampling$elpd_loo_approx, observations = observations - current_obs) } # If sampling without replacement if (object$loo_subsampling$estimator %in% c("diff_srs", "srs")) { current_idxs <- obs_idx(object, rep = FALSE) new_idx <- (1:length(object$loo_subsampling$elpd_loo_approx))[-current_idxs] idxs <- subsample_idxs(estimator = object$loo_subsampling$estimator, elpd_loo_approximation = object$loo_subsampling$elpd_loo_approx[-current_idxs], observations = observations - current_obs) idxs$idx <- new_idx[idxs$idx] } } # Identify how to update object cidxs <- compare_idxs(idxs, object) # Compute new observations if (!is.null(cidxs$new)) { stopifnot(is.data.frame(data) || is.matrix(data) & !is.null(draws)) data_new_subsample <- data[cidxs$new$idx,, drop = FALSE] if (length(r_eff) > 1) r_eff <- r_eff[cidxs$new$idx] if (!is.null(object$approximate_posterior$log_p) & !is.null(object$approximate_posterior$log_g)) { loo_obj <- loo_approximate_posterior.function(x = object$loo_subsampling$.llfun, data = data_new_subsample, draws = draws, log_p = object$approximate_posterior$log_p, log_g = object$approximate_posterior$log_g, save_psis = !is.null(object$psis_object), cores = cores) } else { loo_obj <- loo.function(x = object$loo_subsampling$.llfun, data = data_new_subsample, draws = draws, r_eff = r_eff, save_psis = !is.null(object$psis_object), cores = cores) } # Add stuff to pointwise loo_obj$pointwise <- add_subsampling_vars_to_pointwise(loo_obj$pointwise, cidxs$new, object$loo_subsampling$elpd_loo_approx) } else { loo_obj <- NULL } if (length(observations) == 1) { # Add new samples pointwise and diagnostic object <- rbind.psis_loo_ss(object, x = loo_obj) # Update m_i for current pointwise (diagnostic stay the same) object$pointwise <- update_m_i_in_pointwise(object$pointwise, cidxs$add, type = "add") } else { # Add new samples pointwise and diagnostic object <- rbind.psis_loo_ss(object, loo_obj) # Replace m_i current pointwise and diagnostics object$pointwise <- update_m_i_in_pointwise(object$pointwise, cidxs$add, type = "replace") # Remove samples object <- remove_idx.psis_loo_ss(object, idxs = cidxs$remove) stopifnot(setequal(obs_idx(object), observations)) # Order object as in observations object <- order.psis_loo_ss(object, observations) } } # Compute estimates if (object$loo_subsampling$estimator == "hh_pps") { object <- loo_subsample_estimation_hh(object) } else if (object$loo_subsampling$estimator == "diff_srs") { object <- loo_subsample_estimation_diff_srs(object) } else if (object$loo_subsampling$estimator == "srs") { object <- loo_subsample_estimation_srs(object) } else { stop("No correct estimator used.") } assert_psis_loo_ss(object) object } #' Get observation indices used in subsampling #' #' @param x A `psis_loo_ss` object. #' @param rep If sampling with replacement is used, an observation can have #' multiple samples and these are then repeated in the returned object if #' `rep=TRUE` (e.g., a vector `c(1,1,2)` indicates that observation 1 has been #' subampled two times). If `rep=FALSE` only the unique indices are returned. #' #' @return An integer vector. #' #' @export obs_idx <- function(x, rep = TRUE) { checkmate::assert_class(x, "psis_loo_ss") if (rep) { idxs <- as.integer(rep(x$pointwise[,"idx"], x$pointwise[,"m_i"])) } else { idxs <- as.integer(x$pointwise[,"idx"]) } idxs } #' The number of observations in a `psis_loo_ss` object. #' @importFrom stats nobs #' @param object a `psis_loo_ss` object. #' @param ... Currently unused. #' @export nobs.psis_loo_ss <- function(object, ...) { as.integer(sum(object$pointwise[,"m_i"])) } # internal ---------------------------------------------------------------- #' The possible choices of loo_approximations implemented #' #' @details #' The choice `psis` is returned if a `psis_loo` object #' is converted to a `psis_loo_ss` object with `as.psis_loo_ss()`. #' But `psis` cannot be chosen in the api of `loo_subsample()`. #' #' @noRd #' @param api The choices available in the loo API or all possible choices. #' @return A character vector of allowed choices. loo_approximation_choices <- function(api = TRUE) { lac <- c("plpd", "lpd", "waic", "waic_grad_marginal", "waic_grad", "waic_hess", "tis", "sis", "none") if (!api) lac <- c(lac, "psis") lac } #' The estimators implemented #' #' @noRd #' @return A character vector of allowed choices. estimator_choices <- function() { c("hh_pps", "diff_srs", "srs") } ## Approximate elpd ----- #' Utility function to apply user-specified log-likelihood to a single data point #' @details #' See [elpd_loo_approximation] and [compute_lpds] for usage examples #' @noRd #' #' @return lpd value for a single data point i lpd_i <- function(i, llfun, data, draws) { ll_i <- llfun(data_i = data[i,, drop=FALSE], draws = draws) ll_i <- as.vector(ll_i) lpd_i <- logMeanExp(ll_i) lpd_i } #' Utility function to compute lpd using user-defined likelihood function #' using platform-dependent parallel backends when cores > 1 #' #' @details #' See [elpd_loo_approximation] for usage examples #' #' @noRd #' @return a vector of computed log probability densities compute_lpds <- function(N, data, draws, llfun, cores) { if (cores == 1) { lpds <- lapply(X = seq_len(N), FUN = lpd_i, llfun, data, draws) } else { if (.Platform$OS.type != "windows") { lpds <- mclapply(X = seq_len(N), mc.cores = cores, FUN = lpd_i, llfun, data, draws) } else { cl <- makePSOCKcluster(cores) on.exit(stopCluster(cl)) lpds <- parLapply(cl, X = seq_len(N), fun = lpd_i, llfun, data, draws) } } unlist(lpds) } #' Compute approximation to loo_i:s #' #' @details #' See [loo_subsample.function()] and the `loo_approximation` argument. #' @noRd #' @inheritParams loo_subsample.function #' #' @return a vector with approximations of elpd_{loo,i}s elpd_loo_approximation <- function(.llfun, data, draws, cores, loo_approximation, loo_approximation_draws = NULL, .llgrad = NULL, .llhess = NULL) { checkmate::assert_function(.llfun, args = c("data_i", "draws"), ordered = TRUE) stopifnot(is.data.frame(data) || is.matrix(data), !is.null(draws)) checkmate::assert_choice(loo_approximation, choices = loo_approximation_choices(), null.ok = FALSE) checkmate::assert_int(loo_approximation_draws, lower = 2, null.ok = TRUE) if (!is.null(.llgrad)) { checkmate::assert_function(.llgrad, args = c("data_i", "draws"), ordered = TRUE) } if (!is.null(.llhess)) { checkmate::assert_function(.llhess, args = c("data_i", "draws"), ordered = TRUE) } cores <- loo_cores(cores) N <- dim(data)[1] if (loo_approximation == "none") return(rep(1L,N)) if (loo_approximation %in% c("tis", "sis")) { draws <- .thin_draws(draws, loo_approximation_draws) is_values <- suppressWarnings(loo.function(.llfun, data = data, draws = draws, is_method = loo_approximation)) return(is_values$pointwise[, "elpd_loo"]) } if (loo_approximation == "waic") { draws <- .thin_draws(draws, loo_approximation_draws) waic_full_obj <- waic.function(.llfun, data = data, draws = draws) return(waic_full_obj$pointwise[,"elpd_waic"]) } # Compute the lpd or log p(y_i|y_{-i}) if (loo_approximation == "lpd") { draws <- .thin_draws(draws, loo_approximation_draws) lpds <- compute_lpds(N, data, draws, .llfun, cores) return(lpds) # Use only the lpd } # Compute the point lpd or log p(y_i|\hat{\theta}) - also used in waic_delta approaches if (loo_approximation == "plpd" | loo_approximation == "waic_grad" | loo_approximation == "waic_grad_marginal" | loo_approximation == "waic_hess") { draws <- .thin_draws(draws, loo_approximation_draws) point_est <- .compute_point_estimate(draws) lpds <- compute_lpds(N, data, point_est, .llfun, cores) if (loo_approximation == "plpd") return(lpds) # Use only the lpd } if (loo_approximation == "waic_grad" | loo_approximation == "waic_grad_marginal" | loo_approximation == "waic_hess") { checkmate::assert_true(!is.null(.llgrad)) point_est <- .compute_point_estimate(draws) # Compute the lpds lpds <- compute_lpds(N, data, point_est, .llfun, cores) if (loo_approximation == "waic_grad" | loo_approximation == "waic_hess") { cov_est <- stats::cov(draws) } if (loo_approximation == "waic_grad_marginal") { marg_vars <- apply(draws, MARGIN = 2, var) } p_eff_approx <- numeric(N) if (cores>1) warning("Multicore is not implemented for waic_delta", call. = FALSE) if (loo_approximation == "waic_grad") { for(i in 1:nrow(data)) { grad_i <- t(.llgrad(data[i,,drop = FALSE], point_est)) local_cov <- cov_est[rownames(grad_i), rownames(grad_i)] p_eff_approx[i] <- t(grad_i) %*% local_cov %*% grad_i } } else if (loo_approximation == "waic_grad_marginal") { for(i in 1:nrow(data)) { grad_i <- t(.llgrad(data[i,,drop = FALSE], point_est)) p_eff_approx[i] <- sum(grad_i * marg_vars[rownames(grad_i)] * grad_i) } } else if (loo_approximation == "waic_hess") { checkmate::assert_true(!is.null(.llhess)) for(i in 1:nrow(data)) { grad_i <- t(.llgrad(data[i,,drop = FALSE], point_est)) hess_i <- .llhess(data_i = data[i,,drop = FALSE], draws = point_est[,rownames(grad_i), drop = FALSE])[,,1] local_cov <- cov_est[rownames(grad_i), rownames(grad_i)] p_eff_approx[i] <- t(grad_i) %*% local_cov %*% grad_i + 0.5 * sum(diag(local_cov %*% hess_i %*% local_cov %*% hess_i)) } } else { stop(loo_approximation, " is not implemented!", call. = FALSE) } return(lpds - p_eff_approx) } } #' Compute a point estimate from a draws object #' #' @details This is a generic function to thin draws from arbitrary draws #' objects. The function is internal and should only be used by developers to #' enable [loo_subsample()] for arbitrary draws objects. #' #' @param draws A draws object with draws from the posterior. #' @return A 1 by P matrix with point estimates from a draws object. #' @keywords internal #' @export .compute_point_estimate <- function(draws) { UseMethod(".compute_point_estimate") } .compute_point_estimate.matrix <- function(draws) { t(as.matrix(colMeans(draws))) } .compute_point_estimate.default <- function(draws) { stop(".compute_point_estimate() has not been implemented for objects of class '", class(draws), "'") } #' Thin a draws object #' #' @details This is a generic function to thin draws from arbitrary draws #' objects. The function is internal and should only be used by developers to #' enable [loo_subsample()] for arbitrary draws objects. #' #' @param draws A draws object with posterior draws. #' @param loo_approximation_draws The number of posterior draws to return (ie after thinning). #' @keywords internal #' @export #' @return A thinned draws object. .thin_draws <- function(draws, loo_approximation_draws) { UseMethod(".thin_draws") } .thin_draws.matrix <- function(draws, loo_approximation_draws) { if (is.null(loo_approximation_draws)) return(draws) checkmate::assert_int(loo_approximation_draws, lower = 1, upper = .ndraws(draws), null.ok = TRUE) S <- .ndraws(draws) idx <- 1:loo_approximation_draws * S %/% loo_approximation_draws draws <- draws[idx, , drop = FALSE] draws } .thin_draws.numeric <- function(draws, loo_approximation_draws) { .thin_draws.matrix(as.matrix(draws), loo_approximation_draws) } .thin_draws.default <- function(draws, loo_approximation_draws) { stop(".thin_draws() has not been implemented for objects of class '", class(draws), "'") } #' The number of posterior draws in a draws object. #' #' @details This is a generic function to return the total number of draws from #' an arbitrary draws objects. The function is internal and should only be #' used by developers to enable [loo_subsample()] for arbitrary draws objects. #' #' @param x A draws object with posterior draws. #' @return An integer with the number of draws. #' @keywords internal #' @export .ndraws <- function(x) { UseMethod(".ndraws") } .ndraws.matrix <- function(x) { nrow(x) } .ndraws.default <- function(x) { stop(".ndraws() has not been implemented for objects of class '", class(x), "'") } ## Subsampling ----- #' Subsampling strategy #' #' @noRd #' @param estimator The estimator to use, see `estimator_choices()`. #' @param elpd_loo_approximation A vector of loo approximations, see `elpd_loo_approximation()`. #' @param observations The total number of subsample observations to sample. #' @return A `subsample_idxs` data frame. subsample_idxs <- function(estimator, elpd_loo_approximation, observations) { checkmate::assert_choice(estimator, choices = estimator_choices()) checkmate::assert_numeric(elpd_loo_approximation) checkmate::assert_int(observations) if (estimator == "hh_pps") { pi_values <- pps_elpd_loo_approximation_to_pis(elpd_loo_approximation) idxs_df <- pps_sample(observations, pis = pi_values) } if (estimator == "diff_srs" | estimator == "srs") { if (observations > length(elpd_loo_approximation)) { stop("'observations' is larger than the total sample size in 'data'.", call. = FALSE) } idx <- 1:length(elpd_loo_approximation) idx_m <- idx[order(stats::runif(length(elpd_loo_approximation)))][1:observations] idx_m <- idx_m[order(idx_m)] idxs_df <- data.frame(idx=as.integer(idx_m), m_i=1L) } assert_subsample_idxs(x = idxs_df) idxs_df } #' Compute pis from approximation for use in pps sampling. #' @noRd #' @details pis are the sampling probabilities and sum to 1. #' @inheritParams subsample_idxs #' @return A vector of pis. pps_elpd_loo_approximation_to_pis <- function(elpd_loo_approximation) { checkmate::assert_numeric(elpd_loo_approximation) pi_values <- abs(elpd_loo_approximation) pi_values <- pi_values/sum(pi_values) # \tilde{\pi} pi_values } #' Compute subsampling indices from an observation vector #' @noRd #' @param observation A vector of indices. #' @return A `subsample_idxs` data frame. compute_idxs <- function(observations) { checkmate::assert_integer(observations, lower = 1, min.len = 2, any.missing = FALSE) tab <- table(observations) idxs_df <- data.frame(idx = as.integer(names(tab)), m_i = as.integer(unname(tab))) assert_subsample_idxs(idxs_df) idxs_df } #' Compare the indecies to prepare handling #' #' @details #' The function compares the object and sampled indices into `new` #' (observations not in `object`), `add` (observations in `object`), and #' `remove` (observations in `object` but not in idxs). #' @noRd #' @param idxs A `subsample_idxs` data frame. #' @param object A `psis_loo_ss` object. #' @return A list of three `subsample_idxs` data frames. Elements without any #' observations return `NULL`. compare_idxs <- function(idxs, object) { assert_subsample_idxs(idxs) current_idx <- compute_idxs(obs_idx(object)) result <- list() new_idx <- !(idxs$idx %in% current_idx$idx) remove_idx <- !(current_idx$idx %in% idxs$idx) result$new <- idxs[new_idx, ] if (nrow(result$new) == 0) { result["new"] <- NULL } else { assert_subsample_idxs(result$new) } result$add <- idxs[!new_idx, ] if (nrow(result$add) == 0) { result["add"] <- NULL } else { assert_subsample_idxs(result$add) } result$remove <- current_idx[remove_idx, ] if (nrow(result$remove) == 0) { result["remove"] <- NULL } else { assert_subsample_idxs(result$remove) } result } #' Draw a PPS sample with replacement and return a idx_df #' @noRd #' @details #' We are sampling with replacement, hence we only want to compute elpd #' for each observation once. #' @param m The total sampling size. #' @param pis The probability of selecting each observation. #' @return a `subsample_idxs` data frame. pps_sample <- function(m, pis) { checkmate::assert_int(m) checkmate::assert_numeric(pis, min.len = 2, lower = 0, upper = 1) idx <- sample(1:length(pis), size = m, replace = TRUE, prob = pis) idxs_df <- as.data.frame(table(idx), stringsAsFactors = FALSE) colnames(idxs_df) <- c("idx", "m_i") idxs_df$idx <- as.integer(idxs_df$idx) idxs_df$m_i <- as.integer(idxs_df$m_i) assert_subsample_idxs(idxs_df) idxs_df } ## Constructor --- #' Construct a `psis_loo_ss} object #' #' @noRd #' @param x A `psis_loo` object. #' @param idxs a `subsample_idxs` data frame. #' @param elpd_loo_approximation A vector of loo approximations, see #' `elpd_loo_approximation()`. #' @inheritParams loo_subsample #' @param .llfun,.llgrad,.llhess See llfun, llgrad and llhess in `loo_subsample()`. #' @param data_dim Dimension of the data object. #' @param ndraws Dimension of the draws object. #' @return A `psis_loo_ss` object. psis_loo_ss_object <- function(x, idxs, elpd_loo_approx, loo_approximation, loo_approximation_draws, estimator, .llfun, .llgrad, .llhess, data_dim, ndraws) { # Assertions checkmate::assert_class(x, "psis_loo") assert_subsample_idxs(idxs) checkmate::assert_numeric(elpd_loo_approx, any.missing = FALSE) checkmate::assert_choice(loo_approximation, loo_approximation_choices()) checkmate::assert_int(loo_approximation_draws, null.ok = TRUE) checkmate::assert_choice(estimator, estimator_choices()) checkmate::assert_function(.llfun, args = c("data_i", "draws"), ordered = TRUE) checkmate::assert_function(.llgrad, args = c("data_i", "draws"), ordered = TRUE, null.ok = TRUE) checkmate::assert_function(.llhess, args = c("data_i", "draws"), ordered = TRUE, null.ok = TRUE) checkmate::assert_integer(data_dim, len = 2, lower = 1, any.missing = FALSE) checkmate::assert_int(ndraws, lower = 1) # Construct object class(x) <- c("psis_loo_ss", class(x)) x$pointwise <- add_subsampling_vars_to_pointwise(pointwise = x$pointwise, idxs, elpd_loo_approx) x$estimates <- cbind(x$estimates, matrix(0, nrow = nrow(x$estimates))) colnames(x$estimates)[ncol(x$estimates)] <- "subsampling SE" x$loo_subsampling <- list() x$loo_subsampling$elpd_loo_approx <- elpd_loo_approx x$loo_subsampling$loo_approximation <- loo_approximation x$loo_subsampling["loo_approximation_draws"] <- list(loo_approximation_draws) x$loo_subsampling$estimator <- estimator x$loo_subsampling$.llfun <- .llfun x$loo_subsampling[".llgrad"] <- list(.llgrad) x$loo_subsampling[".llhess"] <- list(.llhess) x$loo_subsampling$data_dim <- data_dim x$loo_subsampling$ndraws <- ndraws # Compute estimates if (estimator == "hh_pps") { x <- loo_subsample_estimation_hh(x) } else if (estimator == "diff_srs") { x <- loo_subsample_estimation_diff_srs(x) } else if (estimator == "srs") { x <- loo_subsample_estimation_srs(x) } else { stop("No correct estimator used.") } assert_psis_loo_ss(x) x } as.psis_loo_ss <- function(x) { UseMethod("as.psis_loo_ss") } as.psis_loo_ss.psis_loo_ss <- function(x) { x } as.psis_loo_ss.psis_loo <- function(x) { class(x) <- c("psis_loo_ss", class(x)) x$estimates <- cbind(x$estimates, matrix(0, nrow = nrow(x$estimates))) colnames(x$estimates)[ncol(x$estimates)] <- "subsampling SE" x$pointwise <- cbind(x$pointwise, matrix(1:nrow(x$pointwise), byrow = FALSE, ncol = 1), matrix(rep(1,nrow(x$pointwise)), byrow = FALSE, ncol = 1), x$pointwise[, "elpd_loo"]) ncp <- ncol(x$pointwise) colnames(x$pointwise)[(ncp-2):ncp] <- c("idx", "m_i", "elpd_loo_approx") x$loo_subsampling <- list(elpd_loo_approx=x$pointwise[, "elpd_loo"], loo_approximation = "psis", loo_approximation_draws = NULL, estimator = "diff_srs", data_dim = c(nrow(x$pointwise), NA), ndraws = NA) assert_psis_loo_ss(x) x } as.psis_loo <- function(x) { UseMethod("as.psis_loo") } as.psis_loo.psis_loo <- function(x) { x } as.psis_loo.psis_loo_ss <- function(x) { if (x$loo_subsampling$data_dim[1] == nrow(x$pointwise)) { x$estimates <- x$estimates[, 1:2] x$pointwise <- x$pointwise[, 1:5] x$loo_subsampling <- NULL loo_obj <- importance_sampling_loo_object(pointwise = x$pointwise[, 1:5], diagnostics = x$diagnostics, dims = attr(x, "dims"), is_method = "psis", is_object = x$psis_object) if (inherits(x, "psis_loo_ap")) { loo_obj$approximate_posterior <- list(log_p = x$approximate_posterior$log_p, log_g = x$approximate_posterior$log_g) class(loo_obj) <- c("psis_loo_ap", class(loo_obj)) assert_psis_loo_ap(loo_obj) } } else { stop("A subsampling loo object can only be coerced to a loo object ", "if all observations in data have been subsampled.", call. = FALSE) } loo_obj } #' Add subsampling information to the pointwise element in a `psis_loo` object. #' @noRd #' @param pointwise The `pointwise` element in a `psis_loo` object. #' @param idxs A `subsample_idxs` data frame. #' @param elpd_loo_approximation A vector of loo approximations, see `elpd_loo_approximation()`. #' @return A `pointwise` matrix with subsampling information. add_subsampling_vars_to_pointwise <- function(pointwise, idxs, elpd_loo_approx) { checkmate::assert_matrix(pointwise, any.missing = FALSE, min.cols = 5) checkmate::assert_names(colnames(pointwise), identical.to = c("elpd_loo","mcse_elpd_loo","p_loo","looic", "influence_pareto_k")) assert_subsample_idxs(idxs) checkmate::assert_numeric(elpd_loo_approx) pw <- cbind(as.data.frame(pointwise), idxs) pw$elpd_loo_approx <- elpd_loo_approx[idxs$idx] pw <- as.matrix(pw) rownames(pw) <- NULL assert_subsampling_pointwise(pw) pw } #' Add `psis_loo` object to a `psis_loo_ss` object #' @noRd #' @param object A `psis_loo_ss` object. #' @param x A `psis_loo` object. #' @return An updated `psis_loo_ss` object. rbind.psis_loo_ss <- function(object, x) { checkmate::assert_class(object, "psis_loo_ss") if (is.null(x)) return(object) # Fallback checkmate::assert_class(x, "psis_loo") assert_subsampling_pointwise(object$pointwise) assert_subsampling_pointwise(x$pointwise) checkmate::assert_disjunct(object$pointwise[, "idx"], x$pointwise[, "idx"]) object$pointwise <- rbind(object$pointwise, x$pointwise) object$diagnostics$pareto_k <- c(object$diagnostics$pareto_k, x$diagnostics$pareto_k) object$diagnostics$n_eff <- c(object$diagnostics$n_eff, x$diagnostics$n_eff) attr(object, "dims")[2] <- nrow(object$pointwise) object } #' Remove observations in `idxs` from object #' @noRd #' @param object A `psis_loo_ss` object. #' @param idxs A `subsample_idxs` data frame. #' @return A `psis_loo_ss` object. remove_idx.psis_loo_ss <- function(object, idxs) { checkmate::assert_class(object, "psis_loo_ss") if (is.null(idxs)) return(object) # Fallback assert_subsample_idxs(idxs) row_map <- data.frame( row_no = 1:nrow(object$pointwise), idx = object$pointwise[, "idx"] ) row_map <- merge(row_map, idxs, by = "idx", all.y = TRUE) object$pointwise <- object$pointwise[-row_map$row_no,,drop = FALSE] object$diagnostics$pareto_k <- object$diagnostics$pareto_k[-row_map$row_no] object$diagnostics$n_eff <- object$diagnostics$n_eff[-row_map$row_no] attr(object, "dims")[2] <- nrow(object$pointwise) object } #' Order object by `observations`. #' @noRd #' @param x A `psis_loo_ss` object. #' @param observations A vector with indices. #' @return An ordered `psis_loo_ss` object. order.psis_loo_ss <- function(x, observations) { checkmate::assert_class(x, "psis_loo_ss") checkmate::assert_integer(observations, len = nobs(x)) if (identical(obs_idx(x), observations)) return(x) # Fallback checkmate::assert_set_equal(obs_idx(x), observations) row_map_x <- data.frame(row_no_x = 1:nrow(x$pointwise), idx = x$pointwise[, "idx"]) row_map_obs <- data.frame(row_no_obs = 1:length(observations), idx = observations) row_map <- merge(row_map_obs, row_map_x, by = "idx", sort = FALSE) x$pointwise <- x$pointwise[row_map$row_no_x,,drop = FALSE] x$diagnostics$pareto_k <- x$diagnostics$pareto_k[row_map$row_no_x] x$diagnostics$n_eff <- x$diagnostics$n_eff[row_map$row_no_x] x } #' Update m_i in a `pointwise` element. #' @noRd #' @param x A `psis_loo_ss` `pointwise` data frame. #' @param idxs A `subsample_idxs` data frame. #' @param type should the m_i:s in `idxs` `"replace"` the current m_i:s or #' `"add"` to them. #' @return An ordered `psis_loo_ss` object. update_m_i_in_pointwise <- function(pointwise, idxs, type = "replace") { assert_subsampling_pointwise(pointwise) if (is.null(idxs)) return(pointwise) # Fallback assert_subsample_idxs(idxs) checkmate::assert_choice(type, choices = c("replace", "add")) row_map <- data.frame(row_no = 1:nrow(pointwise), idx = pointwise[, "idx"]) row_map <- merge(row_map, idxs, by = "idx", all.y = TRUE) if (type == "replace") { pointwise[row_map$row_no, "m_i"] <- row_map$m_i } if (type == "add") { pointwise[row_map$row_no, "m_i"] <- pointwise[row_map$row_no, "m_i"] + row_map$m_i } pointwise } ## Estimation --- #' Estimate the elpd using the Hansen-Hurwitz estimator #' @noRd #' @param x A `psis_loo_ss` object. #' @return A `psis_loo_ss` object. loo_subsample_estimation_hh <- function(x) { checkmate::assert_class(x, "psis_loo_ss") N <- length(x$loo_subsampling$elpd_loo_approx) pis <- pps_elpd_loo_approximation_to_pis(x$loo_subsampling$elpd_loo_approx) pis_sample <- pis[x$pointwise[,"idx"]] hh_elpd_loo <- whhest(z = pis_sample, m_i = x$pointwise[, "m_i"], y = x$pointwise[, "elpd_loo"], N) srs_elpd_loo <- srs_est(y = x$pointwise[, "elpd_loo"], y_approx = pis_sample) x$estimates["elpd_loo", "Estimate"] <- hh_elpd_loo$y_hat_ppz if (hh_elpd_loo$hat_v_y_ppz > 0) { x$estimates["elpd_loo", "SE"] <- sqrt(hh_elpd_loo$hat_v_y_ppz) } else { warning("Negative estimate of SE, more subsampling obs. needed.", call. = FALSE) x$estimates["elpd_loo", "SE"] <- NaN } x$estimates["elpd_loo", "subsampling SE"] <- sqrt(hh_elpd_loo$v_hat_y_ppz) hh_p_loo <- whhest(z = pis_sample, m_i = x$pointwise[,"m_i"], y = x$pointwise[,"p_loo"], N) x$estimates["p_loo", "Estimate"] <- hh_p_loo$y_hat_ppz if (hh_p_loo$hat_v_y_ppz > 0) { x$estimates["p_loo", "SE"] <- sqrt(hh_p_loo$hat_v_y_ppz) } else { warning("Negative estimate of SE, more subsampling obs. needed.", call. = FALSE) x$estimates["elpd_loo", "SE"] <- NaN } x$estimates["p_loo", "subsampling SE"] <- sqrt(hh_p_loo$v_hat_y_ppz) update_psis_loo_ss_estimates(x) } #' Update a `psis_loo_ss} object with generic estimates #' #' @noRd #' @details #' Updates a `psis_loo_ss` with generic estimates (looic) #' and updates components in the object based on x$estimate. #' @param x A `psis_loo_ss` object. #' @return x A `psis_loo_ss` object. update_psis_loo_ss_estimates <- function(x) { checkmate::assert_class(x, "psis_loo_ss") x$estimates["looic", "Estimate"] <- (-2) * x$estimates["elpd_loo", "Estimate"] x$estimates["looic", "SE"] <- 2 * x$estimates["elpd_loo", "SE"] x$estimates["looic", "subsampling SE"] <- 2 * x$estimates["elpd_loo", "subsampling SE"] x$elpd_loo <- x$estimates["elpd_loo", "Estimate"] x$p_loo <- x$estimates["p_loo", "Estimate"] x$looic <- x$estimates["looic", "Estimate"] x$se_elpd_loo <- x$estimates["elpd_loo", "SE"] x$se_p_loo <- x$estimates["p_loo", "SE"] x$se_looic <- x$estimates["looic", "SE"] x } #' Weighted Hansen-Hurwitz estimator #' @noRd #' @param z Normalized probabilities for the observation. #' @param m_i The number of times obs i was selected. #' @param y The values observed. #' @param N The total number of observations in finite population. #' @return A list with estimates. whhest <- function(z, m_i, y, N) { checkmate::assert_numeric(z, lower = 0, upper = 1) checkmate::assert_numeric(y, len = length(z)) checkmate::assert_integerish(m_i, len = length(z)) est_list <- list(m = sum(m_i)) est_list$y_hat_ppz <- sum(m_i*(y/z))/est_list$m est_list$v_hat_y_ppz <- (sum(m_i*((y/z - est_list$y_hat_ppz)^2))/est_list$m)/(est_list$m-1) # See unbiadness proof in supplementary material to the article est_list$hat_v_y_ppz <- (sum(m_i*(y^2/z)) / est_list$m) + est_list$v_hat_y_ppz / N - est_list$y_hat_ppz^2 / N est_list } #' Estimate elpd using the difference estimator and srs wor #' @noRd #' @param x A `psis_loo_ss` object. #' @return A `psis_loo_ss` object. loo_subsample_estimation_diff_srs <- function(x) { checkmate::assert_class(x, "psis_loo_ss") elpd_loo_est <- srs_diff_est(y_approx = x$loo_subsampling$elpd_loo_approx, y = x$pointwise[, "elpd_loo"], y_idx = x$pointwise[, "idx"]) x$estimates["elpd_loo", "Estimate"] <- elpd_loo_est$y_hat x$estimates["elpd_loo", "SE"] <- sqrt(elpd_loo_est$hat_v_y) x$estimates["elpd_loo", "subsampling SE"] <- sqrt(elpd_loo_est$v_y_hat) p_loo_est <- srs_est(y = x$pointwise[, "p_loo"], y_approx = x$loo_subsampling$elpd_loo_approx) x$estimates["p_loo", "Estimate"] <- p_loo_est$y_hat x$estimates["p_loo", "SE"] <- sqrt(p_loo_est$hat_v_y) x$estimates["p_loo", "subsampling SE"] <- sqrt(p_loo_est$v_y_hat) update_psis_loo_ss_estimates(x) } #' Difference estimation using SRS-WOR sampling #' @noRd #' @param y_approx Approximated values of all observations. #' @param y The values observed. #' @param y_idx The index of `y` in `y_approx`. #' @return A list with estimates. srs_diff_est <- function(y_approx, y, y_idx) { checkmate::assert_numeric(y_approx) checkmate::assert_numeric(y, max.len = length(y_approx)) checkmate::assert_integerish(y_idx, len = length(y)) N <- length(y_approx) m <- length(y) y_approx_m <- y_approx[y_idx] e_i <- y - y_approx_m t_pi_tilde <- sum(y_approx) t_pi2_tilde <- sum(y_approx^2) t_e <- N * mean(e_i) t_hat_epsilon <- N * mean(y^2 - y_approx_m^2) est_list <- list(m = length(y), N = N) est_list$y_hat <- t_pi_tilde + t_e est_list$v_y_hat <- N^2 * (1 - m / N) * var(e_i) / m est_list$hat_v_y <- (t_pi2_tilde + t_hat_epsilon) - # a (has been checked) (1/N) * (t_e^2 - est_list$v_y_hat + 2 * t_pi_tilde * est_list$y_hat - t_pi_tilde^2) # b est_list } #' Estimate elpd using the standard SRS estimator and SRS WOR #' @noRd #' @param x A `psis_loo_ss` object. #' @return A `psis_loo_ss` object. loo_subsample_estimation_srs <- function(x) { checkmate::assert_class(x, "psis_loo_ss") elpd_loo_est <- srs_est(y = x$pointwise[, "elpd_loo"], y_approx = x$loo_subsampling$elpd_loo_approx) x$estimates["elpd_loo", "Estimate"] <- elpd_loo_est$y_hat x$estimates["elpd_loo", "SE"] <- sqrt(elpd_loo_est$hat_v_y) x$estimates["elpd_loo", "subsampling SE"] <- sqrt(elpd_loo_est$v_y_hat) p_loo_est <- srs_est(y = x$pointwise[, "p_loo"], y_approx = x$loo_subsampling$elpd_loo_approx) x$estimates["p_loo", "Estimate"] <- p_loo_est$y_hat x$estimates["p_loo", "SE"] <- sqrt(p_loo_est$hat_v_y) x$estimates["p_loo", "subsampling SE"] <- sqrt(p_loo_est$v_y_hat) update_psis_loo_ss_estimates(x) } #' Simple SRS-WOR estimation #' @noRd #' @param y The values observed. #' @param y_approx A vector of length N. #' @return A list of estimates. srs_est <- function(y, y_approx) { checkmate::assert_numeric(y) checkmate::assert_numeric(y_approx, min.len = length(y)) N <- length(y_approx) m <- length(y) est_list <- list(m = m) est_list$y_hat <- N * mean(y) est_list$v_y_hat <- N^2 * (1-m/N) * var(y)/m est_list$hat_v_y <- N * var(y) est_list } ## Specialized assertions of objects --- #' Assert that the object has the expected properties #' @noRd #' @param x An object to assert. #' @param N The total number of data points in data. #' @param estimator The estimator used. #' @return An asserted object of `x`. assert_observations <- function(x, N, estimator) { checkmate::assert_int(N) checkmate::assert_choice(estimator, choices = estimator_choices()) if (is.null(x)) return(x) if (checkmate::test_class(x, "psis_loo_ss")) { x <- obs_idx(x) checkmate::assert_integer(x, lower = 1, upper = N, any.missing = FALSE) return(x) } x <- as.integer(x) if (length(x) > 1) { checkmate::assert_integer(x, lower = 1, upper = N, any.missing = FALSE) if (estimator %in% "hh_pps") { message("Sampling proportional to elpd approximation and with replacement assumed.") } if (estimator %in% c("diff_srs", "srs")) { message("Simple random sampling with replacement assumed.") } } else { checkmate::assert_integer(x, lower = 1, any.missing = FALSE) } x } #' Assert that the object has the expected properties #' @noRd #' @inheritParams assert_observations #' @return An asserted object of `x`. assert_subsample_idxs <- function(x) { checkmate::assert_data_frame(x, types = c("integer", "integer"), any.missing = FALSE, min.rows = 1, col.names = "named") checkmate::assert_names(names(x), identical.to = c("idx", "m_i")) checkmate::assert_integer(x$idx, lower = 1, any.missing = FALSE, unique = TRUE) checkmate::assert_integer(x$m_i, lower = 1, any.missing = FALSE) x } #' Assert that the object has the expected properties #' @noRd #' @inheritParams assert_observations #' @return An asserted object of `x`. assert_psis_loo_ss <- function(x) { checkmate::assert_class(x, "psis_loo_ss") checkmate::assert_names(names(x), must.include = c("estimates", "pointwise", "diagnostics", "psis_object", "loo_subsampling")) checkmate::assert_names(rownames(x$estimates), must.include = c("elpd_loo", "p_loo", "looic")) checkmate::assert_names(colnames(x$estimates), must.include = c("Estimate", "SE", "subsampling SE")) assert_subsampling_pointwise(x$pointwise) checkmate::assert_names(names(x$loo_subsampling), must.include = c("elpd_loo_approx", "loo_approximation", "loo_approximation_draws", "estimator", "data_dim", "ndraws")) checkmate::assert_numeric(x$loo_subsampling$elpd_loo_approx, any.missing = FALSE, len = x$loo_subsampling$data_dim[1]) checkmate::assert_choice(x$loo_subsampling$loo_approximation, choices = loo_approximation_choices(api = FALSE)) checkmate::assert_int(x$loo_subsampling$loo_approximation_draws, null.ok = TRUE) checkmate::assert_choice(x$loo_subsampling$estimator, choices = estimator_choices()) checkmate::assert_integer(x$loo_subsampling$data_dim, any.missing = TRUE, len = 2) checkmate::assert_int(x$loo_subsampling$data_dim[1], na.ok = FALSE) checkmate::assert_integer(x$loo_subsampling$ndraws, len = 1, any.missing = TRUE) x } #' Assert that the object has the expected properties #' @noRd #' @inheritParams assert_observations #' @return An asserted object of `x`. assert_subsampling_pointwise <- function(x) { checkmate::assert_matrix(x, any.missing = FALSE, ncols = 8) checkmate::assert_names(colnames(x), identical.to = c("elpd_loo", "mcse_elpd_loo", "p_loo", "looic", "influence_pareto_k", "idx", "m_i", "elpd_loo_approx")) x } loo/R/loo_compare.R0000644000176200001440000002211214410420140013652 0ustar liggesusers#' Model comparison #' #' @description Compare fitted models based on [ELPD][loo-glossary]. #' #' By default the print method shows only the most important information. Use #' `print(..., simplify=FALSE)` to print a more detailed summary. #' #' @export #' @param x An object of class `"loo"` or a list of such objects. If a list is #' used then the list names will be used as the model names in the output. See #' **Examples**. #' @param ... Additional objects of class `"loo"`, if not passed in as a single #' list. #' #' @return A matrix with class `"compare.loo"` that has its own #' print method. See the **Details** section. #' #' @details #' When comparing two fitted models, we can estimate the difference in their #' expected predictive accuracy by the difference in #' [`elpd_loo`][loo-glossary] or `elpd_waic` (or multiplied by \eqn{-2}, if #' desired, to be on the deviance scale). #' #' When using `loo_compare()`, the returned matrix will have one row per model #' and several columns of estimates. The values in the #' [`elpd_diff`][loo-glossary] and [`se_diff`][loo-glossary] columns of the #' returned matrix are computed by making pairwise comparisons between each #' model and the model with the largest ELPD (the model in the first row). For #' this reason the `elpd_diff` column will always have the value `0` in the #' first row (i.e., the difference between the preferred model and itself) and #' negative values in subsequent rows for the remaining models. #' #' To compute the standard error of the difference in [ELPD][loo-glossary] --- #' which should not be expected to equal the difference of the standard errors #' --- we use a paired estimate to take advantage of the fact that the same #' set of \eqn{N} data points was used to fit both models. These calculations #' should be most useful when \eqn{N} is large, because then non-normality of #' the distribution is not such an issue when estimating the uncertainty in #' these sums. These standard errors, for all their flaws, should give a #' better sense of uncertainty than what is obtained using the current #' standard approach of comparing differences of deviances to a Chi-squared #' distribution, a practice derived for Gaussian linear models or #' asymptotically, and which only applies to nested models in any case. #' #' @seealso #' * The [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. #' @template loo-and-psis-references #' #' @examples #' # very artificial example, just for demonstration! #' LL <- example_loglik_array() #' loo1 <- loo(LL, r_eff = NA) # should be worst model when compared #' loo2 <- loo(LL + 1, r_eff = NA) # should be second best model when compared #' loo3 <- loo(LL + 2, r_eff = NA) # should be best model when compared #' #' comp <- loo_compare(loo1, loo2, loo3) #' print(comp, digits = 2) #' #' # show more details with simplify=FALSE #' # (will be the same for all models in this artificial example) #' print(comp, simplify = FALSE, digits = 3) #' #' # can use a list of objects with custom names #' # will use apple, banana, and cherry, as the names in the output #' loo_compare(list("apple" = loo1, "banana" = loo2, "cherry" = loo3)) #' #' \dontrun{ #' # works for waic (and kfold) too #' loo_compare(waic(LL), waic(LL - 10)) #' } #' loo_compare <- function(x, ...) { UseMethod("loo_compare") } #' @rdname loo_compare #' @export loo_compare.default <- function(x, ...) { if (is.loo(x)) { dots <- list(...) loos <- c(list(x), dots) } else { if (!is.list(x) || !length(x)) { stop("'x' must be a list if not a 'loo' object.") } if (length(list(...))) { stop("If 'x' is a list then '...' should not be specified.") } loos <- x } # If subsampling is used if (any(sapply(loos, inherits, "psis_loo_ss"))) { return(loo_compare.psis_loo_ss_list(loos)) } loo_compare_checks(loos) comp <- loo_compare_matrix(loos) ord <- loo_compare_order(loos) # compute elpd_diff and se_elpd_diff relative to best model rnms <- rownames(comp) diffs <- mapply(FUN = elpd_diffs, loos[ord[1]], loos[ord]) elpd_diff <- apply(diffs, 2, sum) se_diff <- apply(diffs, 2, se_elpd_diff) comp <- cbind(elpd_diff = elpd_diff, se_diff = se_diff, comp) rownames(comp) <- rnms class(comp) <- c("compare.loo", class(comp)) return(comp) } #' @rdname loo_compare #' @export #' @param digits For the print method only, the number of digits to use when #' printing. #' @param simplify For the print method only, should only the essential columns #' of the summary matrix be printed? The entire matrix is always returned, but #' by default only the most important columns are printed. print.compare.loo <- function(x, ..., digits = 1, simplify = TRUE) { xcopy <- x if (inherits(xcopy, "old_compare.loo")) { if (NCOL(xcopy) >= 2 && simplify) { patts <- "^elpd_|^se_diff|^p_|^waic$|^looic$" xcopy <- xcopy[, grepl(patts, colnames(xcopy))] } } else if (NCOL(xcopy) >= 2 && simplify) { xcopy <- xcopy[, c("elpd_diff", "se_diff")] } print(.fr(xcopy, digits), quote = FALSE) invisible(x) } # internal ---------------------------------------------------------------- #' Compute pointwise elpd differences #' @noRd #' @param loo_a,loo_b Two `"loo"` objects. elpd_diffs <- function(loo_a, loo_b) { pt_a <- loo_a$pointwise pt_b <- loo_b$pointwise elpd <- grep("^elpd", colnames(pt_a)) pt_b[, elpd] - pt_a[, elpd] } #' Compute standard error of the elpd difference #' @noRd #' @param diffs Vector of pointwise elpd differences se_elpd_diff <- function(diffs) { N <- length(diffs) # As `elpd_diff` is defined as the sum of N independent components, # we can compute the standard error by using the standard deviation # of the N components and multiplying by `sqrt(N)`. sqrt(N) * sd(diffs) } #' Perform checks on `"loo"` objects before comparison #' @noRd #' @param loos List of `"loo"` objects. #' @return Nothing, just possibly throws errors/warnings. loo_compare_checks <- function(loos) { ## errors if (length(loos) <= 1L) { stop("'loo_compare' requires at least two models.", call.=FALSE) } if (!all(sapply(loos, is.loo))) { stop("All inputs should have class 'loo'.", call.=FALSE) } Ns <- sapply(loos, function(x) nrow(x$pointwise)) if (!all(Ns == Ns[1L])) { stop("Not all models have the same number of data points.", call.=FALSE) } ## warnings yhash <- lapply(loos, attr, which = "yhash") yhash_ok <- sapply(yhash, function(x) { # ok only if all yhash are same (all NULL is ok) isTRUE(all.equal(x, yhash[[1]])) }) if (!all(yhash_ok)) { warning("Not all models have the same y variable. ('yhash' attributes do not match)", call. = FALSE) } if (all(sapply(loos, is.kfold))) { Ks <- unlist(lapply(loos, attr, which = "K")) if (!all(Ks == Ks[1])) { warning("Not all kfold objects have the same K value. ", "For a more accurate comparison use the same number of folds. ", call. = FALSE) } } else if (any(sapply(loos, is.kfold)) && any(sapply(loos, is.psis_loo))) { warning("Comparing LOO-CV to K-fold-CV. ", "For a more accurate comparison use the same number of folds ", "or loo for all models compared.", call. = FALSE) } } #' Find the model names associated with `"loo"` objects #' #' @export #' @keywords internal #' @param x List of `"loo"` objects. #' @return Character vector of model names the same length as `x.` #' find_model_names <- function(x) { stopifnot(is.list(x)) out_names <- character(length(x)) names1 <- names(x) names2 <- lapply(x, "attr", "model_name", exact = TRUE) names3 <- lapply(x, "[[", "model_name") names4 <- paste0("model", seq_along(x)) for (j in seq_along(x)) { if (isTRUE(nzchar(names1[j]))) { out_names[j] <- names1[j] } else if (length(names2[[j]])) { out_names[j] <- names2[[j]] } else if (length(names3[[j]])) { out_names[j] <- names3[[j]] } else { out_names[j] <- names4[j] } } out_names } #' Compute the loo_compare matrix #' @keywords internal #' @noRd #' @param loos List of `"loo"` objects. loo_compare_matrix <- function(loos){ tmp <- sapply(loos, function(x) { est <- x$estimates setNames(c(est), nm = c(rownames(est), paste0("se_", rownames(est)))) }) colnames(tmp) <- find_model_names(loos) rnms <- rownames(tmp) comp <- tmp ord <- loo_compare_order(loos) comp <- t(comp)[ord, ] patts <- c("elpd", "p_", "^waic$|^looic$", "^se_waic$|^se_looic$") col_ord <- unlist(sapply(patts, function(p) grep(p, colnames(comp))), use.names = FALSE) comp <- comp[, col_ord] comp } #' Computes the order of loos for comparison #' @noRd #' @keywords internal #' @param loos List of `"loo"` objects. loo_compare_order <- function(loos){ tmp <- sapply(loos, function(x) { est <- x$estimates setNames(c(est), nm = c(rownames(est), paste0("se_", rownames(est)))) }) colnames(tmp) <- find_model_names(loos) rnms <- rownames(tmp) ord <- order(tmp[grep("^elpd", rnms), ], decreasing = TRUE) ord } loo/R/tis.R0000644000176200001440000001242113701164066012172 0ustar liggesusers#' Truncated importance sampling (TIS) #' #' Implementation of truncated (self-normalized) importance sampling (TIS), #' truncated at S^(1/2) as recommended by Ionides (2008). #' #' @param log_ratios An array, matrix, or vector of importance ratios on the log #' scale (for Importance sampling LOO, these are *negative* log-likelihood #' values). See the **Methods (by class)** section below for a detailed #' description of how to specify the inputs for each method. #' @template cores #' @param ... Arguments passed on to the various methods. #' @param r_eff Vector of relative effective sample size estimates containing #' one element per observation. The values provided should be the relative #' effective sample sizes of `1/exp(log_ratios)` (i.e., `1/ratios`). #' This is related to the relative efficiency of estimating the normalizing #' term in self-normalizing importance sampling. See the [relative_eff()] #' helper function for computing `r_eff`. If using `psis` with #' draws of the `log_ratios` not obtained from MCMC then the warning #' message thrown when not specifying `r_eff` can be disabled by #' setting `r_eff` to `NA`. #' #' @return The `tis()` methods return an object of class `"tis"`, #' which is a named list with the following components: #' #' \describe{ #' \item{`log_weights`}{ #' Vector or matrix of smoothed (and truncated) but *unnormalized* log #' weights. To get normalized weights use the #' [`weights()`][weights.importance_sampling] method provided for objects of #' class `tis`. #' } #' \item{`diagnostics`}{ #' A named list containing one vector: #' * `pareto_k`: Not used in `tis`, all set to 0. #' * `n_eff`: Effective sample size estimates. #' } #' } #' #' Objects of class `"tis"` also have the following [attributes][attributes()]: #' \describe{ #' \item{`norm_const_log`}{ #' Vector of precomputed values of `colLogSumExps(log_weights)` that are #' used internally by the [weights()]method to normalize the log weights. #' } #' \item{`r_eff`}{ #' If specified, the user's `r_eff` argument. #' } #' \item{`tail_len`}{ #' Not used for `tis`. #' } #' \item{`dims`}{ #' Integer vector of length 2 containing `S` (posterior sample size) #' and `N` (number of observations). #' } #' \item{`method`}{ #' Method used for importance sampling, here `tis`. #' } #' } #' #' @seealso #' * [psis()] for approximate LOO-CV using PSIS. #' * [loo()] for approximate LOO-CV. #' * [pareto-k-diagnostic] for PSIS diagnostics. #' #' @references #' Ionides, Edward L. (2008). Truncated importance sampling. #' *Journal of Computational and Graphical Statistics* 17(2): 295--311. #' #' @examples #' log_ratios <- -1 * example_loglik_array() #' r_eff <- relative_eff(exp(-log_ratios)) #' tis_result <- tis(log_ratios, r_eff = r_eff) #' str(tis_result) #' #' # extract smoothed weights #' lw <- weights(tis_result) # default args are log=TRUE, normalize=TRUE #' ulw <- weights(tis_result, normalize=FALSE) # unnormalized log-weights #' #' w <- weights(tis_result, log=FALSE) # normalized weights (not log-weights) #' uw <- weights(tis_result, log=FALSE, normalize = FALSE) # unnormalized weights #' #' @export tis <- function(log_ratios, ...) UseMethod("tis") #' @export #' @templateVar fn tis #' @template array #' tis.array <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.array(log_ratios = log_ratios, ..., r_eff = r_eff, cores = cores, method = "tis") } #' @export #' @templateVar fn tis #' @template matrix #' tis.matrix <- function(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { importance_sampling.matrix(log_ratios, ..., r_eff = r_eff, cores = cores, method = "tis") } #' @export #' @templateVar fn tis #' @template vector #' tis.default <- function(log_ratios, ..., r_eff = NULL) { importance_sampling.default(log_ratios = log_ratios, ..., r_eff = r_eff, method = "tis") } #' @rdname psis #' @export is.tis <- function(x) { inherits(x, "tis") && is.list(x) } # internal ---------------------------------------------------------------- #' Truncated Importance Sampling on a single vector #' #' @noRd #' @param log_ratios_i A vector of log importance ratios (for `loo()`, negative #' log likelihoods). #' @param ... Not used. Included to conform to PSIS API. #' #' @details Implementation of Truncated importance sampling (TIS), a method for #' stabilizing importance ratios. The version of TIS implemented here #' corresponds to the algorithm presented in Ionides (2008) with truncation at #' sqrt(S). #' #' @return A named list containing: #' * `lw`: vector of unnormalized log weights #' * `pareto_k`: scalar Pareto k estimate. For 'tis', this defaults to 0. #' do_tis_i <- function(log_ratios_i, ...) { S <- length(log_ratios_i) log_Z <- logSumExp(log_ratios_i) - log(S) # Normalization term, c-hat in Ionides (2008) appendix log_cutpoint <- log_Z + 0.5 * log(S) lw_i <- pmin(log_ratios_i, log_cutpoint) list(log_weights = lw_i, pareto_k = 0) } loo/R/importance_sampling.R0000644000176200001440000001664013701164066015435 0ustar liggesusers#' A parent class for different importance sampling methods. #' @keywords internal #' @inheritParams psis #' @param method The importance sampling method to use. The following methods #' are implemented: #' * [`"psis"`][psis]: Pareto-Smoothed Importance Sampling (PSIS). Default method. #' * [`"tis"`][tis]: Truncated Importance Sampling (TIS) with truncation at #' `sqrt(S)`, where `S` is the number of posterior draws. #' * [`"sis"`][sis]: Standard Importance Sampling (SIS). #' importance_sampling <- function(log_ratios, method, ...) UseMethod("importance_sampling") #' @noRd #' @keywords internal #' @description #' Currently implemented importance sampling methods assert_importance_sampling_method_is_implemented <- function(x){ if (!x %in% implemented_is_methods()) { stop("Importance sampling method '", x, "' is not implemented. Implemented methods: '", paste0(implemented_is_methods, collapse = "', '"), "'") } } implemented_is_methods <- function() c("psis", "tis", "sis") #' Importance sampling of array #' @inheritParams psis #' @template is_method #' @keywords internal #' @export importance_sampling.array <- function(log_ratios, method, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) stopifnot(length(dim(log_ratios)) == 3) assert_importance_sampling_method_is_implemented(method) log_ratios <- validate_ll(log_ratios) log_ratios <- llarray_to_matrix(log_ratios) r_eff <- prepare_psis_r_eff(r_eff, len = ncol(log_ratios)) do_importance_sampling(log_ratios, r_eff = r_eff, cores = cores, method = method) } #' Importance sampling of matrices #' @inheritParams psis #' @template is_method #' @keywords internal #' @export importance_sampling.matrix <- function(log_ratios, method, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) assert_importance_sampling_method_is_implemented(method) log_ratios <- validate_ll(log_ratios) r_eff <- prepare_psis_r_eff(r_eff, len = ncol(log_ratios)) do_importance_sampling(log_ratios, r_eff = r_eff, cores = cores, method = method) } #' Importance sampling (default) #' @inheritParams psis #' @template is_method #' @keywords internal #' @export importance_sampling.default <- function(log_ratios, method, ..., r_eff = NULL) { stopifnot(is.null(dim(log_ratios)) || length(dim(log_ratios)) == 1) assert_importance_sampling_method_is_implemented(method) dim(log_ratios) <- c(length(log_ratios), 1) r_eff <- prepare_psis_r_eff(r_eff, len = 1) importance_sampling.matrix(log_ratios, r_eff = r_eff, cores = 1, method = method) } #' @export dim.importance_sampling <- function(x) { attr(x, "dims") } #' Extract importance sampling weights #' #' @export #' @export weights.importance_sampling #' @method weights importance_sampling #' @param object An object returned by [psis()], [tis()], or [sis()]. #' @param log Should the weights be returned on the log scale? Defaults to #' `TRUE`. #' @param normalize Should the weights be normalized? Defaults to `TRUE`. #' @param ... Ignored. #' #' @return The `weights()` method returns an object with the same dimensions as #' the `log_weights` component of `object`. The `normalize` and `log` #' arguments control whether the returned weights are normalized and whether #' or not to return them on the log scale. #' #' @examples #' # See the examples at help("psis") #' weights.importance_sampling <- function(object, ..., log = TRUE, normalize = TRUE) { out <- object[["log_weights"]] # smoothed but unnormalized log weights if (normalize) { out <- sweep(out, MARGIN = 2, STATS = attr(object, "norm_const_log"), # colLogSumExp(log_weights) check.margin = FALSE) } if (!log) { out <- exp(out) } return(out) } # internal ---------------------------------------------------------------- #' Structure the object returned by the importance_sampling methods #' #' @noRd #' @param unnormalized_log_weights Smoothed and possibly truncated log weights, #' but unnormalized. #' @param pareto_k Vector of GPD k estimates. #' @param tail_len Vector of tail lengths used to fit GPD. #' @param r_eff Vector of relative MCMC n_eff for `exp(log lik)` #' @template is_method #' @return A list of class `"psis"` with structure described in the main doc at #' the top of this file. #' importance_sampling_object <- function(unnormalized_log_weights, pareto_k, tail_len, r_eff, method) { stopifnot(is.matrix(unnormalized_log_weights)) methods <- unique(method) stopifnot(all(methods %in% implemented_is_methods())) if (length(methods) == 1) { method <- methods classes <- c(tolower(method), "importance_sampling", "list") } else { classes <- c("importance_sampling", "list") } norm_const_log <- matrixStats::colLogSumExps(unnormalized_log_weights) out <- structure( list( log_weights = unnormalized_log_weights, diagnostics = list(pareto_k = pareto_k, n_eff = NULL) ), # attributes norm_const_log = norm_const_log, tail_len = tail_len, r_eff = r_eff, dims = dim(unnormalized_log_weights), method = method, class = classes ) # need normalized weights (not on log scale) for psis_n_eff w <- weights(out, normalize = TRUE, log = FALSE) out$diagnostics[["n_eff"]] <- psis_n_eff(w, r_eff) return(out) } #' Do importance sampling given matrix of log weights #' #' @noRd #' @param lr Matrix of log ratios (`-loglik`) #' @param r_eff Vector of relative effective sample sizes #' @param cores User's integer `cores` argument #' @return A list with class `"psis"` and structure described in the main doc at #' the top of this file. #' do_importance_sampling <- function(log_ratios, r_eff, cores, method) { stopifnot(cores == as.integer(cores)) assert_importance_sampling_method_is_implemented(method) N <- ncol(log_ratios) S <- nrow(log_ratios) tail_len <- n_pareto(r_eff, S) if (method == "psis") { is_fun <- do_psis_i throw_tail_length_warnings(tail_len) } else if (method == "tis") { is_fun <- do_tis_i } else if (method == "sis") { is_fun <- do_sis_i } else { stop("Incorrect IS method.") } if (cores == 1) { lw_list <- lapply(seq_len(N), function(i) is_fun(log_ratios_i = log_ratios[, i], tail_len_i = tail_len[i])) } else { if (!os_is_windows()) { lw_list <- parallel::mclapply( X = seq_len(N), mc.cores = cores, FUN = function(i) is_fun(log_ratios_i = log_ratios[, i], tail_len_i = tail_len[i]) ) } else { cl <- parallel::makePSOCKcluster(cores) on.exit(parallel::stopCluster(cl)) lw_list <- parallel::parLapply( cl = cl, X = seq_len(N), fun = function(i) is_fun(log_ratios_i = log_ratios[, i], tail_len_i = tail_len[i]) ) } } log_weights <- psis_apply(lw_list, "log_weights", fun_val = numeric(S)) pareto_k <- psis_apply(lw_list, "pareto_k") throw_pareto_warnings(pareto_k) importance_sampling_object( unnormalized_log_weights = log_weights, pareto_k = pareto_k, tail_len = tail_len, r_eff = r_eff, method = rep(method, length(pareto_k)) # Conform to other attr that exist per obs. ) } loo/R/example_log_lik_array.R0000644000176200001440000000211013575772017015730 0ustar liggesusers#' Objects to use in examples and tests #' #' Example pointwise log-likelihood objects to use in demonstrations and tests. #' See the **Value** and **Examples** sections below. #' #' @export #' @return #' `example_loglik_array()` returns a 500 (draws) x 2 (chains) x 32 #' (observations) pointwise log-likelihood array. #' #' `example_loglik_matrix()` returns the same pointwise log-likelihood values #' as `example_loglik_array()` but reshaped into a 1000 (draws*chains) x 32 #' (observations) matrix. #' #' @examples #' LLarr <- example_loglik_array() #' (dim_arr <- dim(LLarr)) #' LLmat <- example_loglik_matrix() #' (dim_mat <- dim(LLmat)) #' #' all.equal(dim_mat[1], dim_arr[1] * dim_arr[2]) #' all.equal(dim_mat[2], dim_arr[3]) #' #' all.equal(LLarr[, 1, ], LLmat[1:500, ]) #' all.equal(LLarr[, 2, ], LLmat[501:1000, ]) #' example_loglik_array <- function() { # .example_loglik_array exists in R/sysdata.R return(.example_loglik_array) } #' @rdname example_loglik_array #' @export example_loglik_matrix <- function() { ll <- example_loglik_array() return(llarray_to_matrix(ll)) } loo/R/loo_model_weights.R0000644000176200001440000003567614410420140015101 0ustar liggesusers#' Model averaging/weighting via stacking or pseudo-BMA weighting #' #' Model averaging via stacking of predictive distributions, pseudo-BMA #' weighting or pseudo-BMA+ weighting with the Bayesian bootstrap. See Yao et #' al. (2018), Vehtari, Gelman, and Gabry (2017), and Vehtari, Simpson, #' Gelman, Yao, and Gabry (2019) for background. #' #' @export #' @param x A list of `"psis_loo"` objects (objects returned by [loo()]) or #' pointwise log-likelihood matrices or , one for each model. If the list #' elements are named the names will be used to label the models in the #' results. Each matrix/object should have dimensions \eqn{S} by \eqn{N}, #' where \eqn{S} is the size of the posterior sample (with all chains merged) #' and \eqn{N} is the number of data points. If `x` is a list of #' log-likelihood matrices then [loo()] is called internally on each matrix. #' Currently the `loo_model_weights()` function is not implemented to be used #' with results from K-fold CV, but you can still obtain weights using K-fold #' CV results by calling the `stacking_weights()` function directly. #' @param method Either `"stacking"` (the default) or `"pseudobma"`, indicating which method #' to use for obtaining the weights. `"stacking"` refers to stacking of #' predictive distributions and `"pseudobma"` refers to pseudo-BMA+ weighting #' (or plain pseudo-BMA weighting if argument `BB` is `FALSE`). #' @param BB Logical used when `"method"`=`"pseudobma"`. If #' `TRUE` (the default), the Bayesian bootstrap will be used to adjust #' the pseudo-BMA weighting, which is called pseudo-BMA+ weighting. It helps #' regularize the weight away from 0 and 1, so as to reduce the variance. #' @param BB_n For pseudo-BMA+ weighting only, the number of samples to use for #' the Bayesian bootstrap. The default is `BB_n=1000`. #' @param alpha Positive scalar shape parameter in the Dirichlet distribution #' used for the Bayesian bootstrap. The default is `alpha=1`, which #' corresponds to a uniform distribution on the simplex space. #' @param optim_method If `method="stacking"`, a string passed to the `method` #' argument of [stats::constrOptim()] to specify the optimization algorithm. #' The default is `optim_method="BFGS"`, but other options are available (see #' [stats::optim()]). #' @param optim_control If `method="stacking"`, a list of control parameters for #' optimization passed to the `control` argument of [stats::constrOptim()]. #' @param r_eff_list Optionally, a list of relative effective sample size #' estimates for the likelihood `(exp(log_lik))` of each observation in #' each model. See [psis()] and [relative_eff()] helper #' function for computing `r_eff`. If `x` is a list of `"psis_loo"` #' objects then `r_eff_list` is ignored. #' @template cores #' @param ... Unused, except for the generic to pass arguments to individual #' methods. #' #' @return A numeric vector containing one weight for each model. #' #' @details #' `loo_model_weights()` is a wrapper around the `stacking_weights()` and #' `pseudobma_weights()` functions that implements stacking, pseudo-BMA, and #' pseudo-BMA+ weighting for combining multiple predictive distributions. We can #' use approximate or exact leave-one-out cross-validation (LOO-CV) or K-fold CV #' to estimate the expected log predictive density (ELPD). #' #' The stacking method (`method="stacking"`), which is the default for #' `loo_model_weights()`, combines all models by maximizing the leave-one-out #' predictive density of the combination distribution. That is, it finds the #' optimal linear combining weights for maximizing the leave-one-out log score. #' #' The pseudo-BMA method (`method="pseudobma"`) finds the relative weights #' proportional to the ELPD of each model. However, when #' `method="pseudobma"`, the default is to also use the Bayesian bootstrap #' (`BB=TRUE`), which corresponds to the pseudo-BMA+ method. The Bayesian #' bootstrap takes into account the uncertainty of finite data points and #' regularizes the weights away from the extremes of 0 and 1. #' #' In general, we recommend stacking for averaging predictive distributions, #' while pseudo-BMA+ can serve as a computationally easier alternative. #' #' @seealso #' * The __loo__ package [vignettes](https://mc-stan.org/loo/articles/), particularly #' [Bayesian Stacking and Pseudo-BMA weights using the __loo__ package](https://mc-stan.org/loo/articles/loo2-weights.html). #' * [loo()] for details on leave-one-out ELPD estimation. #' * [constrOptim()] for the choice of optimization methods and control-parameters. #' * [relative_eff()] for computing `r_eff`. #' #' @template loo-and-psis-references #' @template stacking-references #' #' @examples #' \dontrun{ #' ### Demonstrating usage after fitting models with RStan #' library(rstan) #' #' # generate fake data from N(0,1). #' N <- 100 #' y <- rnorm(N, 0, 1) #' #' # Suppose we have three models: N(-1, sigma), N(0.5, sigma) and N(0.6,sigma). #' stan_code <- " #' data { #' int N; #' vector[N] y; #' real mu_fixed; #' } #' parameters { #' real sigma; #' } #' model { #' sigma ~ exponential(1); #' y ~ normal(mu_fixed, sigma); #' } #' generated quantities { #' vector[N] log_lik; #' for (n in 1:N) log_lik[n] = normal_lpdf(y[n]| mu_fixed, sigma); #' }" #' #' mod <- stan_model(model_code = stan_code) #' fit1 <- sampling(mod, data=list(N=N, y=y, mu_fixed=-1)) #' fit2 <- sampling(mod, data=list(N=N, y=y, mu_fixed=0.5)) #' fit3 <- sampling(mod, data=list(N=N, y=y, mu_fixed=0.6)) #' model_list <- list(fit1, fit2, fit3) #' log_lik_list <- lapply(model_list, extract_log_lik) #' #' # optional but recommended #' r_eff_list <- lapply(model_list, function(x) { #' ll_array <- extract_log_lik(x, merge_chains = FALSE) #' relative_eff(exp(ll_array)) #' }) #' #' # stacking method: #' wts1 <- loo_model_weights( #' log_lik_list, #' method = "stacking", #' r_eff_list = r_eff_list, #' optim_control = list(reltol=1e-10) #' ) #' print(wts1) #' #' # can also pass a list of psis_loo objects to avoid recomputing loo #' loo_list <- lapply(1:length(log_lik_list), function(j) { #' loo(log_lik_list[[j]], r_eff = r_eff_list[[j]]) #' }) #' #' wts2 <- loo_model_weights( #' loo_list, #' method = "stacking", #' optim_control = list(reltol=1e-10) #' ) #' all.equal(wts1, wts2) #' #' # can provide names to be used in the results #' loo_model_weights(setNames(loo_list, c("A", "B", "C"))) #' #' #' # pseudo-BMA+ method: #' set.seed(1414) #' loo_model_weights(loo_list, method = "pseudobma") #' #' # pseudo-BMA method (set BB = FALSE): #' loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) #' #' # calling stacking_weights or pseudobma_weights directly #' lpd1 <- loo(log_lik_list[[1]], r_eff = r_eff_list[[1]])$pointwise[,1] #' lpd2 <- loo(log_lik_list[[2]], r_eff = r_eff_list[[2]])$pointwise[,1] #' lpd3 <- loo(log_lik_list[[3]], r_eff = r_eff_list[[3]])$pointwise[,1] #' stacking_weights(cbind(lpd1, lpd2, lpd3)) #' pseudobma_weights(cbind(lpd1, lpd2, lpd3)) #' pseudobma_weights(cbind(lpd1, lpd2, lpd3), BB = FALSE) #' } #' loo_model_weights <- function(x, ...) { UseMethod("loo_model_weights") } #' @rdname loo_model_weights #' @export #' @export loo_model_weights.default loo_model_weights.default <- function(x, ..., method = c("stacking", "pseudobma"), optim_method = "BFGS", optim_control = list(), BB = TRUE, BB_n = 1000, alpha = 1, r_eff_list = NULL, cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) method <- match.arg(method) K <- length(x) # number of models if (is.matrix(x[[1]])) { N <- ncol(x[[1]]) # number of data points validate_log_lik_list(x) validate_r_eff_list(r_eff_list, K, N) lpd_point <- matrix(NA, N, K) elpd_loo <- rep(NA, K) for (k in 1:K) { r_eff_k <- r_eff_list[[k]] # possibly NULL log_likelihood <- x[[k]] loo_object <- loo(log_likelihood, r_eff = r_eff_k, cores = cores) lpd_point[, k] <- loo_object$pointwise[, "elpd_loo"] #calculate log(p_k (y_i | y_-i)) elpd_loo[k] <- loo_object$estimates["elpd_loo", "Estimate"] } } else if (is.psis_loo(x[[1]])) { validate_psis_loo_list(x) lpd_point <- do.call(cbind, lapply(x, function(obj) obj$pointwise[, "elpd_loo"])) elpd_loo <- sapply(x, function(obj) obj$estimates["elpd_loo", "Estimate"]) } else { stop("'x' must be a list of matrices or a list of 'psis_loo' objects.") } ## 1) stacking on log score if (method =="stacking") { wts <- stacking_weights( lpd_point = lpd_point, optim_method = optim_method, optim_control = optim_control ) } else { # method =="pseudobma" wts <- pseudobma_weights( lpd_point = lpd_point, BB = BB, BB_n = BB_n, alpha = alpha ) } if (is.matrix(x[[1]])) { if (!is.null(names(x)) && all(nzchar(names(x)))) { wts <- setNames(wts, names(x)) } } else { # list of loo objects wts <- setNames(wts, find_model_names(x)) } wts } #' @rdname loo_model_weights #' @export #' @param lpd_point If calling `stacking_weights()` or `pseudobma_weights()` #' directly, a matrix of pointwise leave-one-out (or K-fold) log likelihoods #' evaluated for different models. It should be a \eqn{N} by \eqn{K} matrix #' where \eqn{N} is sample size and \eqn{K} is the number of models. Each #' column corresponds to one model. These values can be calculated #' approximately using [loo()] or by running exact leave-one-out or K-fold #' cross-validation. #' #' @importFrom stats constrOptim #' stacking_weights <- function(lpd_point, optim_method = "BFGS", optim_control = list()) { stopifnot(is.matrix(lpd_point)) N <- nrow(lpd_point) K <- ncol(lpd_point) if (K < 2) { stop("At least two models are required for stacking weights.") } exp_lpd_point <- exp(lpd_point) negative_log_score_loo <- function(w) { # objective function: log score stopifnot(length(w) == K - 1) w_full <- c(w, 1 - sum(w)) sum <- 0 for (i in 1:N) { sum <- sum + log(exp(lpd_point[i, ]) %*% w_full) } return(-as.numeric(sum)) } gradient <- function(w) { # gradient of the objective function stopifnot(length(w) == K - 1) w_full <- c(w, 1 - sum(w)) grad <- rep(0, K - 1) for (k in 1:(K - 1)) { for (i in 1:N) { grad[k] <- grad[k] + (exp_lpd_point[i, k] - exp_lpd_point[i, K]) / (exp_lpd_point[i,] %*% w_full) } } return(-grad) } ui <- rbind(rep(-1, K - 1), diag(K - 1)) # K-1 simplex constraint matrix ci <- c(-1, rep(0, K - 1)) w <- constrOptim( theta = rep(1 / K, K - 1), f = negative_log_score_loo, grad = gradient, ui = ui, ci = ci, method = optim_method, control = optim_control )$par wts <- structure( c(w, 1 - sum(w)), names = paste0("model", 1:K), class = c("stacking_weights") ) return(wts) } #' @rdname loo_model_weights #' @export #' pseudobma_weights <- function(lpd_point, BB = TRUE, BB_n = 1000, alpha = 1) { stopifnot(is.matrix(lpd_point)) N <- nrow(lpd_point) K <- ncol(lpd_point) if (K < 2) { stop("At least two models are required for pseudo-BMA weights.") } if (!BB) { elpd <- colSums2(lpd_point) uwts <- exp(elpd - max(elpd)) wts <- structure( uwts / sum(uwts), names = paste0("model", 1:K), class = "pseudobma_weights" ) return(wts) } temp <- matrix(NA, BB_n, K) BB_weighting <- dirichlet_rng(BB_n, rep(alpha, N)) for (bb in 1:BB_n) { z_bb <- BB_weighting[bb, ] %*% lpd_point * N uwts <- exp(z_bb - max(z_bb)) temp[bb, ] <- uwts / sum(uwts) } wts <- structure( colMeans(temp), names = paste0("model", 1:K), class = "pseudobma_bb_weights" ) return(wts) } #' Generate dirichlet simulations, rewritten version #' @importFrom stats rgamma #' @noRd dirichlet_rng <- function(n, alpha) { K <- length(alpha) gamma_sim <- matrix(rgamma(K * n, alpha), ncol = K, byrow = TRUE) gamma_sim / rowSums(gamma_sim) } #' @export print.stacking_weights <- function(x, digits = 3, ...) { cat("Method: stacking\n------\n") print_weight_vector(x, digits = digits) } #' @export print.pseudobma_weights <- function(x, digits = 3, ...) { cat("Method: pseudo-BMA\n------\n") print_weight_vector(x, digits = digits) } #' @export print.pseudobma_bb_weights <- function(x, digits = 3, ...) { cat("Method: pseudo-BMA+ with Bayesian bootstrap\n------\n") print_weight_vector(x, digits = digits) } print_weight_vector <- function(x, digits) { z <- cbind(x) colnames(z) <- "weight" print(.fr(z, digits = digits), quote = FALSE) invisible(x) } #' Validate r_eff_list argument if provided #' #' @noRd #' @param r_eff_list User's `r_eff_list` argument #' @param K Required length of `r_eff_list` (number of models). #' @param N Required length of each element of `r_eff_list` (number of data points). #' @return Either throws an error or returns `TRUE` invisibly. #' validate_r_eff_list <- function(r_eff_list, K, N) { if (is.null(r_eff_list)) return(invisible(TRUE)) if (length(r_eff_list) != K) { stop("If r_eff_list is specified then it must contain ", "one component for each model being compared.", call. = FALSE) } if (any(sapply(r_eff_list, length) != N)) { stop("Each component of r_eff list must have the same length ", "as the number of columns in the log-likelihood matrix.", call. = FALSE) } invisible(TRUE) } #' Validate log-likelihood list argument #' #' Checks that log-likelihood list has at least 2 elements and that each element #' has the same dimensions. #' #' @noRd #' @param log_lik_list User's list of log-likelihood matrices (the `x` argument #' to loo_model_weights). #' @return Either throws an error or returns `TRUE` invisibly. #' validate_log_lik_list <- function(log_lik_list) { stopifnot(is.list(log_lik_list)) if (length(log_lik_list) < 2) { stop("At least two models are required.", call. = FALSE) } if (length(unique(sapply(log_lik_list, ncol))) != 1 | length(unique(sapply(log_lik_list, nrow))) != 1) { stop("Each log-likelihood matrix must have the same dimensions.", call. = FALSE) } invisible(TRUE) } validate_psis_loo_list <- function(psis_loo_list) { stopifnot(is.list(psis_loo_list)) if (length(psis_loo_list) < 2) { stop("At least two models are required.", call. = FALSE) } if (!all(sapply(psis_loo_list, is.psis_loo))) { stop("List elements must all be 'psis_loo' objects or log-likelihood matrices.") } dims <- sapply(psis_loo_list, dim) if (length(unique(dims[1, ])) != 1 | length(unique(dims[2, ])) != 1) { stop("Each object in the list must have the same dimensions.", call. = FALSE) } invisible(TRUE) } loo/R/kfold-generic.R0000644000176200001440000000316213575772017014120 0ustar liggesusers#' Generic function for K-fold cross-validation for developers #' #' @description For developers of Bayesian modeling packages, **loo** includes #' a generic function `kfold()` so that methods may be defined for K-fold #' CV without name conflicts between packages. See, for example, the #' `kfold()` methods in the **rstanarm** and **brms** packages. #' #' The **Value** section below describes the objects that `kfold()` #' methods should return in order to be compatible with #' [loo_compare()] and the **loo** package print methods. #' #' #' @name kfold-generic #' @param x A fitted model object. #' @param ... Arguments to pass to specific methods. #' #' @return For developers defining a `kfold()` method for a class #' `"foo"`, the `kfold.foo()` function should return a list with class #' `c("kfold", "loo")` with at least the following named elements: #' * `"estimates"`: A `1x2` matrix containing the ELPD estimate and its #' standard error. The matrix must have row name "`elpd_kfold`" and column #' names `"Estimate"` and `"SE"`. #' * `"pointwise"`: A `Nx1` matrix with column name `"elpd_kfold"` containing #' the pointwise contributions for each data point. #' #' It is important for the object to have at least these classes and #' components so that it is compatible with other functions like #' [loo_compare()] and `print()` methods. #' NULL #' @rdname kfold-generic #' @export kfold <- function(x, ...) { UseMethod("kfold") } #' @rdname kfold-generic #' @export is.kfold <- function(x) { inherits(x, "kfold") && is.loo(x) } #' @export dim.kfold <- function(x) { attr(x, "dims") } loo/R/compare.R0000644000176200001440000001220613701164066013022 0ustar liggesusers#' Model comparison (deprecated, old version) #' #' **This function is deprecated**. Please use the new [loo_compare()] function #' instead. #' #' @export #' @param ... At least two objects returned by [loo()] (or [waic()]). #' @param x A list of at least two objects returned by [loo()] (or #' [waic()]). This argument can be used as an alternative to #' specifying the objects in `...`. #' #' @return A vector or matrix with class `'compare.loo'` that has its own #' print method. If exactly two objects are provided in `...` or #' `x`, then the difference in expected predictive accuracy and the #' standard error of the difference are returned. If more than two objects are #' provided then a matrix of summary information is returned (see **Details**). #' #' @details #' When comparing two fitted models, we can estimate the difference in their #' expected predictive accuracy by the difference in `elpd_loo` or #' `elpd_waic` (or multiplied by -2, if desired, to be on the #' deviance scale). #' #' *When that difference, `elpd_diff`, is positive then the expected #' predictive accuracy for the second model is higher. A negative #' `elpd_diff` favors the first model.* #' #' When using `compare()` with more than two models, the values in the #' `elpd_diff` and `se_diff` columns of the returned matrix are #' computed by making pairwise comparisons between each model and the model #' with the best ELPD (i.e., the model in the first row). #' Although the `elpd_diff` column is equal to the difference in #' `elpd_loo`, do not expect the `se_diff` column to be equal to the #' the difference in `se_elpd_loo`. #' #' To compute the standard error of the difference in ELPD we use a #' paired estimate to take advantage of the fact that the same set of _N_ #' data points was used to fit both models. These calculations should be most #' useful when _N_ is large, because then non-normality of the #' distribution is not such an issue when estimating the uncertainty in these #' sums. These standard errors, for all their flaws, should give a better #' sense of uncertainty than what is obtained using the current standard #' approach of comparing differences of deviances to a Chi-squared #' distribution, a practice derived for Gaussian linear models or #' asymptotically, and which only applies to nested models in any case. #' #' @template loo-and-psis-references #' #' @examples #' \dontrun{ #' loo1 <- loo(log_lik1) #' loo2 <- loo(log_lik2) #' print(compare(loo1, loo2), digits = 3) #' print(compare(x = list(loo1, loo2))) #' #' waic1 <- waic(log_lik1) #' waic2 <- waic(log_lik2) #' compare(waic1, waic2) #' } #' compare <- function(..., x = list()) { .Deprecated("loo_compare") dots <- list(...) if (length(dots)) { if (length(x)) { stop("If 'x' is specified then '...' should not be specified.", call. = FALSE) } nms <- as.character(match.call(expand.dots = TRUE))[-1L] } else { if (!is.list(x) || !length(x)) { stop("'x' must be a list.", call. = FALSE) } dots <- x nms <- names(dots) if (!length(nms)) { nms <- paste0("model", seq_along(dots)) } } if (!all(sapply(dots, is.loo))) { stop("All inputs should have class 'loo'.") } if (length(dots) <= 1L) { stop("'compare' requires at least two models.") } else if (length(dots) == 2L) { loo1 <- dots[[1]] loo2 <- dots[[2]] comp <- compare_two_models(loo1, loo2) class(comp) <- c(class(comp), "old_compare.loo") return(comp) } else { Ns <- sapply(dots, function(x) nrow(x$pointwise)) if (!all(Ns == Ns[1L])) { stop("Not all models have the same number of data points.", call. = FALSE) } x <- sapply(dots, function(x) { est <- x$estimates setNames(c(est), nm = c(rownames(est), paste0("se_", rownames(est))) ) }) colnames(x) <- nms rnms <- rownames(x) comp <- x ord <- order(x[grep("^elpd", rnms), ], decreasing = TRUE) comp <- t(comp)[ord, ] patts <- c("elpd", "p_", "^waic$|^looic$", "^se_waic$|^se_looic$") col_ord <- unlist(sapply(patts, function(p) grep(p, colnames(comp))), use.names = FALSE) comp <- comp[, col_ord] # compute elpd_diff and se_elpd_diff relative to best model rnms <- rownames(comp) diffs <- mapply(elpd_diffs, dots[ord[1]], dots[ord]) elpd_diff <- apply(diffs, 2, sum) se_diff <- apply(diffs, 2, se_elpd_diff) comp <- cbind(elpd_diff = elpd_diff, se_diff = se_diff, comp) rownames(comp) <- rnms class(comp) <- c("compare.loo", class(comp), "old_compare.loo") comp } } # internal ---------------------------------------------------------------- compare_two_models <- function(loo_a, loo_b, return = c("elpd_diff", "se"), check_dims = TRUE) { if (check_dims) { if (dim(loo_a$pointwise)[1] != dim(loo_b$pointwise)[1]) { stop(paste("Models don't have the same number of data points.", "\nFound N_1 =", dim(loo_a$pointwise)[1], "and N_2 =", dim(loo_b$pointwise)[1]), call. = FALSE) } } diffs <- elpd_diffs(loo_a, loo_b) comp <- c(elpd_diff = sum(diffs), se = se_elpd_diff(diffs)) structure(comp, class = "compare.loo") } loo/R/elpd.R0000644000176200001440000000374213762013700012320 0ustar liggesusers#' Generic (expected) log-predictive density #' #' The `elpd()` methods for arrays and matrices can compute the expected log #' pointwise predictive density for a new dataset or the log pointwise #' predictive density of the observed data (an overestimate of the elpd). #' #' @export #' @param x A log-likelihood array or matrix. The **Methods (by class)** #' section, below, has detailed descriptions of how to specify the inputs for #' each method. #' @param ... Currently ignored. #' #' @details The `elpd()` function is an S3 generic and methods are provided for #' 3-D pointwise log-likelihood arrays and matrices. #' #' @seealso The vignette *Holdout validation and K-fold cross-validation of Stan #' programs with the loo package* for demonstrations of using the `elpd()` #' methods. #' #' @examples #' # Calculate the lpd of the observed data #' LLarr <- example_loglik_array() #' elpd(LLarr) #' elpd <- function(x, ...) { UseMethod("elpd") } #' @export #' @templateVar fn elpd #' @template array #' elpd.array <- function(x, ...) { ll <- llarray_to_matrix(x) elpd.matrix(ll) } #' @export #' @templateVar fn elpd #' @template matrix #' elpd.matrix <- function(x, ...) { pointwise <- pointwise_elpd_calcs(x) elpd_object(pointwise, dim(x)) } # internal ---------------------------------------------------------------- pointwise_elpd_calcs <- function(ll){ elpd <- colLogSumExps(ll) - log(nrow(ll)) ic <- -2 * elpd cbind(elpd, ic) } elpd_object <- function(pointwise, dims) { if (!is.matrix(pointwise)) stop("Internal error ('pointwise' must be a matrix)") cols_to_summarize <- colnames(pointwise) estimates <- table_of_estimates(pointwise[, cols_to_summarize, drop=FALSE]) out <- nlist(estimates, pointwise) structure( out, dims = dims, class = c("elpd_generic", "loo") ) } print_dims.elpd_generic <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), "log-likelihood matrix using the generic elpd function\n" ) } loo/R/psislw.R0000644000176200001440000001400013575772017012721 0ustar liggesusers#' Pareto smoothed importance sampling (deprecated, old version) #' #' As of version `2.0.0` this function is **deprecated**. Please use the #' [psis()] function for the new PSIS algorithm. #' #' @export #' @param lw A matrix or vector of log weights. For computing LOO, `lw = #' -log_lik`, the *negative* of an \eqn{S} (simulations) by \eqn{N} (data #' points) pointwise log-likelihood matrix. #' @param wcp The proportion of importance weights to use for the generalized #' Pareto fit. The `100*wcp`\% largest weights are used as the sample #' from which to estimate the parameters of the generalized Pareto #' distribution. #' @param wtrunc For truncating very large weights to \eqn{S}^`wtrunc`. Set #' to zero for no truncation. #' @param cores The number of cores to use for parallelization. This defaults to #' the option `mc.cores` which can be set for an entire R session by #' `options(mc.cores = NUMBER)`, the old option `loo.cores` is now #' deprecated but will be given precedence over `mc.cores` until it is #' removed. **As of version 2.0.0, the default is now 1 core if #' `mc.cores` is not set, but we recommend using as many (or close to as #' many) cores as possible.** #' @param llfun,llargs See [loo.function()]. #' @param ... Ignored when `psislw()` is called directly. The `...` is #' only used internally when `psislw()` is called by the [loo()] #' function. #' #' @return A named list with components `lw_smooth` (modified log weights) and #' `pareto_k` (estimated generalized Pareto shape parameter(s) k). #' #' @seealso [pareto-k-diagnostic] for PSIS diagnostics. #' #' @template loo-and-psis-references #' #' @importFrom parallel mclapply makePSOCKcluster stopCluster parLapply #' psislw <- function(lw, wcp = 0.2, wtrunc = 3/4, cores = getOption("mc.cores", 1), llfun = NULL, llargs = NULL, ...) { .Deprecated("psis") cores <- loo_cores(cores) .psis <- function(lw_i) { x <- lw_i - max(lw_i) cutoff <- lw_cutpoint(x, wcp, MIN_CUTOFF) above_cut <- x > cutoff x_body <- x[!above_cut] x_tail <- x[above_cut] tail_len <- length(x_tail) if (tail_len < MIN_TAIL_LENGTH || all(x_tail == x_tail[1])) { if (all(x_tail == x_tail[1])) warning( "All tail values are the same. ", "Weights are truncated but not smoothed.", call. = FALSE ) else if (tail_len < MIN_TAIL_LENGTH) warning( "Too few tail samples to fit generalized Pareto distribution.\n", "Weights are truncated but not smoothed.", call. = FALSE ) x_new <- x k <- Inf } else { # store order of tail samples, fit gPd to the right tail samples, compute # order statistics for the fit, remap back to the original order, join # body and gPd smoothed tail tail_ord <- order(x_tail) exp_cutoff <- exp(cutoff) fit <- gpdfit(exp(x_tail) - exp_cutoff, wip=FALSE, min_grid_pts = 80) k <- fit$k sigma <- fit$sigma prb <- (seq_len(tail_len) - 0.5) / tail_len qq <- qgpd(prb, k, sigma) + exp_cutoff smoothed_tail <- rep.int(0, tail_len) smoothed_tail[tail_ord] <- log(qq) x_new <- x x_new[!above_cut] <- x_body x_new[above_cut] <- smoothed_tail } # truncate (if wtrunc > 0) and renormalize, # return log weights and pareto k lw_new <- lw_normalize(lw_truncate(x_new, wtrunc)) nlist(lw_new, k) } .psis_loop <- function(i) { if (LL_FUN) { ll_i <- llfun(i = i, data = llargs$data[i,, drop=FALSE], draws = llargs$draws) lw_i <- -1 * ll_i } else { lw_i <- lw[, i] ll_i <- -1 * lw_i } psis <- .psis(lw_i) if (FROM_LOO) nlist(lse = logSumExp(ll_i + psis$lw_new), k = psis$k) else psis } # minimal cutoff value. there must be at least 5 log-weights larger than this # in order to fit the gPd to the tail MIN_CUTOFF <- -700 MIN_TAIL_LENGTH <- 5 dots <- list(...) FROM_LOO <- if ("COMPUTE_LOOS" %in% names(dots)) dots$COMPUTE_LOOS else FALSE if (!missing(lw)) { if (!is.matrix(lw)) lw <- as.matrix(lw) N <- ncol(lw) LL_FUN <- FALSE } else { if (is.null(llfun) || is.null(llargs)) stop("Either 'lw' or 'llfun' and 'llargs' must be specified.") N <- llargs$N LL_FUN <- TRUE } if (cores == 1) { # don't call functions from parallel package if cores=1 out <- lapply(X = 1:N, FUN = .psis_loop) } else { # parallelize if (.Platform$OS.type != "windows") { out <- mclapply(X = 1:N, FUN = .psis_loop, mc.cores = cores) } else { # nocov start cl <- makePSOCKcluster(cores) on.exit(stopCluster(cl)) out <- parLapply(cl, X = 1:N, fun = .psis_loop) # nocov end } } pareto_k <- vapply(out, "[[", 2L, FUN.VALUE = numeric(1)) psislw_warnings(pareto_k) if (FROM_LOO) { loos <- vapply(out, "[[", 1L, FUN.VALUE = numeric(1)) nlist(loos, pareto_k) } else { funval <- if (LL_FUN) llargs$S else nrow(lw) lw_smooth <- vapply(out, "[[", 1L, FUN.VALUE = numeric(funval)) out <- nlist(lw_smooth, pareto_k) class(out) <- c("psis", "list") return(out) } } # internal ---------------------------------------------------------------- lw_cutpoint <- function(y, wcp, min_cut) { if (min_cut < log(.Machine$double.xmin)) min_cut <- -700 cp <- quantile(y, 1 - wcp, names = FALSE) max(cp, min_cut) } lw_truncate <- function(y, wtrunc) { if (wtrunc == 0) return(y) logS <- log(length(y)) lwtrunc <- wtrunc * logS - logS + logSumExp(y) y[y > lwtrunc] <- lwtrunc y } lw_normalize <- function(y) { y - logSumExp(y) } # warnings about pareto k values ------------------------------------------ psislw_warnings <- function(k) { if (any(k > 0.7)) { .warn( "Some Pareto k diagnostic values are too high. ", .k_help() ) } else if (any(k > 0.5)) { .warn( "Some Pareto k diagnostic values are slightly high. ", .k_help() ) } } loo/R/datasets.R0000644000176200001440000000330314407123455013203 0ustar liggesusers#' Datasets for loo examples and vignettes #' #' Small datasets for use in **loo** examples and vignettes. The `Kline` #' and `milk` datasets are also included in the **rethinking** package #' (McElreath, 2016a), but we include them here as **rethinking** is not #' on CRAN. #' #' @name loo-datasets #' @aliases Kline milk voice voice_loo #' #' @details #' Currently the data sets included are: #' * `Kline`: #' Small dataset from Kline and Boyd (2010) on tool complexity and demography #' in Oceanic islands societies. This data is discussed in detail in #' McElreath (2016a,2016b). [(Link to variable descriptions)](https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/Kline) #' * `milk`: #' Small dataset from Hinde and Milligan (2011) on primate milk #' composition.This data is discussed in detail in McElreath (2016a,2016b). #' [(Link to variable descriptions)](https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/milk) #' * `voice`: #' Voice rehabilitation data from Tsanas et al. (2014). #' @references #' Hinde and Milligan. 2011. *Evolutionary Anthropology* 20:9-23. #' #' Kline, M.A. and R. Boyd. 2010. *Proc R Soc B* 277:2559-2564. #' #' McElreath, R. (2016a). rethinking: Statistical Rethinking book package. #' R package version 1.59. #' #' McElreath, R. (2016b). *Statistical rethinking: A Bayesian course with #' examples in R and Stan*. Chapman & Hall/CRC. #' #' A. Tsanas, M.A. Little, C. Fox, L.O. Ramig: Objective automatic assessment of #' rehabilitative speech treatment in Parkinson's disease, IEEE #' Transactions on Neural Systems and Rehabilitation Engineering, Vol. 22, pp. #' 181-190, January 2014 #' #' @examples #' str(Kline) #' str(milk) #' NULL loo/NEWS.md0000644000176200001440000002427214411130104012135 0ustar liggesusers# loo 2.6.0 ### New features * New `loo_predictive_metric()` function for computing estimates of leave-one-out predictive metrics: mean absolute error, mean squared error and root mean squared error for continuous predictions, and accuracy and balanced accuracy for binary classification. (#202, @LeeviLindgren) * New functions `crps()`, `scrps()`, `loo_crps()`, and `loo_scrps()` for computing the (scaled) continuously ranked probability score. (#203, @LeeviLindgren) * New vignette "Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models." This is a demonstration of the mixture estimators proposed by [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). (#210) ### Bug fixes * Minor fix to model names displayed by `loo_model_weights()` to make them consistent with `loo_compare()`. (#217) # loo 2.5.1 * Fix R CMD check error on M1 Mac # loo 2.5.0 ### Improvements * New [Frequently Asked Questions page](https://mc-stan.org/loo/articles/online-only/faq.html) on the package website. (#143) * Speed improvement from simplifying the normalization when fitting the generalized Pareto distribution. (#187, @sethaxen) * Added parallel likelihood computation to speedup `loo_subsample()` when using posterior approximations. (#171, @kdubovikov) * Switch unit tests from Travis to GitHub Actions. (#164) ### Bug fixes * Fixed a bug causing the normalizing constant of the PSIS (log) weights not to get updated when performing moment matching with `save_psis = TRUE` (#166, @fweber144). # loo 2.4.1 * Fixed issue reported by CRAN where one of the vignettes errored on an M1 Mac due to RStan's dependency on V8. # loo 2.4.0 ### Bug fixes * Fixed a bug in `relative_eff.function()` that caused an error on Windows when using multiple cores. (#152) * Fixed a potential numerical issue in `loo_moment_match()` with `split=TRUE`. (#153) * Fixed potential integer overflow with `loo_moment_match()`. (#155, @ecmerkle) * Fixed `relative_eff()` when used with a `posterior::draws_array`. (#161, @rok-cesnovar) ### New features * New generic function `elpd()` (and methods for matrices and arrays) for computing expected log predictive density of new data or log predictive density of observed data. A new vignette demonstrates using this function when doing K-fold CV with rstan. (#159, @bnicenboim) # loo 2.3.1 * Fixed a bug in `loo_moment_match()` that prevented `...` arguments from being used correctly. (#149) # loo 2.3.0 * Added Topi Paananen and Paul Bürkner as coauthors. * New function `loo_moment_match()` (and new vignette), which can be used to update a `loo` object when Pareto k estimates are large. (#130) * The log weights provided by the importance sampling functions `psis()`, `tis()`, and `sis()` no longer have the largest log ratio subtracted from them when returned to the user. This should be less confusing for anyone using the `weights()` method to make an importance sampler. (#112, #146) * MCSE calculation is now deterministic (#116, #147) # loo 2.2.0 * Added Mans Magnusson as a coauthor. * New functions `loo_subsample()` and `loo_approximate_posterior()` (and new vignette) for doing PSIS-LOO with large data. (#113) * Added support for standard importance sampling and truncated importance sampling (functions `sis()` and `tis()`). (#125) * `compare()` now throws a deprecation warning suggesting `loo_compare()`. (#93) * A smaller threshold is used when checking the uniqueness of tail values. (#124) * For WAIC, warnings are only thrown when running `waic()` and not when printing a `waic` object. (#117, @mcol) * Use markdown syntax in roxygen documentation wherever possible. (#108) # loo 2.1.0 * New function `loo_compare()` for model comparison that will eventually replace the existing `compare()` function. (#93) * New vignette on LOO for non-factorizable joint Gaussian models. (#75) * New vignette on "leave-future-out" cross-validation for time series models. (#90) * New glossary page (use `help("loo-glossary")`) with definitions of key terms. (#81) * New `se_diff` column in model comparison results. (#78) * Improved stability of `psis()` when `log_ratios` are very small. (#74) * Allow `r_eff=NA` to suppress warning when specifying `r_eff` is not applicable (i.e., draws not from MCMC). (#72) * Update effective sample size calculations to match RStan's version. (#85) * Naming of k-fold helper functions now matches scikit-learn. (#96) # loo 2.0.0 This is a major release with many changes. Whenever possible we have opted to deprecate rather than remove old functionality, but it is possible that old code that accesses elements inside loo objects by position rather than name may error. * New package documentation website http://mc-stan.org/loo/ with vignettes, function reference, news. * Updated existing vignette and added two new vignettes demonstrating how to use the package. * New function `psis()` replaces `psislw()` (now deprecated). This version implements the improvements to the PSIS algorithm described in the latest version of https://arxiv.org/abs/1507.02646. Additional diagnostic information is now also provided, including PSIS effective sample sizes. * New `weights()` method for extracting smoothed weights from a `psis` object. Arguments `log` and `normalize` control whether the weights are returned on the log scale and whether they are normalized. * Updated the interface for the `loo()` methods to integrate nicely with the new PSIS algorithm. Methods for log-likelihood arrays, matrices, and functions are provided. Several arguments have changed, particularly for the `loo.function` method. The documentation at `help("loo")` has been updated to describe the new behavior. * The structure of the objects returned by the `loo()` function has also changed slightly, as described in the __Value__ section at `help("loo", package = "loo")`. * New function `loo_model_weights()` computes weights for model averaging as described in https://arxiv.org/abs/1704.02030. Implemented methods include stacking of predictive distributions, pseudo-BMA weighting or pseudo-BMA+ weighting with the Bayesian bootstrap. * Setting `options(loo.cores=...)` is now deprecated in favor of `options(mc.cores=...)`. For now, if both the `loo.cores` and `mc.cores` options have been set, preference will be given to `loo.cores` until it is removed in a future release. (thanks to @cfhammill) * New functions `example_loglik_array()` and `example_loglik_matrix()` that provide objects to use in examples and tests. * When comparing more than two models with `compare()`, the first column of the output is now the `elpd` difference from the model in the first row. * New helper functions for splitting observations for K-fold CV: `kfold_split_random()`, `kfold_split_balanced()`, `kfold_split_stratified()`. Additional helper functions for implementing K-fold CV will be included in future releases. # loo 1.1.0 * Introduce the `E_loo()` function for computing weighted expectations (means, variances, quantiles). # loo 1.0.0 * `pareto_k_table()` and `pareto_k_ids()` convenience functions for quickly identifying problematic observations * pareto k values now grouped into `(-Inf, 0.5]`, `(0.5, 0.7]`, `(0.7, 1]`, `(1, Inf)` (didn't used to include 0.7) * warning messages are now issued by `psislw()` instead of `print.loo` * `print.loo()` shows a table of pareto k estimates (if any k > 0.7) * Add argument to `compare()` to allow loo objects to be provided in a list rather than in `'...'` * Update references to point to published paper # loo 0.1.6 * GitHub repository moved from @jgabry to @stan-dev * Better error messages from `extract_log_lik()` * Fix example code in vignette (thanks to GitHub user @krz) # loo 0.1.5 * Add warnings if any p_waic estimates are greather than 0.4 * Improve line coverage of tests to 100% * Update references in documentation * Remove model weights from `compare()`. In previous versions of __loo__ model weights were also reported by `compare()`. We have removed the weights because they were based only on the point estimate of the elpd values ignoring the uncertainty. We are currently working on something similar to these weights that also accounts for uncertainty, which will be included in future versions of __loo__. # loo 0.1.4 This update makes it easier for other package authors using __loo__ to write tests that involve running the `loo` function. It also includes minor bug fixes and additional unit tests. Highlights: * Don't call functions from __parallel__ package if `cores=1`. * Return entire vector/matrix of smoothed weights rather than a summary statistic when `psislw` function is called in an interactive session. * Test coverage > 80% # loo 0.1.3 This update provides several important improvements, most notably an alternative method for specifying the pointwise log-likelihood that reduces memory usage and allows for __loo__ to be used with larger datasets. This update also makes it easier to to incorporate __loo__'s functionality into other packages. * Add Ben Goodrich as contributor * S3 generics and `matrix` and `function` methods for both `loo()` and `waic()`. The matrix method provide the same functionality as in previous versions of __loo__ (taking a log-likelihood matrix as the input). The function method allows the user to provide a function for computing the log-likelihood from the data and posterior draws (which are also provided by the user). The function method is less memory intensive and should make it possible to use __loo__ for models fit to larger amounts of data than before. * Separate `plot` and `print` methods. `plot` also provides `label_points` argument, which, if `TRUE`, will label any Pareto `k` points greater than 1/2 by the index number of the corresponding observation. The plot method also now warns about `Inf`/`NA`/`NaN` values of `k` that are not shown in the plot. * `compare` now returns model weights and accepts more than two inputs. * Allow setting number of cores using `options(loo.cores = NUMBER)`. # loo 0.1.2 * Updates names in package to reflect name changes in the accompanying paper. # loo 0.1.1 * Better handling of special cases * Deprecates `loo_and_waic` function in favor of separate functions `loo` and `waic` * Deprecates `loo_and_waic_diff`. 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Holdout validation and K-fold cross-validation of Stan programs with the loo package

Bruno Nicenboim

2023-03-30

Introduction

This vignette demonstrates how to do holdout validation and K-fold cross-validation with loo for a Stan program.

Example: Eradication of Roaches using holdout validation approach

This vignette uses the same example as in the vignettes Using the loo package (version >= 2.0.0) and Avoiding model refits in leave-one-out cross-validation with moment matching.

Coding the Stan model

Here is the Stan code for fitting a Poisson regression model:

stancode <- "
data {
  int<lower=1> K;
  int<lower=1> N;
  matrix[N,K] x;
  int y[N];
  vector[N] offset;

  real beta_prior_scale;
  real alpha_prior_scale;
}
parameters {
  vector[K] beta;
  real intercept;
}
model {
  y ~ poisson(exp(x * beta + intercept + offset));
  beta ~ normal(0,beta_prior_scale);
  intercept ~ normal(0,alpha_prior_scale);
}
generated quantities {
  vector[N] log_lik;
  for (n in 1:N)
    log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n]));
}
"

Following the usual approach recommended in Writing Stan programs for use with the loo package, we compute the log-likelihood for each observation in the generated quantities block of the Stan program.

Setup

In addition to loo, we load the rstan package for fitting the model. We will also need the rstanarm package for the data.

library("rstan")
library("loo")
seed <- 9547
set.seed(seed)

Holdout validation

For this approach, the model is first fit to the “train” data and then is evaluated on the held-out “test” data.

Splitting the data between train and test

The data is divided between train (80% of the data) and test (20%):

# Prepare data
data(roaches, package = "rstanarm")
roaches$roach1 <- sqrt(roaches$roach1)
roaches$offset <- log(roaches[,"exposure2"])
# 20% of the data goes to the test set:
roaches$test <- 0
roaches$test[sample(.2 * seq_len(nrow(roaches)))] <- 1
# data to "train" the model
data_train <- list(y = roaches$y[roaches$test == 0],
                   x = as.matrix(roaches[roaches$test == 0,
                                         c("roach1", "treatment", "senior")]),
                   N = nrow(roaches[roaches$test == 0,]),
                   K = 3,
                   offset = roaches$offset[roaches$test == 0],
                   beta_prior_scale = 2.5,
                   alpha_prior_scale = 5.0
                   )
# data to "test" the model
data_test <- list(y = roaches$y[roaches$test == 1],
                   x = as.matrix(roaches[roaches$test == 1,
                                         c("roach1", "treatment", "senior")]),
                   N = nrow(roaches[roaches$test == 1,]),
                   K = 3,
                   offset = roaches$offset[roaches$test == 1],
                   beta_prior_scale = 2.5,
                   alpha_prior_scale = 5.0
                   )

Fitting the model with RStan

Next we fit the model to the “test” data in Stan using the rstan package:

# Compile
stanmodel <- stan_model(model_code = stancode)
# Fit model
fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0)

We recompute the generated quantities using the posterior draws conditional on the training data, but we now pass in the held-out data to get the log predictive densities for the test data. Because we are using independent data, the log predictive density coincides with the log likelihood of the test data.

gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test)
log_pd <- extract_log_lik(gen_test)

Computing holdout elpd:

Now we evaluate the predictive performance of the model on the test data using elpd().

(elpd_holdout <- elpd(log_pd))

Computed from 4000 by 52 log-likelihood matrix using the generic elpd function

     Estimate    SE
elpd  -1736.7 288.9
ic     3473.5 577.9

When one wants to compare different models, the function loo_compare() can be used to assess the difference in performance.

K-fold cross validation

For this approach the data is divided into folds, and each time one fold is tested while the rest of the data is used to fit the model (see Vehtari et al., 2017).

Splitting the data in folds

We use the data that is already pre-processed and we divide it in 10 random folds using kfold_split_random

# Prepare data
roaches$fold <- kfold_split_random(K = 10, N = nrow(roaches))

Fitting and extracting the log pointwise predictive densities for each fold

We now loop over the 10 folds. In each fold we do the following. First, we fit the model to all the observations except the ones belonging to the left-out fold. Second, we compute the log pointwise predictive densities for the left-out fold. Last, we store the predictive density for the observations of the left-out fold in a matrix. The output of this loop is a matrix of the log pointwise predictive densities of all the observations.

# Prepare a matrix with the number of post-warmup iterations by number of observations:
log_pd_kfold <- matrix(nrow = 4000, ncol = nrow(roaches))
# Loop over the folds
for(k in 1:10){
  data_train <- list(y = roaches$y[roaches$fold != k],
                   x = as.matrix(roaches[roaches$fold != k,
                                         c("roach1", "treatment", "senior")]),
                   N = nrow(roaches[roaches$fold != k,]),
                   K = 3,
                   offset = roaches$offset[roaches$fold != k],
                   beta_prior_scale = 2.5,
                   alpha_prior_scale = 5.0
                   )
  data_test <- list(y = roaches$y[roaches$fold == k],
                   x = as.matrix(roaches[roaches$fold == k,
                                         c("roach1", "treatment", "senior")]),
                   N = nrow(roaches[roaches$fold == k,]),
                   K = 3,
                   offset = roaches$offset[roaches$fold == k],
                   beta_prior_scale = 2.5,
                   alpha_prior_scale = 5.0
                   )
  fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0)
  gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test)
  log_pd_kfold[, roaches$fold == k] <- extract_log_lik(gen_test)
}

Computing K-fold elpd:

Now we evaluate the predictive performance of the model on the 10 folds using elpd().

(elpd_kfold <- elpd(log_pd_kfold))

Computed from 4000 by 262 log-likelihood matrix using the generic elpd function

     Estimate     SE
elpd  -5552.7  727.9
ic    11105.5 1455.8

If one wants to compare several models (with loo_compare), one should use the same folds for all the different models.

References

Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge University Press.

Stan Development Team (2020) RStan: the R interface to Stan, Version 2.21.1 https://mc-stan.org

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

loo/inst/doc/loo2-weights.Rmd0000644000176200001440000003777114407123455015600 0ustar liggesusers--- title: "Bayesian Stacking and Pseudo-BMA weights using the loo package" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates the new functionality in __loo__ v2.0.0 for Bayesian stacking and Pseudo-BMA weighting. In this vignette we can't provide all of the necessary background on this topic, so we encourage readers to refer to the paper * Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, \doi:10.1214/17-BA1091. [Online](https://projecteuclid.org/euclid.ba/1516093227) which provides important details on the methods demonstrated in this vignette. Here we just quote from the abstract of the paper: > **Abstract**: Bayesian model averaging is flawed in the $\mathcal{M}$-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions. We extend the utility function to any proper scoring rule and use Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions. We compare stacking of predictive distributions to several alternatives: stacking of means, Bayesian model averaging (BMA), Pseudo-BMA, and a variant of Pseudo-BMA that is stabilized using the Bayesian bootstrap. Based on simulations and real-data applications, we recommend stacking of predictive distributions, with bootstrapped-Pseudo-BMA as an approximate alternative when computation cost is an issue. Ideally, we would avoid the Bayesian model combination problem by extending the model to include the separate models as special cases, and preferably as a continuous expansion of the model space. For example, instead of model averaging over different covariate combinations, all potentially relevant covariates should be included in a predictive model (for causal analysis more care is needed) and a prior assumption that only some of the covariates are relevant can be presented with regularized horseshoe prior (Piironen and Vehtari, 2017a). For variable selection we recommend projective predictive variable selection (Piironen and Vehtari, 2017a; [__projpred__ package](https://cran.r-project.org/package=projpred)). To demonstrate how to use __loo__ package to compute Bayesian stacking and Pseudo-BMA weights, we repeat two simple model averaging examples from Chapters 6 and 10 of _Statistical Rethinking_ by Richard McElreath. In _Statistical Rethinking_ WAIC is used to form weights which are similar to classical "Akaike weights". Pseudo-BMA weighting using PSIS-LOO for computation is close to these WAIC weights, but named after the Pseudo Bayes Factor by Geisser and Eddy (1979). As discussed below, in general we prefer using stacking rather than WAIC weights or the similar pseudo-BMA weights. # Setup In addition to the __loo__ package we will also load the __rstanarm__ package for fitting the models. ```{r setup, message=FALSE} library(rstanarm) library(loo) ``` # Example: Primate milk In _Statistical Rethinking_, McElreath describes the data for the primate milk example as follows: > A popular hypothesis has it that primates with larger brains produce more energetic milk, so that brains can grow quickly. ... The question here is to what extent energy content of milk, measured here by kilocalories, is related to the percent of the brain mass that is neocortex. ... We'll end up needing female body mass as well, to see the masking that hides the relationships among the variables. ```{r data} data(milk) d <- milk[complete.cases(milk),] d$neocortex <- d$neocortex.perc /100 str(d) ``` We repeat the analysis in Chapter 6 of _Statistical Rethinking_ using the following four models (here we use the default weakly informative priors in __rstanarm__, while flat priors were used in _Statistical Rethinking_). ```{r fits, results="hide"} fit1 <- stan_glm(kcal.per.g ~ 1, data = d, seed = 2030) fit2 <- update(fit1, formula = kcal.per.g ~ neocortex) fit3 <- update(fit1, formula = kcal.per.g ~ log(mass)) fit4 <- update(fit1, formula = kcal.per.g ~ neocortex + log(mass)) ``` McElreath uses WAIC for model comparison and averaging, so we'll start by also computing WAIC for these models so we can compare the results to the other options presented later in the vignette. The __loo__ package provides `waic` methods for log-likelihood arrays, matrices and functions. Since we fit our model with rstanarm we can use the `waic` method provided by the __rstanarm__ package (a wrapper around `waic` from the __loo__ package), which allows us to just pass in our fitted model objects instead of first extracting the log-likelihood values. ```{r waic} waic1 <- waic(fit1) waic2 <- waic(fit2) waic3 <- waic(fit3) waic4 <- waic(fit4) waics <- c( waic1$estimates["elpd_waic", 1], waic2$estimates["elpd_waic", 1], waic3$estimates["elpd_waic", 1], waic4$estimates["elpd_waic", 1] ) ``` We get some warnings when computing WAIC for models 3 and 4, indicating that we shouldn't trust the WAIC weights we will compute later. Following the recommendation in the warning, we next use the `loo` methods to compute PSIS-LOO instead. The __loo__ package provides `loo` methods for log-likelihood arrays, matrices, and functions, but since we fit our model with __rstanarm__ we can just pass the fitted model objects directly and __rstanarm__ will extract the needed values to pass to the __loo__ package. (Like __rstanarm__, some other R packages for fitting Stan models, e.g. __brms__, also provide similar methods for interfacing with the __loo__ package.) ```{r loo} # note: the loo function accepts a 'cores' argument that we recommend specifying # when working with bigger datasets loo1 <- loo(fit1) loo2 <- loo(fit2) loo3 <- loo(fit3) loo4 <- loo(fit4) lpd_point <- cbind( loo1$pointwise[,"elpd_loo"], loo2$pointwise[,"elpd_loo"], loo3$pointwise[,"elpd_loo"], loo4$pointwise[,"elpd_loo"] ) ``` With `loo` we don't get any warnings for models 3 and 4, but for illustration of good results, we display the diagnostic details for these models anyway. ```{r print-loo} print(loo3) print(loo4) ``` One benefit of PSIS-LOO over WAIC is better diagnostics. Here for both models 3 and 4 all $k<0.7$ and the Monte Carlo SE of `elpd_loo` is 0.1 or less, and we can expect the model comparison to be reliable. Next we compute and compare 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights. ```{r weights} waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ``` With all approaches Model 4 with `neocortex` and `log(mass)` gets most of the weight. Based on theory, Pseudo-BMA weights without Bayesian bootstrap should be close to WAIC weights, and we can also see that here. Pseudo-BMA+ weights with Bayesian bootstrap provide more cautious weights further away from 0 and 1 (see Yao et al. (2018) for a discussion of why this can be beneficial and results from related experiments). In this particular example, the Bayesian stacking weights are not much different from the other weights. One of the benefits of stacking is that it manages well if there are many similar models. Consider for example that there could be many irrelevant covariates that when included would produce a similar model to one of the existing models. To emulate this situation here we simply copy the first model a bunch of times, but you can imagine that instead we would have ten alternative models with about the same predictive performance. WAIC weights for such a scenario would be close to the following: ```{r waic_wts_demo} waic_wts_demo <- exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) / sum(exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)])) round(waic_wts_demo, 3) ``` Notice how much the weight for model 4 is lowered now that more models similar to model 1 (or in this case identical) have been added. Both WAIC weights and Pseudo-BMA approaches first estimate the predictive performance separately for each model and then compute weights based on estimated relative predictive performances. Similar models share similar weights so the weights of other models must be reduced for the total sum of the weights to remain the same. On the other hand, stacking optimizes the weights _jointly_, allowing for the very similar models (in this toy example repeated models) to share their weight while more unique models keep their original weights. In our example we can see this difference clearly: ```{r stacking_weights} stacking_weights(lpd_point[,c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) ``` Using stacking, the weight for the best model stays essentially unchanged. # Example: Oceanic tool complexity Another example we consider is the Kline oceanic tool complexity data, which McElreath describes as follows: >Different historical island populations possessed tool kits of different size. These kits include fish hooks, axes, boats, hand plows, and many other types of tools. A number of theories predict that larger populations will both develop and sustain more complex tool kits. ... It's also suggested that contact rates among populations effectively increases population [sic, probably should be tool kit] size, as it's relevant to technological evolution. We build models predicting the total number of tools given the log population size and the contact rate (high vs. low). ```{r Kline} data(Kline) d <- Kline d$log_pop <- log(d$population) d$contact_high <- ifelse(d$contact=="high", 1, 0) str(d) ``` We start with a Poisson regression model with the log population size, the contact rate, and an interaction term between them (priors are informative priors as in _Statistical Rethinking_). ```{r fit10, results="hide"} fit10 <- stan_glm( total_tools ~ log_pop + contact_high + log_pop * contact_high, family = poisson(link = "log"), data = d, prior = normal(0, 1, autoscale = FALSE), prior_intercept = normal(0, 100, autoscale = FALSE), seed = 2030 ) ``` Before running other models, we check whether Poisson is good choice as the conditional observation model. ```{r loo10} loo10 <- loo(fit10) print(loo10) ``` We get at least one observation with $k>0.7$ and the estimated effective number of parameters `p_loo` is larger than the total number of parameters in the model. This indicates that Poisson might be too narrow. A negative binomial model might be better, but with so few observations it is not so clear. We can compute LOO more accurately by running Stan again for the leave-one-out folds with high $k$ estimates. When using __rstanarm__ this can be done by specifying the `k_threshold` argument: ```{r loo10-threshold} loo10 <- loo(fit10, k_threshold=0.7) print(loo10) ``` In this case we see that there is not much difference, and thus it is relatively safe to continue. As a comparison we also compute WAIC: ```{r waic10} waic10 <- waic(fit10) print(waic10) ``` The WAIC computation is giving warnings and the estimated ELPD is slightly more optimistic. We recommend using the PSIS-LOO results instead. To assess whether the contact rate and interaction term are useful, we can make a comparison to models without these terms. ```{r contact_high, results="hide"} fit11 <- update(fit10, formula = total_tools ~ log_pop + contact_high) fit12 <- update(fit10, formula = total_tools ~ log_pop) ``` ```{r loo-contact_high} (loo11 <- loo(fit11)) (loo12 <- loo(fit12)) ``` ```{r relo-contact_high} loo11 <- loo(fit11, k_threshold=0.7) loo12 <- loo(fit12, k_threshold=0.7) lpd_point <- cbind( loo10$pointwise[, "elpd_loo"], loo11$pointwise[, "elpd_loo"], loo12$pointwise[, "elpd_loo"] ) ``` For comparison we'll also compute WAIC values for these additional models: ```{r waic-contact_high} waic11 <- waic(fit11) waic12 <- waic(fit12) waics <- c( waic10$estimates["elpd_waic", 1], waic11$estimates["elpd_waic", 1], waic12$estimates["elpd_waic", 1] ) ``` The WAIC computation again gives warnings, and we recommend using PSIS-LOO instead. Finally, we compute 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights. ```{r weights-contact_high} waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ``` All weights favor the second model with the log population and the contact rate. WAIC weights and Pseudo-BMA weights (without Bayesian bootstrap) are similar, while Pseudo-BMA+ is more cautious and closer to stacking weights. It may seem surprising that Bayesian stacking is giving zero weight to the first model, but this is likely due to the fact that the estimated effect for the interaction term is close to zero and thus models 1 and 2 give very similar predictions. In other words, incorporating the model with the interaction (model 1) into the model average doesn't improve the predictions at all and so model 1 is given a weight of 0. On the other hand, models 2 and 3 are giving slightly different predictions and thus their combination may be slightly better than either alone. This behavior is related to the repeated similar model illustration in the milk example above. # Simpler coding using `loo_model_weights` function Although in the examples above we called the `stacking_weights` and `pseudobma_weights` functions directly, we can also use the `loo_model_weights` wrapper, which takes as its input either a list of pointwise log-likelihood matrices or a list of precomputed loo objects. There are also `loo_model_weights` methods for stanreg objects (fitted model objects from __rstanarm__) as well as fitted model objects from other packages (e.g. __brms__) that do the preparation work for the user (see, e.g., the examples at `help("loo_model_weights", package = "rstanarm")`). ```{r loo_model_weights} # using list of loo objects loo_list <- list(loo10, loo11, loo12) loo_model_weights(loo_list) loo_model_weights(loo_list, method = "pseudobma") loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) ``` # References McElreath, R. (2016). _Statistical rethinking: A Bayesian course with examples in R and Stan_. Chapman & Hall/CRC. http://xcelab.net/rm/statistical-rethinking/ Piironen, J. and Vehtari, A. (2017a). Sparsity information and regularization in the horseshoe and other shrinkage priors. In Electronic Journal of Statistics, 11(2):5018-5051. [Online](https://projecteuclid.org/euclid.ejs/1513306866). Piironen, J. and Vehtari, A. (2017b). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3):711-735. \doi:10.1007/s11222-016-9649-y. [Online](https://link.springer.com/article/10.1007/s11222-016-9649-y). Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, \doi:10.1214/17-BA1091. [Online](https://projecteuclid.org/euclid.ba/1516093227). loo/inst/doc/loo2-elpd.R0000644000176200001440000001063314411455326014515 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----stancode----------------------------------------------------------------- stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ## ----setup, message=FALSE----------------------------------------------------- library("rstan") library("loo") seed <- 9547 set.seed(seed) ## ----modelfit-holdout, message=FALSE------------------------------------------ # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) roaches$offset <- log(roaches[,"exposure2"]) # 20% of the data goes to the test set: roaches$test <- 0 roaches$test[sample(.2 * seq_len(nrow(roaches)))] <- 1 # data to "train" the model data_train <- list(y = roaches$y[roaches$test == 0], x = as.matrix(roaches[roaches$test == 0, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 0,]), K = 3, offset = roaches$offset[roaches$test == 0], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) # data to "test" the model data_test <- list(y = roaches$y[roaches$test == 1], x = as.matrix(roaches[roaches$test == 1, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 1,]), K = 3, offset = roaches$offset[roaches$test == 1], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) ## ----fit-train---------------------------------------------------------------- # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) ## ----gen-test----------------------------------------------------------------- gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd <- extract_log_lik(gen_test) ## ----elpd-holdout------------------------------------------------------------- (elpd_holdout <- elpd(log_pd)) ## ----prepare-folds, message=FALSE--------------------------------------------- # Prepare data roaches$fold <- kfold_split_random(K = 10, N = nrow(roaches)) ## ----------------------------------------------------------------------------- # Prepare a matrix with the number of post-warmup iterations by number of observations: log_pd_kfold <- matrix(nrow = 4000, ncol = nrow(roaches)) # Loop over the folds for(k in 1:10){ data_train <- list(y = roaches$y[roaches$fold != k], x = as.matrix(roaches[roaches$fold != k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold != k,]), K = 3, offset = roaches$offset[roaches$fold != k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) data_test <- list(y = roaches$y[roaches$fold == k], x = as.matrix(roaches[roaches$fold == k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold == k,]), K = 3, offset = roaches$offset[roaches$fold == k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd_kfold[, roaches$fold == k] <- extract_log_lik(gen_test) } ## ----elpd-kfold--------------------------------------------------------------- (elpd_kfold <- elpd(log_pd_kfold)) loo/inst/doc/loo2-with-rstan.html0000644000176200001440000005774614411465510016450 0ustar liggesusers Writing Stan programs for use with the loo package

Writing Stan programs for use with the loo package

Aki Vehtari and Jonah Gabry

2023-03-30

Introduction

This vignette demonstrates how to write a Stan program that computes and stores the pointwise log-likelihood required for using the loo package. The other vignettes included with the package demonstrate additional functionality.

Some sections from this vignette are excerpted from our papers

  • Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

  • Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.04544.

which provide important background for understanding the methods implemented in the package.

Example: Well water in Bangladesh

This example comes from a survey of residents from a small area in Bangladesh that was affected by arsenic in drinking water. Respondents with elevated arsenic levels in their wells were asked if they were interested in getting water from a neighbor’s well, and a series of logistic regressions were fit to predict this binary response given various information about the households (Gelman and Hill, 2007). Here we fit a model for the well-switching response given two predictors: the arsenic level of the water in the resident’s home, and the distance of the house from the nearest safe well.

The sample size in this example is \(N=3020\), which is not huge but is large enough that it is important to have a computational method for LOO that is fast for each data point. On the plus side, with such a large dataset, the influence of any given observation is small, and so the computations should be stable.

Coding the Stan model

Here is the Stan code for fitting the logistic regression model, which we save in a file called logistic.stan:

data {
  int<lower=0> N;             // number of data points
  int<lower=0> P;             // number of predictors (including intercept)
  matrix[N,P] X;              // predictors (including 1s for intercept)
  int<lower=0,upper=1> y[N];  // binary outcome
}
parameters {
  vector[P] beta;
}
model {
  beta ~ normal(0, 1);
  y ~ bernoulli_logit(X * beta);
}
generated quantities {
  vector[N] log_lik;
  for (n in 1:N) {
    log_lik[n] = bernoulli_logit_lpmf(y[n] | X[n] * beta);
  }
}

We have defined the log likelihood as a vector named log_lik in the generated quantities block so that the individual terms will be saved by Stan. After running Stan, log_lik can be extracted (using the extract_log_lik function provided in the loo package) as an \(S \times N\) matrix, where \(S\) is the number of simulations (posterior draws) and \(N\) is the number of data points.

Fitting the model with RStan

Next we fit the model in Stan using the rstan package:

library("rstan")

# Prepare data 
url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat"
wells <- read.table(url)
wells$dist100 <- with(wells, dist / 100)
X <- model.matrix(~ dist100 + arsenic, wells)
standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X))

# Fit model
fit_1 <- stan("logistic.stan", data = standata)
print(fit_1, pars = "beta")
         mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
beta[1]  0.00       0 0.08 -0.16 -0.05  0.00  0.05  0.15  1964    1
beta[2] -0.89       0 0.10 -1.09 -0.96 -0.89 -0.82 -0.68  2048    1
beta[3]  0.46       0 0.04  0.38  0.43  0.46  0.49  0.54  2198    1

Computing approximate leave-one-out cross-validation using PSIS-LOO

We can then use the loo package to compute the efficient PSIS-LOO approximation to exact LOO-CV:

library("loo")

# Extract pointwise log-likelihood
# using merge_chains=FALSE returns an array, which is easier to 
# use with relative_eff()
log_lik_1 <- extract_log_lik(fit_1, merge_chains = FALSE)

# as of loo v2.0.0 we can optionally provide relative effective sample sizes
# when calling loo, which allows for better estimates of the PSIS effective
# sample sizes and Monte Carlo error
r_eff <- relative_eff(exp(log_lik_1), cores = 2) 

# preferably use more than 2 cores (as many cores as possible)
# will use value of 'mc.cores' option if cores is not specified
loo_1 <- loo(log_lik_1, r_eff = r_eff, cores = 2)
print(loo_1)
Computed from 4000 by 3020 log-likelihood matrix

         Estimate   SE
elpd_loo  -1968.5 15.6
p_loo         3.2  0.1
looic      3937.0 31.2
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

The printed output from the loo function shows the estimates \(\widehat{\mbox{elpd}}_{\rm loo}\) (expected log predictive density), \(\widehat{p}_{\rm loo}\) (effective number of parameters), and \({\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}\) (the LOO information criterion).

The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the \(k\) parameter is explained in help('pareto-k-diagnostic') and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case the message tells us that all of the estimates for \(k\) are fine.

Comparing models

To compare this model to an alternative model for the same data we can use the loo_compare function in the loo package. First we’ll fit a second model to the well-switching data, using log(arsenic) instead of arsenic as a predictor:

standata$X[, "arsenic"] <- log(standata$X[, "arsenic"])
fit_2 <- stan(fit = fit_1, data = standata) 

log_lik_2 <- extract_log_lik(fit_2, merge_chains = FALSE)
r_eff_2 <- relative_eff(exp(log_lik_2))
loo_2 <- loo(log_lik_2, r_eff = r_eff_2, cores = 2)
print(loo_2)
Computed from 4000 by 3020 log-likelihood matrix

         Estimate   SE
elpd_loo  -1952.3 16.2
p_loo         3.1  0.1
looic      3904.6 32.4
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

We can now compare the models on LOO using the loo_compare function:

# Compare
comp <- loo_compare(loo_1, loo_2)

This new object, comp, contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error:

print(comp) # can set simplify=FALSE for more detailed print output
       elpd_diff se_diff
model2   0.0       0.0  
model1 -16.3       4.4

The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in elpd and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (model2).

References

Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge University Press.

Stan Development Team (2017). The Stan C++ Library, Version 2.17.0. https://mc-stan.org/

Stan Development Team (2018) RStan: the R interface to Stan, Version 2.17.3. https://mc-stan.org/

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. online, arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

loo/inst/doc/loo2-lfo.html0000644000176200001440000155515014411464745015132 0ustar liggesusers Approximate leave-future-out cross-validation for Bayesian time series models

Approximate leave-future-out cross-validation for Bayesian time series models

Paul Bürkner, Jonah Gabry, Aki Vehtari

2023-03-30

Introduction

One of the most common goals of a time series analysis is to use the observed series to inform predictions for future observations. We will refer to this task of predicting a sequence of \(M\) future observations as \(M\)-step-ahead prediction (\(M\)-SAP). Fortunately, once we have fit a model and can sample from the posterior predictive distribution, it is straightforward to generate predictions as far into the future as we want. It is also straightforward to evaluate the \(M\)-SAP performance of a time series model by comparing the predictions to the observed sequence of \(M\) future data points once they become available.

Unfortunately, we are often in the position of having to use a model to inform decisions before we can collect the future observations required for assessing the predictive performance. If we have many competing models we may also need to first decide which of the models (or which combination of the models) we should rely on for predictions. In these situations the best we can do is to use methods for approximating the expected predictive performance of our models using only the observations of the time series we already have.

If there were no time dependence in the data or if the focus is to assess the non-time-dependent part of the model, we could use methods like leave-one-out cross-validation (LOO-CV). For a data set with \(N\) observations, we refit the model \(N\) times, each time leaving out one of the \(N\) observations and assessing how well the model predicts the left-out observation. LOO-CV is very expensive computationally in most realistic settings, but the Pareto smoothed importance sampling (PSIS, Vehtari et al, 2017, 2019) algorithm provided by the loo package allows for approximating exact LOO-CV with PSIS-LOO-CV. PSIS-LOO-CV requires only a single fit of the full model and comes with diagnostics for assessing the validity of the approximation.

With a time series we can do something similar to LOO-CV but, except in a few cases, it does not make sense to leave out observations one at a time because then we are allowing information from the future to influence predictions of the past (i.e., times \(t + 1, t+2, \ldots\) should not be used to predict for time \(t\)). To apply the idea of cross-validation to the \(M\)-SAP case, instead of leave-one-out cross-validation we need some form of leave-future-out cross-validation (LFO-CV). As we will demonstrate in this case study, LFO-CV does not refer to one particular prediction task but rather to various possible cross-validation approaches that all involve some form of prediction for new time series data. Like exact LOO-CV, exact LFO-CV requires refitting the model many times to different subsets of the data, which is computationally very costly for most nontrivial examples, in particular for Bayesian analyses where refitting the model means estimating a new posterior distribution rather than a point estimate.

Although PSIS-LOO-CV provides an efficient approximation to exact LOO-CV, until now there has not been an analogous approximation to exact LFO-CV that drastically reduces the computational burden while also providing informative diagnostics about the quality of the approximation. In this case study we present PSIS-LFO-CV, an algorithm that typically only requires refitting the time-series model a small number times and will make LFO-CV tractable for many more realistic applications than previously possible.

More details can be found in our paper about approximate LFO-CV (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1902.06281).

\(M\)-step-ahead predictions

Assume we have a time series of observations \(y = (y_1, y_2, \ldots, y_N)\) and let \(L\) be the minimum number of observations from the series that we will require before making predictions for future data. Depending on the application and how informative the data is, it may not be possible to make reasonable predictions for \(y_{i+1}\) based on \((y_1, \dots, y_{i})\) until \(i\) is large enough so that we can learn enough about the time series to predict future observations. Setting \(L=10\), for example, means that we will only assess predictive performance starting with observation \(y_{11}\), so that we always have at least 10 previous observations to condition on.

In order to assess \(M\)-SAP performance we would like to compute the predictive densities

\[ p(y_{i+1:M} \,|\, y_{1:i}) = p(y_{i+1}, \ldots, y_{i + M} \,|\, y_{1},...,y_{i}) \]

for each \(i \in \{L, \ldots, N - M\}\). The quantities \(p(y_{i+1:M} \,|\, y_{1:i})\) can be computed with the help of the posterior distribution \(p(\theta \,|\, y_{1:i})\) of the parameters \(\theta\) conditional on only the first \(i\) observations of the time-series:

\[ p(y_{i+1:M} \,| \, y_{1:i}) = \int p(y_{i+1:M} \,| \, y_{1:i}, \theta) \, p(\theta\,|\,y_{1:i}) \,d\theta. \]

Having obtained \(S\) draws \((\theta_{1:i}^{(1)}, \ldots, \theta_{1:i}^{(S)})\) from the posterior distribution \(p(\theta\,|\,y_{1:i})\), we can estimate \(p(y_{i+1:M} | y_{1:i})\) as

\[ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{1}{S}\sum_{s=1}^S p(y_{i+1:M} \,|\, y_{1:i}, \theta_{1:i}^{(s)}). \]

Approximate \(M\)-SAP using importance-sampling

Unfortunately, the math above makes use of the posterior distributions from many different fits of the model to different subsets of the data. That is, to obtain the predictive density \(p(y_{i+1:M} \,|\, y_{1:i})\) requires fitting a model to only the first \(i\) data points, and we will need to do this for every value of \(i\) under consideration (all \(i \in \{L, \ldots, N - M\}\)).

To reduce the number of models that need to be fit for the purpose of obtaining each of the densities \(p(y_{i+1:M} \,|\, y_{1:i})\), we propose the following algorithm. First, we refit the model using the first \(L\) observations of the time series and then perform a single exact \(M\)-step-ahead prediction step for \(p(y_{L+1:M} \,|\, y_{1:L})\). Recall that \(L\) is the minimum number of observations we have deemed acceptable for making predictions (setting \(L=0\) means the first data point will be predicted only based on the prior). We define \(i^\star = L\) as the current point of refit. Next, starting with \(i = i^\star + 1\), we approximate each \(p(y_{i+1:M} \,|\, y_{1:i})\) via

\[ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{ \sum_{s=1}^S w_i^{(s)}\, p(y_{i+1:M} \,|\, y_{1:i}, \theta^{(s)})} { \sum_{s=1}^S w_i^{(s)}}, \]

where \(\theta^{(s)} = \theta^{(s)}_{1:i^\star}\) are draws from the posterior distribution based on the first \(i^\star\) observations and \(w_i^{(s)}\) are the PSIS weights obtained in two steps. First, we compute the raw importance ratios

\[ r_i^{(s)} = \frac{f_{1:i}(\theta^{(s)})}{f_{1:i^\star}(\theta^{(s)})} \propto \prod_{j \in (i^\star + 1):i} p(y_j \,|\, y_{1:(j-1)}, \theta^{(s)}), \]

and then stabilize them using PSIS. The function \(f_{1:i}\) denotes the posterior distribution based on the first \(i\) observations, that is, \(f_{1:i} = p(\theta \,|\, y_{1:i})\), with \(f_{1:i^\star}\) defined analogously. The index set \((i^\star + 1):i\) indicates all observations which are part of the data for the model \(f_{1:i}\) whose predictive performance we are trying to approximate but not for the actually fitted model \(f_{1:i^\star}\). The proportional statement arises from the fact that we ignore the normalizing constants \(p(y_{1:i})\) and \(p(y_{1:i^\star})\) of the compared posteriors, which leads to a self-normalized variant of PSIS (see Vehtari et al, 2017).

Continuing with the next observation, we gradually increase \(i\) by \(1\) (we move forward in time) and repeat the process. At some observation \(i\), the variability of the importance ratios \(r_i^{(s)}\) will become too large and importance sampling will fail. We will refer to this particular value of \(i\) as \(i^\star_1\). To identify the value of \(i^\star_1\), we check for which value of \(i\) does the estimated shape parameter \(k\) of the generalized Pareto distribution first cross a certain threshold \(\tau\) (Vehtari et al, 2019). Only then do we refit the model using the observations up to \(i^\star_1\) and restart the process from there by setting \(\theta^{(s)} = \theta^{(s)}_{1:i^\star_1}\) and \(i^\star = i^\star_1\) until the next refit.

In some cases we may only need to refit once and in other cases we will find a value \(i^\star_2\) that requires a second refitting, maybe an \(i^\star_3\) that requires a third refitting, and so on. We refit as many times as is required (only when \(k > \tau\)) until we arrive at observation \(i = N - M\). For LOO, we recommend to use a threshold of \(\tau = 0.7\) (Vehtari et al, 2017, 2019) and it turns out this is a reasonable threshold for LFO as well (Bürkner et al. 2020).

Autoregressive models

Autoregressive (AR) models are some of the most commonly used time-series models. An AR(p) model —an autoregressive model of order \(p\)— can be defined as

\[ y_i = \eta_i + \sum_{k = 1}^p \varphi_k y_{i - k} + \varepsilon_i, \]

where \(\eta_i\) is the linear predictor for the \(i\)th observation, \(\phi_k\) are the autoregressive parameters and \(\varepsilon_i\) are pairwise independent errors, which are usually assumed to be normally distributed with equal variance \(\sigma^2\). The model implies a recursive formula that allows for computing the right-hand side of the above equation for observation \(i\) based on the values of the equations for previous observations.

Case Study: Annual measurements of the level of Lake Huron

To illustrate the application of PSIS-LFO-CV for estimating expected \(M\)-SAP performance, we will fit a model for 98 annual measurements of the water level (in feet) of Lake Huron from the years 1875–1972. This data set is found in the datasets R package, which is installed automatically with R.

In addition to the loo package, for this analysis we will use the brms interface to Stan to generate a Stan program and fit the model, and also the bayesplot and ggplot2 packages for plotting.

library("loo")
library("brms")
library("bayesplot")
library("ggplot2")
color_scheme_set("brightblue")
theme_set(theme_default())

CHAINS <- 4
SEED <- 5838296
set.seed(SEED)

Before fitting a model, we will first put the data into a data frame and then look at the time series.

N <- length(LakeHuron)
df <- data.frame(
  y = as.numeric(LakeHuron),
  year = as.numeric(time(LakeHuron)),
  time = 1:N
)

ggplot(df, aes(x = year, y = y)) + 
  geom_point(size = 1) +
  labs(
    y = "Water Level (ft)", 
    x = "Year",
    title = "Water Level in Lake Huron (1875-1972)"
  )

The above plot shows rather strong autocorrelation of the time-series as well as some trend towards lower levels for later points in time.

We can specify an AR(4) model for these data using the brms package as follows:

fit <- brm(
  y ~ ar(time, p = 4), 
  data = df, 
  prior = prior(normal(0, 0.5), class = "ar"),
  control = list(adapt_delta = 0.99), 
  seed = SEED, 
  chains = CHAINS
)

The model implied predictions along with the observed values can be plotted, which reveals a rather good fit to the data.

preds <- posterior_predict(fit)
preds <- cbind(
  Estimate = colMeans(preds), 
  Q5 = apply(preds, 2, quantile, probs = 0.05),
  Q95 = apply(preds, 2, quantile, probs = 0.95)
)

ggplot(cbind(df, preds), aes(x = year, y = Estimate)) +
  geom_smooth(aes(ymin = Q5, ymax = Q95), stat = "identity", size = 0.5) +
  geom_point(aes(y = y)) + 
  labs(
    y = "Water Level (ft)", 
    x = "Year",
    title = "Water Level in Lake Huron (1875-1972)",
    subtitle = "Mean (blue) and 90% predictive intervals (gray) vs. observed data (black)"
  )

To allow for reasonable predictions of future values, we will require at least \(L = 20\) historical observations (20 years) to make predictions.

L <- 20

We first perform approximate leave-one-out cross-validation (LOO-CV) for the purpose of later comparison with exact and approximate LFO-CV for the 1-SAP case.

loo_cv <- loo(log_lik(fit)[, (L + 1):N])
print(loo_cv)

Computed from 4000 by 78 log-likelihood matrix

         Estimate   SE
elpd_loo    -88.6  6.4
p_loo         4.8  1.0
looic       177.2 12.8
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

1-step-ahead predictions leaving out all future values

The most basic version of \(M\)-SAP is 1-SAP, in which we predict only one step ahead. In this case, \(y_{i+1:M}\) simplifies to \(y_{i}\) and the LFO-CV algorithm becomes considerably simpler than for larger values of \(M\).

Exact 1-step-ahead predictions

Before we compute approximate LFO-CV using PSIS we will first compute exact LFO-CV for the 1-SAP case so we can use it as a benchmark later. The initial step for the exact computation is to calculate the log-predictive densities by refitting the model many times:

loglik_exact <- matrix(nrow = nsamples(fit), ncol = N)
for (i in L:(N - 1)) {
  past <- 1:i
  oos <- i + 1
  df_past <- df[past, , drop = FALSE]
  df_oos <- df[c(past, oos), , drop = FALSE]
  fit_i <- update(fit, newdata = df_past, recompile = FALSE)
  loglik_exact[, i + 1] <- log_lik(fit_i, newdata = df_oos, oos = oos)[, oos]
}

Then we compute the exact expected log predictive density (ELPD):

# some helper functions we'll use throughout

# more stable than log(sum(exp(x))) 
log_sum_exp <- function(x) {
  max_x <- max(x)  
  max_x + log(sum(exp(x - max_x)))
}

# more stable than log(mean(exp(x)))
log_mean_exp <- function(x) {
  log_sum_exp(x) - log(length(x))
}

# compute log of raw importance ratios
# sums over observations *not* over posterior samples
sum_log_ratios <- function(loglik, ids = NULL) {
  if (!is.null(ids)) loglik <- loglik[, ids, drop = FALSE]
  rowSums(loglik)
}

# for printing comparisons later
rbind_print <- function(...) {
  round(rbind(...), digits = 2)
}
exact_elpds_1sap <- apply(loglik_exact, 2, log_mean_exp)
exact_elpd_1sap <- c(ELPD = sum(exact_elpds_1sap[-(1:L)]))

rbind_print(
  "LOO" = loo_cv$estimates["elpd_loo", "Estimate"],
  "LFO" = exact_elpd_1sap
)
      ELPD
LOO -88.61
LFO -92.49

We see that the ELPD from LFO-CV for 1-step-ahead predictions is lower than the ELPD estimate from LOO-CV, which should be expected since LOO-CV is making use of more of the time series. That is, since the LFO-CV approach only uses observations from before the left-out data point but LOO-CV uses all data points other than the left-out observation, we should expect to see the larger ELPD from LOO-CV.

Approximate 1-step-ahead predictions

We compute approximate 1-SAP with refit at observations where the Pareto \(k\) estimate exceeds the threshold of \(0.7\).

k_thres <- 0.7

The code becomes a little bit more involved as compared to the exact LFO-CV. Note that we can compute exact 1-SAP at the refitting points, which comes with no additional computational costs since we had to refit the model anyway.

approx_elpds_1sap <- rep(NA, N)

# initialize the process for i = L
past <- 1:L
oos <- L + 1
df_past <- df[past, , drop = FALSE]
df_oos <- df[c(past, oos), , drop = FALSE]
fit_past <- update(fit, newdata = df_past, recompile = FALSE)
loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
approx_elpds_1sap[L + 1] <- log_mean_exp(loglik[, oos])

# iterate over i > L
i_refit <- L
refits <- L
ks <- NULL
for (i in (L + 1):(N - 1)) {
  past <- 1:i
  oos <- i + 1
  df_past <- df[past, , drop = FALSE]
  df_oos <- df[c(past, oos), , drop = FALSE]
  loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
  
  logratio <- sum_log_ratios(loglik, (i_refit + 1):i)
  psis_obj <- suppressWarnings(psis(logratio))
  k <- pareto_k_values(psis_obj)
  ks <- c(ks, k)
  if (k > k_thres) {
    # refit the model based on the first i observations
    i_refit <- i
    refits <- c(refits, i)
    fit_past <- update(fit_past, newdata = df_past, recompile = FALSE)
    loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
    approx_elpds_1sap[i + 1] <- log_mean_exp(loglik[, oos])
  } else {
    lw <- weights(psis_obj, normalize = TRUE)[, 1]
    approx_elpds_1sap[i + 1] <- log_sum_exp(lw + loglik[, oos])
  }
}

We see that the final Pareto-\(k\)-estimates are mostly well below the threshold and that we only needed to refit the model a few times:

plot_ks <- function(ks, ids, thres = 0.6) {
  dat_ks <- data.frame(ks = ks, ids = ids)
  ggplot(dat_ks, aes(x = ids, y = ks)) + 
    geom_point(aes(color = ks > thres), shape = 3, show.legend = FALSE) + 
    geom_hline(yintercept = thres, linetype = 2, color = "red2") + 
    scale_color_manual(values = c("cornflowerblue", "darkblue")) + 
    labs(x = "Data point", y = "Pareto k") + 
    ylim(-0.5, 1.5)
}
cat("Using threshold ", k_thres, 
    ", model was refit ", length(refits), 
    " times, at observations", refits)
Using threshold  0.7 , model was refit  3  times, at observations 20 43 84
plot_ks(ks, (L + 1):(N - 1))

The approximate 1-SAP ELPD is remarkably similar to the exact 1-SAP ELPD computed above, which indicates our algorithm to compute approximate 1-SAP worked well for the present data and model.

approx_elpd_1sap <- sum(approx_elpds_1sap, na.rm = TRUE)
rbind_print(
  "approx LFO" = approx_elpd_1sap,
  "exact LFO" = exact_elpd_1sap
)
             ELPD
approx LFO -92.88
exact LFO  -92.49

Plotting exact against approximate predictions, we see that no approximation value deviates far from its exact counterpart, providing further evidence for the good quality of our approximation.

dat_elpd <- data.frame(
  approx_elpd = approx_elpds_1sap,
  exact_elpd = exact_elpds_1sap
)

ggplot(dat_elpd, aes(x = approx_elpd, y = exact_elpd)) +
  geom_abline(color = "gray30") +
  geom_point(size = 2) +
  labs(x = "Approximate ELPDs", y = "Exact ELPDs")

We can also look at the maximum difference and average difference between the approximate and exact ELPD calculations, which also indicate a ver close approximation:

max_diff <- with(dat_elpd, max(abs(approx_elpd - exact_elpd), na.rm = TRUE))
mean_diff <- with(dat_elpd, mean(abs(approx_elpd - exact_elpd), na.rm = TRUE))

rbind_print(
  "Max diff" = round(max_diff, 2), 
  "Mean diff" =  round(mean_diff, 3)
)
          [,1]
Max diff  0.14
Mean diff 0.01

\(M\)-step-ahead predictions leaving out all future values

To illustrate the application of \(M\)-SAP for \(M > 1\), we next compute exact and approximate LFO-CV for the 4-SAP case.

Exact \(M\)-step-ahead predictions

The necessary steps are the same as for 1-SAP with the exception that the log-density values of interest are now the sums of the log predictive densities of four consecutive observations. Further, the stability of the PSIS approximation actually stays the same for all \(M\) as it only depends on the number of observations we leave out, not on the number of observations we predict.

M <- 4
loglikm <- matrix(nrow = nsamples(fit), ncol = N)
for (i in L:(N - M)) {
  past <- 1:i
  oos <- (i + 1):(i + M)
  df_past <- df[past, , drop = FALSE]
  df_oos <- df[c(past, oos), , drop = FALSE]
  fit_past <- update(fit, newdata = df_past, recompile = FALSE)
  loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
  loglikm[, i + 1] <- rowSums(loglik[, oos])
}
exact_elpds_4sap <- apply(loglikm, 2, log_mean_exp)
(exact_elpd_4sap <- c(ELPD = sum(exact_elpds_4sap, na.rm = TRUE)))
     ELPD 
-405.4043

Approximate \(M\)-step-ahead predictions

Computing the approximate PSIS-LFO-CV for the 4-SAP case is a little bit more involved than the approximate version for the 1-SAP case, although the underlying principles remain the same.

approx_elpds_4sap <- rep(NA, N)

# initialize the process for i = L
past <- 1:L
oos <- (L + 1):(L + M)
df_past <- df[past, , drop = FALSE]
df_oos <- df[c(past, oos), , drop = FALSE]
fit_past <- update(fit, newdata = df_past, recompile = FALSE)
loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
loglikm <- rowSums(loglik[, oos])
approx_elpds_1sap[L + 1] <- log_mean_exp(loglikm)

# iterate over i > L
i_refit <- L
refits <- L
ks <- NULL
for (i in (L + 1):(N - M)) {
  past <- 1:i
  oos <- (i + 1):(i + M)
  df_past <- df[past, , drop = FALSE]
  df_oos <- df[c(past, oos), , drop = FALSE]
  loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
  
  logratio <- sum_log_ratios(loglik, (i_refit + 1):i)
  psis_obj <- suppressWarnings(psis(logratio))
  k <- pareto_k_values(psis_obj)
  ks <- c(ks, k)
  if (k > k_thres) {
    # refit the model based on the first i observations
    i_refit <- i
    refits <- c(refits, i)
    fit_past <- update(fit_past, newdata = df_past, recompile = FALSE)
    loglik <- log_lik(fit_past, newdata = df_oos, oos = oos)
    loglikm <- rowSums(loglik[, oos])
    approx_elpds_4sap[i + 1] <- log_mean_exp(loglikm)
  } else {
    lw <- weights(psis_obj, normalize = TRUE)[, 1]
    loglikm <- rowSums(loglik[, oos])
    approx_elpds_4sap[i + 1] <- log_sum_exp(lw + loglikm)
  }
}

Again, we see that the final Pareto-\(k\)-estimates are mostly well below the threshold and that we only needed to refit the model a few times:

cat("Using threshold ", k_thres, 
    ", model was refit ", length(refits), 
    " times, at observations", refits)
Using threshold  0.7 , model was refit  3  times, at observations 20 46 82
plot_ks(ks, (L + 1):(N - M))

The approximate ELPD computed for the 4-SAP case is not as close to its exact counterpart as in the 1-SAP case. In general, the larger \(M\), the larger the variation of the approximate ELPD around the exact ELPD. It turns out that the ELPD estimates of AR-models with \(M>1\) show particular variation due to their predictions’ dependency on other predicted values. In Bürkner et al. (2020) we provide further explanation and simulations for these cases.

approx_elpd_4sap <- sum(approx_elpds_4sap, na.rm = TRUE)
rbind_print(
  "Approx LFO" = approx_elpd_4sap,
  "Exact LFO" = exact_elpd_4sap
)
              ELPD
Approx LFO -396.87
Exact LFO  -405.40

Plotting exact against approximate pointwise predictions confirms that, for a few specific data points, the approximate predictions underestimate the exact predictions.

dat_elpd_4sap <- data.frame(
  approx_elpd = approx_elpds_4sap,
  exact_elpd = exact_elpds_4sap
)

ggplot(dat_elpd_4sap, aes(x = approx_elpd, y = exact_elpd)) +
  geom_abline(color = "gray30") +
  geom_point(size = 2) +
  labs(x = "Approximate ELPDs", y = "Exact ELPDs")

Conclusion

In this case study we have shown how to do carry out exact and approximate leave-future-out cross-validation for \(M\)-step-ahead prediction tasks. For the data and model used in our example, the PSIS-LFO-CV algorithm provides reasonably stable and accurate results despite not requiring us to refit the model nearly as many times. For more details on approximate LFO-CV, we refer to Bürkner et al. (2020).


References

Bürkner P. C., Gabry J., & Vehtari A. (2020). Approximate leave-future-out cross-validation for time series models. Journal of Statistical Computation and Simulation, 90(14):2499-2523. :/10.1080/00949655.2020.1783262. Online. arXiv preprint.

Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Online. arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.


Appendix

Appendix: Session information

sessionInfo()
R version 4.2.2 (2022-10-31)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur ... 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib

locale:
[1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_3.4.1           brms_2.18.0             bayesplot_1.10.0.9001  
[4] rstanarm_2.21.3         Rcpp_1.0.10             loo_2.6.0              
[7] rstan_2.31.0.9000       StanHeaders_2.31.0.9000 knitr_1.40             

loaded via a namespace (and not attached):
  [1] minqa_1.2.5          colorspace_2.1-0     ellipsis_0.3.2      
  [4] estimability_1.4.1   markdown_1.3         base64enc_0.1-3     
  [7] rstudioapi_0.14      farver_2.1.1         DT_0.26             
 [10] fansi_1.0.4          mvtnorm_1.1-3        bridgesampling_1.1-2
 [13] codetools_0.2-18     splines_4.2.2        cachem_1.0.6        
 [16] shinythemes_1.2.0    projpred_2.2.2       jsonlite_1.8.4      
 [19] nloptr_2.0.3         shiny_1.7.3          compiler_4.2.2      
 [22] emmeans_1.8.2        backports_1.4.1      Matrix_1.5-1        
 [25] fastmap_1.1.1        cli_3.6.0            later_1.3.0         
 [28] htmltools_0.5.3      prettyunits_1.1.1    tools_4.2.2         
 [31] igraph_1.3.5         coda_0.19-4          gtable_0.3.2        
 [34] glue_1.6.2           reshape2_1.4.4       dplyr_1.1.0         
 [37] posterior_1.4.1      V8_4.2.2             jquerylib_0.1.4     
 [40] vctrs_0.6.0          nlme_3.1-160         crosstalk_1.2.0     
 [43] tensorA_0.36.2       xfun_0.34            stringr_1.5.0       
 [46] ps_1.7.2             lme4_1.1-31          mime_0.12           
 [49] miniUI_0.1.1.1       lifecycle_1.0.3      gtools_3.9.3        
 [52] MASS_7.3-58.1        zoo_1.8-11           scales_1.2.1        
 [55] colourpicker_1.2.0   promises_1.2.0.1     Brobdingnag_1.2-9   
 [58] parallel_4.2.2       inline_0.3.19        shinystan_2.6.0     
 [61] gamm4_0.2-6          yaml_2.3.7           curl_5.0.0          
 [64] gridExtra_2.3        sass_0.4.2           stringi_1.7.12      
 [67] highr_0.10           dygraphs_1.1.1.6     checkmate_2.1.0     
 [70] boot_1.3-28          pkgbuild_1.4.0       rlang_1.1.0         
 [73] pkgconfig_2.0.3      matrixStats_0.63.0   distributional_0.3.1
 [76] evaluate_0.20        lattice_0.20-45      rstantools_2.3.1    
 [79] htmlwidgets_1.5.4    labeling_0.4.2       processx_3.8.0      
 [82] tidyselect_1.2.0     plyr_1.8.8           magrittr_2.0.3      
 [85] R6_2.5.1             generics_0.1.3       mgcv_1.8-41         
 [88] pillar_1.8.1         withr_2.5.0          xts_0.12.2          
 [91] survival_3.4-0       abind_1.4-5          tibble_3.2.0        
 [94] crayon_1.5.2         utf8_1.2.3           rmarkdown_2.18      
 [97] grid_4.2.2           callr_3.7.3          threejs_0.3.3       
[100] digest_0.6.30        xtable_1.8-4         httpuv_1.6.6        
[103] RcppParallel_5.1.7   stats4_4.2.2         munsell_0.5.0       
[106] bslib_0.4.1          shinyjs_2.1.0

Appendix: Licenses

  • Code © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under BSD-3).
  • Text © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under CC-BY-NC 4.0).
loo/inst/doc/loo2-weights.R0000644000176200001440000001141614411465507015245 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----setup, message=FALSE----------------------------------------------------- library(rstanarm) library(loo) ## ----data--------------------------------------------------------------------- data(milk) d <- milk[complete.cases(milk),] d$neocortex <- d$neocortex.perc /100 str(d) ## ----fits, results="hide"----------------------------------------------------- fit1 <- stan_glm(kcal.per.g ~ 1, data = d, seed = 2030) fit2 <- update(fit1, formula = kcal.per.g ~ neocortex) fit3 <- update(fit1, formula = kcal.per.g ~ log(mass)) fit4 <- update(fit1, formula = kcal.per.g ~ neocortex + log(mass)) ## ----waic--------------------------------------------------------------------- waic1 <- waic(fit1) waic2 <- waic(fit2) waic3 <- waic(fit3) waic4 <- waic(fit4) waics <- c( waic1$estimates["elpd_waic", 1], waic2$estimates["elpd_waic", 1], waic3$estimates["elpd_waic", 1], waic4$estimates["elpd_waic", 1] ) ## ----loo---------------------------------------------------------------------- # note: the loo function accepts a 'cores' argument that we recommend specifying # when working with bigger datasets loo1 <- loo(fit1) loo2 <- loo(fit2) loo3 <- loo(fit3) loo4 <- loo(fit4) lpd_point <- cbind( loo1$pointwise[,"elpd_loo"], loo2$pointwise[,"elpd_loo"], loo3$pointwise[,"elpd_loo"], loo4$pointwise[,"elpd_loo"] ) ## ----print-loo---------------------------------------------------------------- print(loo3) print(loo4) ## ----weights------------------------------------------------------------------ waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ## ----waic_wts_demo------------------------------------------------------------ waic_wts_demo <- exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) / sum(exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)])) round(waic_wts_demo, 3) ## ----stacking_weights--------------------------------------------------------- stacking_weights(lpd_point[,c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) ## ----Kline-------------------------------------------------------------------- data(Kline) d <- Kline d$log_pop <- log(d$population) d$contact_high <- ifelse(d$contact=="high", 1, 0) str(d) ## ----fit10, results="hide"---------------------------------------------------- fit10 <- stan_glm( total_tools ~ log_pop + contact_high + log_pop * contact_high, family = poisson(link = "log"), data = d, prior = normal(0, 1, autoscale = FALSE), prior_intercept = normal(0, 100, autoscale = FALSE), seed = 2030 ) ## ----loo10-------------------------------------------------------------------- loo10 <- loo(fit10) print(loo10) ## ----loo10-threshold---------------------------------------------------------- loo10 <- loo(fit10, k_threshold=0.7) print(loo10) ## ----waic10------------------------------------------------------------------- waic10 <- waic(fit10) print(waic10) ## ----contact_high, results="hide"--------------------------------------------- fit11 <- update(fit10, formula = total_tools ~ log_pop + contact_high) fit12 <- update(fit10, formula = total_tools ~ log_pop) ## ----loo-contact_high--------------------------------------------------------- (loo11 <- loo(fit11)) (loo12 <- loo(fit12)) ## ----relo-contact_high-------------------------------------------------------- loo11 <- loo(fit11, k_threshold=0.7) loo12 <- loo(fit12, k_threshold=0.7) lpd_point <- cbind( loo10$pointwise[, "elpd_loo"], loo11$pointwise[, "elpd_loo"], loo12$pointwise[, "elpd_loo"] ) ## ----waic-contact_high-------------------------------------------------------- waic11 <- waic(fit11) waic12 <- waic(fit12) waics <- c( waic10$estimates["elpd_waic", 1], waic11$estimates["elpd_waic", 1], waic12$estimates["elpd_waic", 1] ) ## ----weights-contact_high----------------------------------------------------- waic_wts <- exp(waics) / sum(exp(waics)) pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE) pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE stacking_wts <- stacking_weights(lpd_point) round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2) ## ----loo_model_weights-------------------------------------------------------- # using list of loo objects loo_list <- list(loo10, loo11, loo12) loo_model_weights(loo_list) loo_model_weights(loo_list, method = "pseudobma") loo_model_weights(loo_list, method = "pseudobma", BB = FALSE) loo/inst/doc/loo2-large-data.R0000644000176200001440000001615214411455345015575 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----setup, message=FALSE----------------------------------------------------- library("rstan") library("loo") set.seed(4711) ## ----llfun_logistic----------------------------------------------------------- # we'll add an argument log to toggle whether this is a log-likelihood or # likelihood function. this will be useful later in the vignette. llfun_logistic <- function(data_i, draws, log = TRUE) { x_i <- as.matrix(data_i[, which(grepl(colnames(data_i), pattern = "X")), drop=FALSE]) logit_pred <- draws %*% t(x_i) dbinom(x = data_i$y, size = 1, prob = 1/(1 + exp(-logit_pred)), log = log) } ## ---- eval=FALSE-------------------------------------------------------------- # # Prepare data # url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" # wells <- read.table(url) # wells$dist100 <- with(wells, dist / 100) # X <- model.matrix(~ dist100 + arsenic, wells) # standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # # # Compile # stan_mod <- stan_model("logistic.stan") # # # Fit model # fit_1 <- sampling(stan_mod, data = standata, seed = 4711) # print(fit_1, pars = "beta") ## ---- eval=FALSE-------------------------------------------------------------- # # used for draws argument to loo_i # parameter_draws_1 <- extract(fit_1)$beta # # # used for data argument to loo_i # stan_df_1 <- as.data.frame(standata) # # # compute relative efficiency (this is slow and optional but is recommended to allow # # for adjusting PSIS effective sample size based on MCMC effective sample size) # r_eff <- relative_eff(llfun_logistic, # log = FALSE, # relative_eff wants likelihood not log-likelihood values # chain_id = rep(1:4, each = 1000), # data = stan_df_1, # draws = parameter_draws_1, # cores = 2) # # loo_i(i = 1, llfun_logistic, r_eff = r_eff, data = stan_df_1, draws = parameter_draws_1) ## ---- eval=FALSE-------------------------------------------------------------- # set.seed(4711) # loo_ss_1 <- # loo_subsample( # llfun_logistic, # observations = 100, # take a subsample of size 100 # cores = 2, # # these next objects were computed above # r_eff = r_eff, # draws = parameter_draws_1, # data = stan_df_1 # ) # print(loo_ss_1) ## ---- eval=FALSE-------------------------------------------------------------- # set.seed(4711) # loo_ss_1b <- # update( # loo_ss_1, # observations = 200, # subsample 200 instead of 100 # r_eff = r_eff, # draws = parameter_draws_1, # data = stan_df_1 # ) # print(loo_ss_1b) ## ---- eval=FALSE-------------------------------------------------------------- # set.seed(4711) # loo_ss_1c <- # loo_subsample( # x = llfun_logistic, # r_eff = r_eff, # draws = parameter_draws_1, # data = stan_df_1, # observations = 100, # estimator = "hh_pps", # use Hansen-Hurwitz # loo_approximation = "lpd", # use lpd instead of plpd # loo_approximation_draws = 100, # cores = 2 # ) # print(loo_ss_1c) ## ---- eval=FALSE-------------------------------------------------------------- # fit_laplace <- optimizing(stan_mod, data = standata, draws = 2000, # importance_resampling = TRUE) # parameter_draws_laplace <- fit_laplace$theta_tilde # draws from approximate posterior # log_p <- fit_laplace$log_p # log density of the posterior # log_g <- fit_laplace$log_g # log density of the approximation ## ---- eval=FALSE-------------------------------------------------------------- # set.seed(4711) # loo_ap_1 <- # loo_approximate_posterior( # x = llfun_logistic, # draws = parameter_draws_laplace, # data = stan_df_1, # log_p = log_p, # log_g = log_g, # cores = 2 # ) # print(loo_ap_1) ## ---- eval=FALSE-------------------------------------------------------------- # set.seed(4711) # loo_ap_ss_1 <- # loo_subsample( # x = llfun_logistic, # draws = parameter_draws_laplace, # data = stan_df_1, # log_p = log_p, # log_g = log_g, # observations = 100, # cores = 2 # ) # print(loo_ap_ss_1) ## ---- eval=FALSE-------------------------------------------------------------- # standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) # fit_2 <- sampling(stan_mod, data = standata) # parameter_draws_2 <- extract(fit_2)$beta # stan_df_2 <- as.data.frame(standata) # # # recompute subsampling loo for first model for demonstration purposes # # # compute relative efficiency (this is slow and optional but is recommended to allow # # for adjusting PSIS effective sample size based on MCMC effective sample size) # r_eff_1 <- relative_eff( # llfun_logistic, # log = FALSE, # relative_eff wants likelihood not log-likelihood values # chain_id = rep(1:4, each = 1000), # data = stan_df_1, # draws = parameter_draws_1, # cores = 2 # ) # # set.seed(4711) # loo_ss_1 <- loo_subsample( # x = llfun_logistic, # r_eff = r_eff_1, # draws = parameter_draws_1, # data = stan_df_1, # observations = 200, # cores = 2 # ) # # # compute subsampling loo for a second model (with log-arsenic) # # r_eff_2 <- relative_eff( # llfun_logistic, # log = FALSE, # relative_eff wants likelihood not log-likelihood values # chain_id = rep(1:4, each = 1000), # data = stan_df_2, # draws = parameter_draws_2, # cores = 2 # ) # loo_ss_2 <- loo_subsample( # x = llfun_logistic, # r_eff = r_eff_2, # draws = parameter_draws_2, # data = stan_df_2, # observations = 200, # cores = 2 # ) # # print(loo_ss_2) ## ---- eval=FALSE-------------------------------------------------------------- # # Compare # comp <- loo_compare(loo_ss_1, loo_ss_2) # print(comp) ## ---- eval=FALSE-------------------------------------------------------------- # loo_ss_2 <- # loo_subsample( # x = llfun_logistic, # r_eff = r_eff_2, # draws = parameter_draws_2, # data = stan_df_2, # observations = loo_ss_1, # cores = 2 # ) ## ---- eval=FALSE-------------------------------------------------------------- # idx <- obs_idx(loo_ss_1) # loo_ss_2 <- loo_subsample( # x = llfun_logistic, # r_eff = r_eff_2, # draws = parameter_draws_2, # data = stan_df_2, # observations = idx, # cores = 2 # ) ## ---- eval=FALSE-------------------------------------------------------------- # comp <- loo_compare(loo_ss_1, loo_ss_2) # print(comp) ## ---- eval=FALSE-------------------------------------------------------------- # # use loo() instead of loo_subsample() to compute full PSIS-LOO for model 2 # loo_full_2 <- loo( # x = llfun_logistic, # r_eff = r_eff_2, # draws = parameter_draws_2, # data = stan_df_2, # cores = 2 # ) # loo_compare(loo_ss_1, loo_full_2) loo/inst/doc/loo2-mixis.R0000644000176200001440000001034514411465131014715 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ---- warnings=FALSE, message=FALSE------------------------------------------- library("rstan") library("loo") library("matrixStats") options(mc.cores = parallel::detectCores(), parallel=FALSE) set.seed(24877) ## ----stancode_horseshoe------------------------------------------------------- stancode_horseshoe <- " data { int n; int p; int y[n]; matrix [n,p] X; real scale_global; int mixis; } transformed data { real nu_global=1; // degrees of freedom for the half-t priors for tau real nu_local=1; // degrees of freedom for the half-t priors for lambdas // (nu_local = 1 corresponds to the horseshoe) real slab_scale=2;// for the regularized horseshoe real slab_df=100; // for the regularized horseshoe } parameters { vector[p] z; // for non-centered parameterization real tau; // global shrinkage parameter vector [p] lambda; // local shrinkage parameter real caux; } transformed parameters { vector[p] beta; { vector[p] lambda_tilde; // 'truncated' local shrinkage parameter real c = slab_scale * sqrt(caux); // slab scale lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda))); beta = z .* lambda_tilde*tau; } } model { vector[n] means=X*beta; vector[n] log_lik; target += std_normal_lpdf(z); target += student_t_lpdf(lambda | nu_local, 0, 1); target += student_t_lpdf(tau | nu_global, 0, scale_global); target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df); for (index in 1:n) { log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]); } target += sum(log_lik); if (mixis) { target += log_sum_exp(-log_lik); } } generated quantities { vector[n] means=X*beta; vector[n] log_lik; for (index in 1:n) { log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]); } } " ## ---- results='hide', warning=FALSE, message=FALSE, error=FALSE--------------- data(voice) y <- voice$y X <- voice[2:length(voice)] n <- dim(X)[1] p <- dim(X)[2] p0 <- 10 scale_global <- 2*p0/(p-p0)/sqrt(n-1) standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) ## ---- results='hide', warning=FALSE------------------------------------------- chains <- 4 n_iter <- 2000 warm_iter <- 1000 stanmodel <- stan_model(model_code = stancode_horseshoe) fit_post <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0) loo_post <-loo(fit_post) ## ----------------------------------------------------------------------------- print(loo_post) ## ---- results='hide', warnings=FALSE------------------------------------------ standata$mixis <- 1 fit_mix <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0, pars = "log_lik") log_lik_mix <- extract(fit_mix)$log_lik ## ----------------------------------------------------------------------------- l_common_mix <- rowLogSumExps(-log_lik_mix) log_weights <- -log_lik_mix - l_common_mix elpd_mixis <- logSumExp(-l_common_mix) - rowLogSumExps(t(log_weights)) ## ----------------------------------------------------------------------------- data(voice_loo) elpd_loo <- voice_loo$elpd_loo ## ----------------------------------------------------------------------------- elpd_psis <- loo_post$pointwise[,1] print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2))) print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2))) ## ----------------------------------------------------------------------------- elpd_psis <- loo_post$pointwise[,1] print(paste("ELPD (PSIS)=",round(sum(elpd_psis),2))) print(paste("ELPD (MixIS)=",round(sum(elpd_mixis),2))) print(paste("ELPD (brute force)=",round(sum(elpd_loo),2))) loo/inst/doc/loo2-lfo.Rmd0000644000176200001440000006042614407123455014677 0ustar liggesusers--- title: "Approximate leave-future-out cross-validation for Bayesian time series models" author: "Paul Bürkner, Jonah Gabry, Aki Vehtari" date: "`r Sys.Date()`" output: html_vignette: toc: yes encoding: "UTF-8" params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r more-knitr-ops, include=FALSE} knitr::opts_chunk$set( cache = TRUE, message = FALSE, warning = FALSE ) ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` ## Introduction One of the most common goals of a time series analysis is to use the observed series to inform predictions for future observations. We will refer to this task of predicting a sequence of $M$ future observations as $M$-step-ahead prediction ($M$-SAP). Fortunately, once we have fit a model and can sample from the posterior predictive distribution, it is straightforward to generate predictions as far into the future as we want. It is also straightforward to evaluate the $M$-SAP performance of a time series model by comparing the predictions to the observed sequence of $M$ future data points once they become available. Unfortunately, we are often in the position of having to use a model to inform decisions _before_ we can collect the future observations required for assessing the predictive performance. If we have many competing models we may also need to first decide which of the models (or which combination of the models) we should rely on for predictions. In these situations the best we can do is to use methods for approximating the expected predictive performance of our models using only the observations of the time series we already have. If there were no time dependence in the data or if the focus is to assess the non-time-dependent part of the model, we could use methods like leave-one-out cross-validation (LOO-CV). For a data set with $N$ observations, we refit the model $N$ times, each time leaving out one of the $N$ observations and assessing how well the model predicts the left-out observation. LOO-CV is very expensive computationally in most realistic settings, but the Pareto smoothed importance sampling (PSIS, Vehtari et al, 2017, 2019) algorithm provided by the *loo* package allows for approximating exact LOO-CV with PSIS-LOO-CV. PSIS-LOO-CV requires only a single fit of the full model and comes with diagnostics for assessing the validity of the approximation. With a time series we can do something similar to LOO-CV but, except in a few cases, it does not make sense to leave out observations one at a time because then we are allowing information from the future to influence predictions of the past (i.e., times $t + 1, t+2, \ldots$ should not be used to predict for time $t$). To apply the idea of cross-validation to the $M$-SAP case, instead of leave-*one*-out cross-validation we need some form of leave-*future*-out cross-validation (LFO-CV). As we will demonstrate in this case study, LFO-CV does not refer to one particular prediction task but rather to various possible cross-validation approaches that all involve some form of prediction for new time series data. Like exact LOO-CV, exact LFO-CV requires refitting the model many times to different subsets of the data, which is computationally very costly for most nontrivial examples, in particular for Bayesian analyses where refitting the model means estimating a new posterior distribution rather than a point estimate. Although PSIS-LOO-CV provides an efficient approximation to exact LOO-CV, until now there has not been an analogous approximation to exact LFO-CV that drastically reduces the computational burden while also providing informative diagnostics about the quality of the approximation. In this case study we present PSIS-LFO-CV, an algorithm that typically only requires refitting the time-series model a small number times and will make LFO-CV tractable for many more realistic applications than previously possible. More details can be found in our paper about approximate LFO-CV (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1902.06281). ## $M$-step-ahead predictions Assume we have a time series of observations $y = (y_1, y_2, \ldots, y_N)$ and let $L$ be the _minimum_ number of observations from the series that we will require before making predictions for future data. Depending on the application and how informative the data is, it may not be possible to make reasonable predictions for $y_{i+1}$ based on $(y_1, \dots, y_{i})$ until $i$ is large enough so that we can learn enough about the time series to predict future observations. Setting $L=10$, for example, means that we will only assess predictive performance starting with observation $y_{11}$, so that we always have at least 10 previous observations to condition on. In order to assess $M$-SAP performance we would like to compute the predictive densities $$ p(y_{i+1:M} \,|\, y_{1:i}) = p(y_{i+1}, \ldots, y_{i + M} \,|\, y_{1},...,y_{i}) $$ for each $i \in \{L, \ldots, N - M\}$. The quantities $p(y_{i+1:M} \,|\, y_{1:i})$ can be computed with the help of the posterior distribution $p(\theta \,|\, y_{1:i})$ of the parameters $\theta$ conditional on only the first $i$ observations of the time-series: $$ p(y_{i+1:M} \,| \, y_{1:i}) = \int p(y_{i+1:M} \,| \, y_{1:i}, \theta) \, p(\theta\,|\,y_{1:i}) \,d\theta. $$ Having obtained $S$ draws $(\theta_{1:i}^{(1)}, \ldots, \theta_{1:i}^{(S)})$ from the posterior distribution $p(\theta\,|\,y_{1:i})$, we can estimate $p(y_{i+1:M} | y_{1:i})$ as $$ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{1}{S}\sum_{s=1}^S p(y_{i+1:M} \,|\, y_{1:i}, \theta_{1:i}^{(s)}). $$ ## Approximate $M$-SAP using importance-sampling {#approximate_MSAP} Unfortunately, the math above makes use of the posterior distributions from many different fits of the model to different subsets of the data. That is, to obtain the predictive density $p(y_{i+1:M} \,|\, y_{1:i})$ requires fitting a model to only the first $i$ data points, and we will need to do this for every value of $i$ under consideration (all $i \in \{L, \ldots, N - M\}$). To reduce the number of models that need to be fit for the purpose of obtaining each of the densities $p(y_{i+1:M} \,|\, y_{1:i})$, we propose the following algorithm. First, we refit the model using the first $L$ observations of the time series and then perform a single exact $M$-step-ahead prediction step for $p(y_{L+1:M} \,|\, y_{1:L})$. Recall that $L$ is the minimum number of observations we have deemed acceptable for making predictions (setting $L=0$ means the first data point will be predicted only based on the prior). We define $i^\star = L$ as the current point of refit. Next, starting with $i = i^\star + 1$, we approximate each $p(y_{i+1:M} \,|\, y_{1:i})$ via $$ p(y_{i+1:M} \,|\, y_{1:i}) \approx \frac{ \sum_{s=1}^S w_i^{(s)}\, p(y_{i+1:M} \,|\, y_{1:i}, \theta^{(s)})} { \sum_{s=1}^S w_i^{(s)}}, $$ where $\theta^{(s)} = \theta^{(s)}_{1:i^\star}$ are draws from the posterior distribution based on the first $i^\star$ observations and $w_i^{(s)}$ are the PSIS weights obtained in two steps. First, we compute the raw importance ratios $$ r_i^{(s)} = \frac{f_{1:i}(\theta^{(s)})}{f_{1:i^\star}(\theta^{(s)})} \propto \prod_{j \in (i^\star + 1):i} p(y_j \,|\, y_{1:(j-1)}, \theta^{(s)}), $$ and then stabilize them using PSIS. The function $f_{1:i}$ denotes the posterior distribution based on the first $i$ observations, that is, $f_{1:i} = p(\theta \,|\, y_{1:i})$, with $f_{1:i^\star}$ defined analogously. The index set $(i^\star + 1):i$ indicates all observations which are part of the data for the model $f_{1:i}$ whose predictive performance we are trying to approximate but not for the actually fitted model $f_{1:i^\star}$. The proportional statement arises from the fact that we ignore the normalizing constants $p(y_{1:i})$ and $p(y_{1:i^\star})$ of the compared posteriors, which leads to a self-normalized variant of PSIS (see Vehtari et al, 2017). Continuing with the next observation, we gradually increase $i$ by $1$ (we move forward in time) and repeat the process. At some observation $i$, the variability of the importance ratios $r_i^{(s)}$ will become too large and importance sampling will fail. We will refer to this particular value of $i$ as $i^\star_1$. To identify the value of $i^\star_1$, we check for which value of $i$ does the estimated shape parameter $k$ of the generalized Pareto distribution first cross a certain threshold $\tau$ (Vehtari et al, 2019). Only then do we refit the model using the observations up to $i^\star_1$ and restart the process from there by setting $\theta^{(s)} = \theta^{(s)}_{1:i^\star_1}$ and $i^\star = i^\star_1$ until the next refit. In some cases we may only need to refit once and in other cases we will find a value $i^\star_2$ that requires a second refitting, maybe an $i^\star_3$ that requires a third refitting, and so on. We refit as many times as is required (only when $k > \tau$) until we arrive at observation $i = N - M$. For LOO, we recommend to use a threshold of $\tau = 0.7$ (Vehtari et al, 2017, 2019) and it turns out this is a reasonable threshold for LFO as well (Bürkner et al. 2020). ## Autoregressive models Autoregressive (AR) models are some of the most commonly used time-series models. An AR(p) model ---an autoregressive model of order $p$--- can be defined as $$ y_i = \eta_i + \sum_{k = 1}^p \varphi_k y_{i - k} + \varepsilon_i, $$ where $\eta_i$ is the linear predictor for the $i$th observation, $\phi_k$ are the autoregressive parameters and $\varepsilon_i$ are pairwise independent errors, which are usually assumed to be normally distributed with equal variance $\sigma^2$. The model implies a recursive formula that allows for computing the right-hand side of the above equation for observation $i$ based on the values of the equations for previous observations. ## Case Study: Annual measurements of the level of Lake Huron To illustrate the application of PSIS-LFO-CV for estimating expected $M$-SAP performance, we will fit a model for 98 annual measurements of the water level (in feet) of [Lake Huron](https://en.wikipedia.org/wiki/Lake_Huron) from the years 1875--1972. This data set is found in the **datasets** R package, which is installed automatically with **R**. In addition to the **loo** package, for this analysis we will use the **brms** interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting. ```{r pkgs, cache=FALSE} library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) CHAINS <- 4 SEED <- 5838296 set.seed(SEED) ``` Before fitting a model, we will first put the data into a data frame and then look at the time series. ```{r hurondata} N <- length(LakeHuron) df <- data.frame( y = as.numeric(LakeHuron), year = as.numeric(time(LakeHuron)), time = 1:N ) ggplot(df, aes(x = year, y = y)) + geom_point(size = 1) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)" ) ``` The above plot shows rather strong autocorrelation of the time-series as well as some trend towards lower levels for later points in time. We can specify an AR(4) model for these data using the **brms** package as follows: ```{r fit, results = "hide"} fit <- brm( y ~ ar(time, p = 4), data = df, prior = prior(normal(0, 0.5), class = "ar"), control = list(adapt_delta = 0.99), seed = SEED, chains = CHAINS ) ``` The model implied predictions along with the observed values can be plotted, which reveals a rather good fit to the data. ```{r plotpreds, cache = FALSE} preds <- posterior_predict(fit) preds <- cbind( Estimate = colMeans(preds), Q5 = apply(preds, 2, quantile, probs = 0.05), Q95 = apply(preds, 2, quantile, probs = 0.95) ) ggplot(cbind(df, preds), aes(x = year, y = Estimate)) + geom_smooth(aes(ymin = Q5, ymax = Q95), stat = "identity", size = 0.5) + geom_point(aes(y = y)) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)", subtitle = "Mean (blue) and 90% predictive intervals (gray) vs. observed data (black)" ) ``` To allow for reasonable predictions of future values, we will require at least $L = 20$ historical observations (20 years) to make predictions. ```{r setL} L <- 20 ``` We first perform approximate leave-one-out cross-validation (LOO-CV) for the purpose of later comparison with exact and approximate LFO-CV for the 1-SAP case. ```{r loo1sap, cache = FALSE} loo_cv <- loo(log_lik(fit)[, (L + 1):N]) print(loo_cv) ``` ## 1-step-ahead predictions leaving out all future values The most basic version of $M$-SAP is 1-SAP, in which we predict only one step ahead. In this case, $y_{i+1:M}$ simplifies to $y_{i}$ and the LFO-CV algorithm becomes considerably simpler than for larger values of $M$. ### Exact 1-step-ahead predictions Before we compute approximate LFO-CV using PSIS we will first compute exact LFO-CV for the 1-SAP case so we can use it as a benchmark later. The initial step for the exact computation is to calculate the log-predictive densities by refitting the model many times: ```{r exact_loglik, results="hide"} loglik_exact <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_i <- update(fit, newdata = df_past, recompile = FALSE) loglik_exact[, i + 1] <- log_lik(fit_i, newdata = df_oos, oos = oos)[, oos] } ``` Then we compute the exact expected log predictive density (ELPD): ```{r helpers} # some helper functions we'll use throughout # more stable than log(sum(exp(x))) log_sum_exp <- function(x) { max_x <- max(x) max_x + log(sum(exp(x - max_x))) } # more stable than log(mean(exp(x))) log_mean_exp <- function(x) { log_sum_exp(x) - log(length(x)) } # compute log of raw importance ratios # sums over observations *not* over posterior samples sum_log_ratios <- function(loglik, ids = NULL) { if (!is.null(ids)) loglik <- loglik[, ids, drop = FALSE] rowSums(loglik) } # for printing comparisons later rbind_print <- function(...) { round(rbind(...), digits = 2) } ``` ```{r exact1sap, cache = FALSE} exact_elpds_1sap <- apply(loglik_exact, 2, log_mean_exp) exact_elpd_1sap <- c(ELPD = sum(exact_elpds_1sap[-(1:L)])) rbind_print( "LOO" = loo_cv$estimates["elpd_loo", "Estimate"], "LFO" = exact_elpd_1sap ) ``` We see that the ELPD from LFO-CV for 1-step-ahead predictions is lower than the ELPD estimate from LOO-CV, which should be expected since LOO-CV is making use of more of the time series. That is, since the LFO-CV approach only uses observations from before the left-out data point but LOO-CV uses _all_ data points other than the left-out observation, we should expect to see the larger ELPD from LOO-CV. ### Approximate 1-step-ahead predictions We compute approximate 1-SAP with refit at observations where the Pareto $k$ estimate exceeds the threshold of $0.7$. ```{r setkthresh} k_thres <- 0.7 ``` The code becomes a little bit more involved as compared to the exact LFO-CV. Note that we can compute exact 1-SAP at the refitting points, which comes with no additional computational costs since we had to refit the model anyway. ```{r refit_loglik, results="hide"} approx_elpds_1sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- L + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[L + 1] <- log_mean_exp(loglik[, oos]) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[i + 1] <- log_mean_exp(loglik[, oos]) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] approx_elpds_1sap[i + 1] <- log_sum_exp(lw + loglik[, oos]) } } ``` We see that the final Pareto-$k$-estimates are mostly well below the threshold and that we only needed to refit the model a few times: ```{r plot_ks} plot_ks <- function(ks, ids, thres = 0.6) { dat_ks <- data.frame(ks = ks, ids = ids) ggplot(dat_ks, aes(x = ids, y = ks)) + geom_point(aes(color = ks > thres), shape = 3, show.legend = FALSE) + geom_hline(yintercept = thres, linetype = 2, color = "red2") + scale_color_manual(values = c("cornflowerblue", "darkblue")) + labs(x = "Data point", y = "Pareto k") + ylim(-0.5, 1.5) } ``` ```{r refitsummary1sap, cache=FALSE} cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - 1)) ``` The approximate 1-SAP ELPD is remarkably similar to the exact 1-SAP ELPD computed above, which indicates our algorithm to compute approximate 1-SAP worked well for the present data and model. ```{r lfosummary1sap, cache = FALSE} approx_elpd_1sap <- sum(approx_elpds_1sap, na.rm = TRUE) rbind_print( "approx LFO" = approx_elpd_1sap, "exact LFO" = exact_elpd_1sap ) ``` Plotting exact against approximate predictions, we see that no approximation value deviates far from its exact counterpart, providing further evidence for the good quality of our approximation. ```{r plot1sap, cache = FALSE} dat_elpd <- data.frame( approx_elpd = approx_elpds_1sap, exact_elpd = exact_elpds_1sap ) ggplot(dat_elpd, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ``` We can also look at the maximum difference and average difference between the approximate and exact ELPD calculations, which also indicate a ver close approximation: ```{r diffs1sap, cache=FALSE} max_diff <- with(dat_elpd, max(abs(approx_elpd - exact_elpd), na.rm = TRUE)) mean_diff <- with(dat_elpd, mean(abs(approx_elpd - exact_elpd), na.rm = TRUE)) rbind_print( "Max diff" = round(max_diff, 2), "Mean diff" = round(mean_diff, 3) ) ``` ## $M$-step-ahead predictions leaving out all future values To illustrate the application of $M$-SAP for $M > 1$, we next compute exact and approximate LFO-CV for the 4-SAP case. ### Exact $M$-step-ahead predictions The necessary steps are the same as for 1-SAP with the exception that the log-density values of interest are now the sums of the log predictive densities of four consecutive observations. Further, the stability of the PSIS approximation actually stays the same for all $M$ as it only depends on the number of observations we leave out, not on the number of observations we predict. ```{r exact_loglikm, results="hide"} M <- 4 loglikm <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm[, i + 1] <- rowSums(loglik[, oos]) } ``` ```{r exact4sap, cache = FALSE} exact_elpds_4sap <- apply(loglikm, 2, log_mean_exp) (exact_elpd_4sap <- c(ELPD = sum(exact_elpds_4sap, na.rm = TRUE))) ``` ### Approximate $M$-step-ahead predictions Computing the approximate PSIS-LFO-CV for the 4-SAP case is a little bit more involved than the approximate version for the 1-SAP case, although the underlying principles remain the same. ```{r refit_loglikm, results="hide"} approx_elpds_4sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- (L + 1):(L + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_1sap[L + 1] <- log_mean_exp(loglikm) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_mean_exp(loglikm) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_sum_exp(lw + loglikm) } } ``` Again, we see that the final Pareto-$k$-estimates are mostly well below the threshold and that we only needed to refit the model a few times: ```{r refitsummary4sap, cache = FALSE} cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - M)) ``` The approximate ELPD computed for the 4-SAP case is not as close to its exact counterpart as in the 1-SAP case. In general, the larger $M$, the larger the variation of the approximate ELPD around the exact ELPD. It turns out that the ELPD estimates of AR-models with $M>1$ show particular variation due to their predictions' dependency on other predicted values. In Bürkner et al. (2020) we provide further explanation and simulations for these cases. ```{r lfosummary4sap, cache = FALSE} approx_elpd_4sap <- sum(approx_elpds_4sap, na.rm = TRUE) rbind_print( "Approx LFO" = approx_elpd_4sap, "Exact LFO" = exact_elpd_4sap ) ``` Plotting exact against approximate pointwise predictions confirms that, for a few specific data points, the approximate predictions underestimate the exact predictions. ```{r plot4sap, cache = FALSE} dat_elpd_4sap <- data.frame( approx_elpd = approx_elpds_4sap, exact_elpd = exact_elpds_4sap ) ggplot(dat_elpd_4sap, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ``` ## Conclusion In this case study we have shown how to do carry out exact and approximate leave-future-out cross-validation for $M$-step-ahead prediction tasks. For the data and model used in our example, the PSIS-LFO-CV algorithm provides reasonably stable and accurate results despite not requiring us to refit the model nearly as many times. For more details on approximate LFO-CV, we refer to Bürkner et al. (2020).
## References Bürkner P. C., Gabry J., & Vehtari A. (2020). Approximate leave-future-out cross-validation for time series models. *Journal of Statistical Computation and Simulation*, 90(14):2499-2523. \doi:/10.1080/00949655.2020.1783262. [Online](https://www.tandfonline.com/doi/full/10.1080/00949655.2020.1783262). [arXiv preprint](https://arxiv.org/abs/1902.06281). Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646).
## Appendix ### Appendix: Session information ```{r sessioninfo} sessionInfo() ``` ### Appendix: Licenses * Code © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under BSD-3). * Text © 2018, Paul Bürkner, Jonah Gabry, Aki Vehtari (licensed under CC-BY-NC 4.0). loo/inst/doc/loo2-with-rstan.Rmd0000644000176200001440000002030714407123455016211 0ustar liggesusers--- title: "Writing Stan programs for use with the loo package" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to write a Stan program that computes and stores the pointwise log-likelihood required for using the __loo__ package. The other vignettes included with the package demonstrate additional functionality. Some sections from this vignette are excerpted from our papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.02646). which provide important background for understanding the methods implemented in the package. # Example: Well water in Bangladesh This example comes from a survey of residents from a small area in Bangladesh that was affected by arsenic in drinking water. Respondents with elevated arsenic levels in their wells were asked if they were interested in getting water from a neighbor's well, and a series of logistic regressions were fit to predict this binary response given various information about the households (Gelman and Hill, 2007). Here we fit a model for the well-switching response given two predictors: the arsenic level of the water in the resident's home, and the distance of the house from the nearest safe well. The sample size in this example is $N=3020$, which is not huge but is large enough that it is important to have a computational method for LOO that is fast for each data point. On the plus side, with such a large dataset, the influence of any given observation is small, and so the computations should be stable. ## Coding the Stan model Here is the Stan code for fitting the logistic regression model, which we save in a file called `logistic.stan`: ``` data { int N; // number of data points int P; // number of predictors (including intercept) matrix[N,P] X; // predictors (including 1s for intercept) int y[N]; // binary outcome } parameters { vector[P] beta; } model { beta ~ normal(0, 1); y ~ bernoulli_logit(X * beta); } generated quantities { vector[N] log_lik; for (n in 1:N) { log_lik[n] = bernoulli_logit_lpmf(y[n] | X[n] * beta); } } ``` We have defined the log likelihood as a vector named `log_lik` in the generated quantities block so that the individual terms will be saved by Stan. After running Stan, `log_lik` can be extracted (using the `extract_log_lik` function provided in the **loo** package) as an $S \times N$ matrix, where $S$ is the number of simulations (posterior draws) and $N$ is the number of data points. ## Fitting the model with RStan Next we fit the model in Stan using the **rstan** package: ```{r, eval=FALSE} library("rstan") # Prepare data url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" wells <- read.table(url) wells$dist100 <- with(wells, dist / 100) X <- model.matrix(~ dist100 + arsenic, wells) standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # Fit model fit_1 <- stan("logistic.stan", data = standata) print(fit_1, pars = "beta") ``` ``` mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat beta[1] 0.00 0 0.08 -0.16 -0.05 0.00 0.05 0.15 1964 1 beta[2] -0.89 0 0.10 -1.09 -0.96 -0.89 -0.82 -0.68 2048 1 beta[3] 0.46 0 0.04 0.38 0.43 0.46 0.49 0.54 2198 1 ``` ## Computing approximate leave-one-out cross-validation using PSIS-LOO We can then use the **loo** package to compute the efficient PSIS-LOO approximation to exact LOO-CV: ```{r, eval=FALSE} library("loo") # Extract pointwise log-likelihood # using merge_chains=FALSE returns an array, which is easier to # use with relative_eff() log_lik_1 <- extract_log_lik(fit_1, merge_chains = FALSE) # as of loo v2.0.0 we can optionally provide relative effective sample sizes # when calling loo, which allows for better estimates of the PSIS effective # sample sizes and Monte Carlo error r_eff <- relative_eff(exp(log_lik_1), cores = 2) # preferably use more than 2 cores (as many cores as possible) # will use value of 'mc.cores' option if cores is not specified loo_1 <- loo(log_lik_1, r_eff = r_eff, cores = 2) print(loo_1) ``` ``` Computed from 4000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1968.5 15.6 p_loo 3.2 0.1 looic 3937.0 31.2 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` The printed output from the `loo` function shows the estimates $\widehat{\mbox{elpd}}_{\rm loo}$ (expected log predictive density), $\widehat{p}_{\rm loo}$ (effective number of parameters), and ${\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}$ (the LOO information criterion). The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the $k$ parameter is explained in `help('pareto-k-diagnostic')` and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case the message tells us that all of the estimates for $k$ are fine. ## Comparing models To compare this model to an alternative model for the same data we can use the `loo_compare` function in the **loo** package. First we'll fit a second model to the well-switching data, using `log(arsenic)` instead of `arsenic` as a predictor: ```{r, eval=FALSE} standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) fit_2 <- stan(fit = fit_1, data = standata) log_lik_2 <- extract_log_lik(fit_2, merge_chains = FALSE) r_eff_2 <- relative_eff(exp(log_lik_2)) loo_2 <- loo(log_lik_2, r_eff = r_eff_2, cores = 2) print(loo_2) ``` ``` Computed from 4000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1952.3 16.2 p_loo 3.1 0.1 looic 3904.6 32.4 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` We can now compare the models on LOO using the `loo_compare` function: ```{r, eval=FALSE} # Compare comp <- loo_compare(loo_1, loo_2) ``` This new object, `comp`, contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error: ```{r, eval=FALSE} print(comp) # can set simplify=FALSE for more detailed print output ``` ``` elpd_diff se_diff model2 0.0 0.0 model1 -16.3 4.4 ``` The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in `elpd` and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (`model2`). # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2017). _The Stan C++ Library, Version 2.17.0._ https://mc-stan.org/ Stan Development Team (2018) _RStan: the R interface to Stan, Version 2.17.3._ https://mc-stan.org/ Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/inst/doc/loo2-mixis.html0000644000176200001440000010175514411465131015466 0ustar liggesusers Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models

Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models

Luca Silva and Giacomo Zanella

2023-03-30

Introduction

This vignette shows how to perform Bayesian leave-one-out cross-validation (LOO-CV) using the mixture estimators proposed in the paper Silva and Zanella (2022). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-\(k\) diagnostics and potential instability of PSIS-LOO estimators.

For this illustration we consider a high-dimensional Bayesian Logistic regression model applied to the Voice dataset.

Setup: load packages and set seed

library("rstan")
library("loo")
library("matrixStats")
options(mc.cores = parallel::detectCores(), parallel=FALSE)
set.seed(24877)

Model

This is the Stan code for a logistic regression model with regularized horseshoe prior. The code includes an if statement to include a code line needed later for the MixIS approach.

stancode_horseshoe <- "
data {
  int <lower=0> n;                
  int <lower=0> p;                
  int <lower=0, upper=1> y[n];    
  matrix [n,p] X;                
  real <lower=0> scale_global;
  int <lower=0,upper=1> mixis;                
}
transformed data {
  real<lower=1> nu_global=1; // degrees of freedom for the half-t priors for tau
  real<lower=1> nu_local=1;  // degrees of freedom for the half-t priors for lambdas
                             // (nu_local = 1 corresponds to the horseshoe)
  real<lower=0> slab_scale=2;// for the regularized horseshoe
  real<lower=0> slab_df=100; // for the regularized horseshoe
}
parameters {
  vector[p] z;                // for non-centered parameterization
  real <lower=0> tau;         // global shrinkage parameter
  vector <lower=0>[p] lambda; // local shrinkage parameter
  real<lower=0> caux;
}
transformed parameters {
  vector[p] beta;
  { 
    vector[p] lambda_tilde;   // 'truncated' local shrinkage parameter
    real c = slab_scale * sqrt(caux); // slab scale
    lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda)));
    beta = z .* lambda_tilde*tau;
  }
}
model {
  vector[n] means=X*beta;
  vector[n] log_lik;
  target += std_normal_lpdf(z);
  target += student_t_lpdf(lambda | nu_local, 0, 1);
  target += student_t_lpdf(tau | nu_global, 0, scale_global);
  target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df);
  for (index in 1:n) {
    log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]);
  }
  target += sum(log_lik);
  if (mixis) {
    target += log_sum_exp(-log_lik);
  }
}
generated quantities {
  vector[n] means=X*beta;
  vector[n] log_lik;
  for (index in 1:n) {
    log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]);
  }
}
"

Dataset

The LSVT Voice Rehabilitation Data Set (see link for details) has \(p=312\) covariates and \(n=126\) observations with binary response. We construct data list for Stan.

data(voice)
y <- voice$y
X <- voice[2:length(voice)]
n <- dim(X)[1]
p <- dim(X)[2]
p0 <- 10
scale_global <- 2*p0/(p-p0)/sqrt(n-1)
standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0)

Note that in our prior specification we divide the prior variance by the number of covariates \(p\). This is often done in high-dimensional contexts to have a prior variance for the linear predictors \(X\beta\) that remains bounded as \(p\) increases.

PSIS estimators and Pareto-\(k\) diagnostics

LOO-CV computations are challenging in this context due to high-dimensionality of the parameter space. To show that, we compute PSIS-LOO estimators, which require sampling from the posterior distribution, and inspect the associated Pareto-\(k\) diagnostics.

chains <- 4
n_iter <- 2000
warm_iter <- 1000
stanmodel <- stan_model(model_code = stancode_horseshoe)
fit_post <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0)
loo_post <-loo(fit_post)
print(loo_post)

Computed from 4000 by 126 log-likelihood matrix

         Estimate   SE
elpd_loo    -42.5  7.1
p_loo        23.8  5.2
looic        85.0 14.2
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     29    23.0%   386       
 (0.5, 0.7]   (ok)       68    54.0%   82        
   (0.7, 1]   (bad)      23    18.3%   13        
   (1, Inf)   (very bad)  6     4.8%   4         
See help('pareto-k-diagnostic') for details.

As we can see the diagnostics signal either “bad” or “very bad” Pareto-\(k\) values for roughly \(15-30\%\) of the observations which is a significant portion of the dataset.

Mixture estimators

We now compute the mixture estimators proposed in Silva and Zanella (2022). These require to sample from the following mixture of leave-one-out posteriors \[\begin{equation} q_{mix}(\theta) = \frac{\sum_{i=1}^n p(y_{-i}|\theta)p(\theta)}{\sum_{i=1}^np(y_{-i})}\propto p(\theta|y)\cdot \left(\sum_{i=1}^np(y_i|\theta)^{-1}\right). \end{equation}\] The code to generate a Stan model for the above mixture distribution is the same to the one for the posterior, just enabling one line of code with a LogSumExp contribution to account for the last term in the equation above.

  if (mixis) {
    target += log_sum_exp(-log_lik);
  }

We sample from the mixture and collect the log-likelihoods term.

standata$mixis <- 1
fit_mix <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0, pars = "log_lik")
log_lik_mix <- extract(fit_mix)$log_lik

We now compute the mixture estimators, following the numerically stable implementation in Appendix A.2 of Silva and Zanella (2022). The code below makes use of the package “matrixStats”.

l_common_mix <- rowLogSumExps(-log_lik_mix)
log_weights <- -log_lik_mix - l_common_mix
elpd_mixis <- logSumExp(-l_common_mix) - rowLogSumExps(t(log_weights))

Comparison with benchmark values obtained with long simulations

To evaluate the performance of mixture estimators (MixIS) we also generate benchmark values, i.e. accurate approximations of the LOO predictives \(\{p(y_i|y_{-i})\}_{i=1,\dots,n}\), obtained by brute-force sampling from the leave-one-out posteriors directly, getting \(90k\) samples from each and discarding the first \(10k\) as warmup. This is computationally heavy, hence we have saved the results and we just load them in the current vignette.

data(voice_loo)
elpd_loo <- voice_loo$elpd_loo

We can then compute the root mean squared error (RMSE) of the PSIS and mixture estimators relative to such benchmark values.

elpd_psis <- loo_post$pointwise[,1]
print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2)))
[1] "RMSE(PSIS) = 0.08"
print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2)))
[1] "RMSE(MixIS) = 0.05"

Here mixture estimator provides a reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results).

We then compare the overall ELPD estimates with the brute force one.

elpd_psis <- loo_post$pointwise[,1]
print(paste("ELPD (PSIS)=",round(sum(elpd_psis),2)))
[1] "ELPD (PSIS)= -42.51"
print(paste("ELPD (MixIS)=",round(sum(elpd_mixis),2)))
[1] "ELPD (MixIS)= -44.94"
print(paste("ELPD (brute force)=",round(sum(elpd_loo),2)))
[1] "ELPD (brute force)= -45.63"

In this example, MixIS provides a more accurate ELPD estimate closer to the brute force estimate, while PSIS severely overestimates the ELPD. Note that low accuracy of the PSIS ELPD estimate is expected in this example given the large number of large Pareto-\(k\) values. In this example, the accuracy of MixIS estimate will also improve with bigger MCMC sample size.

More generally, mixture estimators can be useful in situations where standard PSIS estimators struggle and return many large Pareto-\(k\) values. In these contexts MixIS often provides more accurate LOO-CV and ELPD estimates with a single sampling routine (i.e. with a cost comparable to sampling from the original posterior).

References

Silva L. and Zanella G. (2022). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Preprint at arXiv:2209.09190

Vehtari A., Gelman A., and Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432. Preprint at arXiv:1507.04544

Vehtari A., Simpson D., Gelman A., Yao Y., and Gabry J. (2022). Pareto smoothed importance sampling. Preprint at arXiv:1507.02646

loo/inst/doc/loo2-example.html0000644000176200001440000141436614411455345016004 0ustar liggesusers Using the loo package (version >= 2.0.0)

Using the loo package (version >= 2.0.0)

Aki Vehtari and Jonah Gabry

2023-03-30

Introduction

This vignette demonstrates how to use the loo package to carry out Pareto smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) for purposes of model checking and model comparison.

In this vignette we can’t provide all necessary background information on PSIS-LOO and its diagnostics (Pareto \(k\) and effective sample size), so we encourage readers to refer to the following papers for more details:

  • Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

  • Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

Setup

In addition to the loo package, we’ll also be using rstanarm and bayesplot:

library("rstanarm")
library("bayesplot")
library("loo")

Example: Poisson vs negative binomial for the roaches dataset

Background and model fitting

The Poisson and negative binomial regression models used below in our example, as well as the stan_glm function used to fit the models, are covered in more depth in the rstanarm vignette Estimating Generalized Linear Models for Count Data with rstanarm. In the rest of this vignette we will assume the reader is already familiar with these kinds of models.

Roaches data

The example data we’ll use comes from Chapter 8.3 of Gelman and Hill (2007). We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. Here is how Gelman and Hill describe the experiment and data (pg. 161):

the treatment and control were applied to 160 and 104 apartments, respectively, and the outcome measurement \(y_i\) in each apartment \(i\) was the number of roaches caught in a set of traps. Different apartments had traps for different numbers of days

In addition to an intercept, the regression predictors for the model are roach1, the pre-treatment number of roaches (rescaled above to be in units of hundreds), the treatment indicator treatment, and a variable indicating whether the apartment is in a building restricted to elderly residents senior. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we use the offset argument to specify that log(exposure2) should be added to the linear predictor.

# the 'roaches' data frame is included with the rstanarm package
data(roaches)
str(roaches)
'data.frame':   262 obs. of  5 variables:
 $ y        : int  153 127 7 7 0 0 73 24 2 2 ...
 $ roach1   : num  308 331.25 1.67 3 2 ...
 $ treatment: int  1 1 1 1 1 1 1 1 0 0 ...
 $ senior   : int  0 0 0 0 0 0 0 0 0 0 ...
 $ exposure2: num  0.8 0.6 1 1 1.14 ...
# rescale to units of hundreds of roaches
roaches$roach1 <- roaches$roach1 / 100

Fit Poisson model

We’ll fit a simple Poisson regression model using the stan_glm function from the rstanarm package.

fit1 <-
  stan_glm(
    formula = y ~ roach1 + treatment + senior,
    offset = log(exposure2),
    data = roaches,
    family = poisson(link = "log"),
    prior = normal(0, 2.5, autoscale = TRUE),
    prior_intercept = normal(0, 5, autoscale = TRUE),
    seed = 12345
  )

Usually we would also run posterior predictive checks as shown in the rstanarm vignette Estimating Generalized Linear Models for Count Data with rstanarm, but here we focus only on methods provided by the loo package.


Using the loo package for model checking and comparison

Although cross-validation is mostly used for model comparison, it is also useful for model checking.

Computing PSIS-LOO and checking diagnostics

We start by computing PSIS-LOO with the loo function. Since we fit our model using rstanarm we can use the loo method for stanreg objects (fitted model objects from rstanarm), which doesn’t require us to first extract the pointwise log-likelihood values. If we had written our own Stan program instead of using rstanarm we would pass an array or matrix of log-likelihood values to the loo function (see, e.g. help("loo.array", package = "loo")). We’ll also use the argument save_psis = TRUE to save some intermediate results to be re-used later.

loo1 <- loo(fit1, save_psis = TRUE)
Warning: Found 17 observations with a pareto_k > 0.7. With this many problematic observations we recommend calling 'kfold' with argument 'K=10' to perform 10-fold cross-validation rather than LOO.

loo gives us warnings about the Pareto diagnostics, which indicate that for some observations the leave-one-out posteriors are different enough from the full posterior that importance-sampling is not able to correct the difference. We can see more details by printing the loo object.

print(loo1)

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -6247.8  728.0
p_loo       292.4   73.3
looic     12495.5 1455.9
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     239   91.2%   200       
 (0.5, 0.7]   (ok)         6    2.3%   56        
   (0.7, 1]   (bad)        8    3.1%   25        
   (1, Inf)   (very bad)   9    3.4%   1         
See help('pareto-k-diagnostic') for details.

The table shows us a summary of Pareto \(k\) diagnostic, which is used to assess the reliability of the estimates. In addition to the proportion of leave-one-out folds with \(k\) values in different intervals, the minimum of the effective sample sizes in that category is shown to give idea why higher \(k\) values are bad. Since we have some \(k>1\), we are not able to compute an estimate for the Monte Carlo standard error (SE) of the expected log predictive density (elpd_loo) and NA is displayed. (Full details on the interpretation of the Pareto \(k\) diagnostics are available in the Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) papers referenced at the top of this vignette.)

In this case the elpd_loo estimate should not be considered reliable. If we had a well-specified model we would expect the estimated effective number of parameters (p_loo) to be smaller than or similar to the total number of parameters in the model. Here p_loo is almost 300, which is about 70 times the total number of parameters in the model, indicating severe model misspecification.

Plotting Pareto \(k\) diagnostics

Using the plot method on our loo1 object produces a plot of the \(k\) values (in the same order as the observations in the dataset used to fit the model) with horizontal lines corresponding to the same categories as in the printed output above.

plot(loo1)

This plot is useful to quickly see the distribution of \(k\) values, but it’s often also possible to see structure with respect to data ordering. In our case this is mild, but there seems to be a block of data that is somewhat easier to predict (indices around 90–150). Unfortunately even for these data points we see some high \(k\) values.

Marginal posterior predictive checks

The loo package can be used in combination with the bayesplot package for leave-one-out cross-validation marginal posterior predictive checks Gabry et al (2018). LOO-PIT values are cumulative probabilities for \(y_i\) computed using the LOO marginal predictive distributions \(p(y_i|y_{-i})\). For a good model, the distribution of LOO-PIT values should be uniform. In the following plot the distribution (smoothed density estimate) of the LOO-PIT values for our model (thick curve) is compared to many independently generated samples (each the same size as our dataset) from the standard uniform distribution (thin curves).

yrep <- posterior_predict(fit1)

ppc_loo_pit_overlay(
  y = roaches$y,
  yrep = yrep,
  lw = weights(loo1$psis_object)
)
NOTE: The kernel density estimate assumes continuous observations and is not optimal for discrete observations.

The excessive number of values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion.

Try alternative model with more flexibility

Here we will try negative binomial regression, which is commonly used for overdispersed count data.
Unlike the Poisson distribution, the negative binomial distribution allows the conditional mean and variance of \(y\) to differ.

fit2 <- update(fit1, family = neg_binomial_2)
loo2 <- loo(fit2, save_psis = TRUE, cores = 2)
Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly.
print(loo2)

Computed from 4000 by 262 log-likelihood matrix

         Estimate   SE
elpd_loo   -895.6 37.8
p_loo         6.7  2.7
looic      1791.3 75.5
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     260   99.2%   2615      
 (0.5, 0.7]   (ok)         1    0.4%   377       
   (0.7, 1]   (bad)        1    0.4%   28        
   (1, Inf)   (very bad)   0    0.0%   <NA>      
See help('pareto-k-diagnostic') for details.
plot(loo2, label_points = TRUE)

Using the label_points argument will label any \(k\) values larger than 0.7 with the index of the corresponding data point. These high values are often the result of model misspecification and frequently correspond to data points that would be considered ``outliers’’ in the data and surprising according to the model Gabry et al (2019). Unfortunately, while large \(k\) values are a useful indicator of model misspecification, small \(k\) values are not a guarantee that a model is well-specified.

If there are a small number of problematic \(k\) values then we can use a feature in rstanarm that lets us refit the model once for each of these problematic observations. Each time the model is refit, one of the observations with a high \(k\) value is omitted and the LOO calculations are performed exactly for that observation. The results are then recombined with the approximate LOO calculations already carried out for the observations without problematic \(k\) values:

if (any(pareto_k_values(loo2) > 0.7)) {
  loo2 <- loo(fit2, save_psis = TRUE, k_threshold = 0.7)
}
1 problematic observation(s) found.
Model will be refit 1 times.

Fitting model 1 out of 1 (leaving out observation 93)
print(loo2)

Computed from 4000 by 262 log-likelihood matrix

         Estimate   SE
elpd_loo   -895.5 37.7
p_loo         6.6  2.6
looic      1791.1 75.4
------
Monte Carlo SE of elpd_loo is 0.2.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     260   99.6%   2615      
 (0.5, 0.7]   (ok)         1    0.4%   377       
   (0.7, 1]   (bad)        0    0.0%   <NA>      
   (1, Inf)   (very bad)   0    0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

In the print output we can see that the Monte Carlo SE is small compared to the other uncertainties.

On the other hand, p_loo is about 7 and still a bit higher than the total number of parameters in the model. This indicates that there is almost certainly still some degree of model misspecification, but this is much better than the p_loo estimate for the Poisson model.

For further model checking we again examine the LOO-PIT values.

yrep <- posterior_predict(fit2)
ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2$psis_object))
NOTE: The kernel density estimate assumes continuous observations and is not optimal for discrete observations.

The plot for the negative binomial model looks better than the Poisson plot, but we still see that this model is not capturing all of the essential features in the data.

Comparing the models on expected log predictive density

We can use the loo_compare function to compare our two models on expected log predictive density (ELPD) for new data:

loo_compare(loo1, loo2)
     elpd_diff se_diff
fit2     0.0       0.0
fit1 -5352.2     709.2

The difference in ELPD is much larger than several times the estimated standard error of the difference again indicating that the negative-binomial model is expected to have better predictive performance than the Poisson model. However, according to the LOO-PIT checks there is still some misspecification, and a reasonable guess is that a hurdle or zero-inflated model would be an improvement (we leave that for another case study).


References

Gabry, J., Simpson, D., Vehtari, A., Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. :10.1111/rssa.12378. (journal version, arXiv preprint, code on GitHub)

Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. online, arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

loo/inst/doc/loo2-example.Rmd0000644000176200001440000003015314407123455015544 0ustar liggesusers--- title: "Using the loo package (version >= 2.0.0)" author: "Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to use the __loo__ package to carry out Pareto smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) for purposes of model checking and model comparison. In this vignette we can't provide all necessary background information on PSIS-LOO and its diagnostics (Pareto $k$ and effective sample size), so we encourage readers to refer to the following papers for more details: * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). # Setup In addition to the __loo__ package, we'll also be using __rstanarm__ and __bayesplot__: ```{r setup, message=FALSE} library("rstanarm") library("bayesplot") library("loo") ``` # Example: Poisson vs negative binomial for the roaches dataset ## Background and model fitting The Poisson and negative binomial regression models used below in our example, as well as the `stan_glm` function used to fit the models, are covered in more depth in the __rstanarm__ vignette [_Estimating Generalized Linear Models for Count Data with rstanarm_](http://mc-stan.org/rstanarm/articles/count.html). In the rest of this vignette we will assume the reader is already familiar with these kinds of models. ### Roaches data The example data we'll use comes from Chapter 8.3 of [Gelman and Hill (2007)](http://www.stat.columbia.edu/~gelman/arm/). We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. Here is how Gelman and Hill describe the experiment and data (pg. 161): > the treatment and control were applied to 160 and 104 apartments, respectively, and the outcome measurement $y_i$ in each apartment $i$ was the number of roaches caught in a set of traps. Different apartments had traps for different numbers of days In addition to an intercept, the regression predictors for the model are `roach1`, the pre-treatment number of roaches (rescaled above to be in units of hundreds), the treatment indicator `treatment`, and a variable indicating whether the apartment is in a building restricted to elderly residents `senior`. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we use the `offset` argument to specify that `log(exposure2)` should be added to the linear predictor. ```{r data} # the 'roaches' data frame is included with the rstanarm package data(roaches) str(roaches) # rescale to units of hundreds of roaches roaches$roach1 <- roaches$roach1 / 100 ``` ### Fit Poisson model We'll fit a simple Poisson regression model using the `stan_glm` function from the __rstanarm__ package. ```{r count-roaches-mcmc, results="hide"} fit1 <- stan_glm( formula = y ~ roach1 + treatment + senior, offset = log(exposure2), data = roaches, family = poisson(link = "log"), prior = normal(0, 2.5, autoscale = TRUE), prior_intercept = normal(0, 5, autoscale = TRUE), seed = 12345 ) ``` Usually we would also run posterior predictive checks as shown in the __rstanarm__ vignette [Estimating Generalized Linear Models for Count Data with rstanarm](http://mc-stan.org/rstanarm/articles/count.html), but here we focus only on methods provided by the __loo__ package.
## Using the __loo__ package for model checking and comparison _Although cross-validation is mostly used for model comparison, it is also useful for model checking._ ### Computing PSIS-LOO and checking diagnostics We start by computing PSIS-LOO with the `loo` function. Since we fit our model using __rstanarm__ we can use the `loo` method for `stanreg` objects (fitted model objects from __rstanarm__), which doesn't require us to first extract the pointwise log-likelihood values. If we had written our own Stan program instead of using __rstanarm__ we would pass an array or matrix of log-likelihood values to the `loo` function (see, e.g. `help("loo.array", package = "loo")`). We'll also use the argument `save_psis = TRUE` to save some intermediate results to be re-used later. ```{r loo1} loo1 <- loo(fit1, save_psis = TRUE) ``` `loo` gives us warnings about the Pareto diagnostics, which indicate that for some observations the leave-one-out posteriors are different enough from the full posterior that importance-sampling is not able to correct the difference. We can see more details by printing the `loo` object. ```{r print-loo1} print(loo1) ``` The table shows us a summary of Pareto $k$ diagnostic, which is used to assess the reliability of the estimates. In addition to the proportion of leave-one-out folds with $k$ values in different intervals, the minimum of the effective sample sizes in that category is shown to give idea why higher $k$ values are bad. Since we have some $k>1$, we are not able to compute an estimate for the Monte Carlo standard error (SE) of the expected log predictive density (`elpd_loo`) and `NA` is displayed. (Full details on the interpretation of the Pareto $k$ diagnostics are available in the Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) papers referenced at the top of this vignette.) In this case the `elpd_loo` estimate should not be considered reliable. If we had a well-specified model we would expect the estimated effective number of parameters (`p_loo`) to be smaller than or similar to the total number of parameters in the model. Here `p_loo` is almost 300, which is about 70 times the total number of parameters in the model, indicating severe model misspecification. ### Plotting Pareto $k$ diagnostics Using the `plot` method on our `loo1` object produces a plot of the $k$ values (in the same order as the observations in the dataset used to fit the model) with horizontal lines corresponding to the same categories as in the printed output above. ```{r plot-loo1, out.width = "70%"} plot(loo1) ``` This plot is useful to quickly see the distribution of $k$ values, but it's often also possible to see structure with respect to data ordering. In our case this is mild, but there seems to be a block of data that is somewhat easier to predict (indices around 90--150). Unfortunately even for these data points we see some high $k$ values. ### Marginal posterior predictive checks The `loo` package can be used in combination with the `bayesplot` package for leave-one-out cross-validation marginal posterior predictive checks [Gabry et al (2018)](https://arxiv.org/abs/1709.01449). LOO-PIT values are cumulative probabilities for $y_i$ computed using the LOO marginal predictive distributions $p(y_i|y_{-i})$. For a good model, the distribution of LOO-PIT values should be uniform. In the following plot the distribution (smoothed density estimate) of the LOO-PIT values for our model (thick curve) is compared to many independently generated samples (each the same size as our dataset) from the standard uniform distribution (thin curves). ```{r ppc_loo_pit_overlay} yrep <- posterior_predict(fit1) ppc_loo_pit_overlay( y = roaches$y, yrep = yrep, lw = weights(loo1$psis_object) ) ``` The excessive number of values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion. ## Try alternative model with more flexibility Here we will try [negative binomial](https://en.wikipedia.org/wiki/Negative_binomial_distribution) regression, which is commonly used for overdispersed count data. Unlike the Poisson distribution, the negative binomial distribution allows the conditional mean and variance of $y$ to differ. ```{r count-roaches-negbin, results="hide"} fit2 <- update(fit1, family = neg_binomial_2) ``` ```{r loo2} loo2 <- loo(fit2, save_psis = TRUE, cores = 2) print(loo2) ``` ```{r plot-loo2} plot(loo2, label_points = TRUE) ``` Using the `label_points` argument will label any $k$ values larger than 0.7 with the index of the corresponding data point. These high values are often the result of model misspecification and frequently correspond to data points that would be considered ``outliers'' in the data and surprising according to the model [Gabry et al (2019)](https://arxiv.org/abs/1709.01449). Unfortunately, while large $k$ values are a useful indicator of model misspecification, small $k$ values are not a guarantee that a model is well-specified. If there are a small number of problematic $k$ values then we can use a feature in __rstanarm__ that lets us refit the model once for each of these problematic observations. Each time the model is refit, one of the observations with a high $k$ value is omitted and the LOO calculations are performed exactly for that observation. The results are then recombined with the approximate LOO calculations already carried out for the observations without problematic $k$ values: ```{r reloo} if (any(pareto_k_values(loo2) > 0.7)) { loo2 <- loo(fit2, save_psis = TRUE, k_threshold = 0.7) } print(loo2) ``` In the print output we can see that the Monte Carlo SE is small compared to the other uncertainties. On the other hand, `p_loo` is about 7 and still a bit higher than the total number of parameters in the model. This indicates that there is almost certainly still some degree of model misspecification, but this is much better than the `p_loo` estimate for the Poisson model. For further model checking we again examine the LOO-PIT values. ```{r ppc_loo_pit_overlay-negbin} yrep <- posterior_predict(fit2) ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2$psis_object)) ``` The plot for the negative binomial model looks better than the Poisson plot, but we still see that this model is not capturing all of the essential features in the data. ## Comparing the models on expected log predictive density We can use the `loo_compare` function to compare our two models on expected log predictive density (ELPD) for new data: ```{r loo_compare} loo_compare(loo1, loo2) ``` The difference in ELPD is much larger than several times the estimated standard error of the difference again indicating that the negative-binomial model is expected to have better predictive performance than the Poisson model. However, according to the LOO-PIT checks there is still some misspecification, and a reasonable guess is that a hurdle or zero-inflated model would be an improvement (we leave that for another case study).
# References Gabry, J., Simpson, D., Vehtari, A., Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. _J. R. Stat. Soc. A_, 182: 389-402. \doi:10.1111/rssa.12378. ([journal version](https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssa.12378), [arXiv preprint](https://arxiv.org/abs/1709.01449), [code on GitHub](https://github.com/jgabry/bayes-vis-paper)) Gelman, A. and Hill, J. (2007). _Data Analysis Using Regression and Multilevel/Hierarchical Models._ Cambridge University Press, Cambridge, UK. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/inst/doc/loo2-moment-matching.Rmd0000644000176200001440000002774214407123455017212 0ustar liggesusers--- title: "Avoiding model refits in leave-one-out cross-validation with moment matching" author: "Topi Paananen, Paul Bürkner, Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to improve the Monte Carlo sampling accuracy of leave-one-out cross-validation with the __loo__ package and Stan. The __loo__ package automatically monitors the sampling accuracy using Pareto $k$ diagnostics for each observation. Here, we present a method for quickly improving the accuracy when the Pareto diagnostics indicate problems. This is done by performing some additional computations using the existing posterior sample. If successful, this will decrease the Pareto $k$ values, making the model assessment more reliable. __loo__ also stores the original Pareto $k$ values with the name `influence_pareto_k` which are not changed. They can be used as a diagnostic of how much each observation influences the posterior distribution. The methodology presented is based on the paper * Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2020). Implicitly Adaptive Importance Sampling. [arXiv preprint arXiv:1906.08850](https://arxiv.org/abs/1906.08850). More information about the Pareto $k$ diagnostics is given in the following papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). # Example: Eradication of Roaches We will use the same example as in the vignette [_Using the loo package (version >= 2.0.0)_](https://mc-stan.org/loo/articles/loo2-example.html). See the demo for a description of the problem and data. We will use the same Poisson regression model as in the case study. ## Coding the Stan model Here is the Stan code for fitting the Poisson regression model, which we will use for modeling the number of roaches. ```{r stancode} stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ``` Following the usual approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. ## Setup In addition to __loo__, we load the __rstan__ package for fitting the model, and the __rstanarm__ package for the data. ```{r setup, message=FALSE} library("rstan") library("loo") seed <- 9547 set.seed(seed) ``` ## Fitting the model with RStan Next we fit the model in Stan using the __rstan__ package: ```{r modelfit, message=FALSE} # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) y <- roaches$y x <- roaches[,c("roach1", "treatment", "senior")] offset <- log(roaches[,"exposure2"]) n <- dim(x)[1] k <- dim(x)[2] standata <- list(N = n, K = k, x = as.matrix(x), y = y, offset = offset, beta_prior_scale = 2.5, alpha_prior_scale = 5.0) # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = standata, seed = seed, refresh = 0) print(fit, pars = "beta") ``` Let us now evaluate the predictive performance of the model using `loo()`. ```{r loo1} loo1 <- loo(fit) loo1 ``` The `loo()` function output warnings that there are some observations which are highly influential, and thus the accuracy of importance sampling is compromised as indicated by the large Pareto $k$ diagnostic values (> 0.7). As discussed in the vignette [_Using the loo package (version >= 2.0.0)_](https://mc-stan.org/loo/articles/loo2-example.html), this may be an indication of model misspecification. Despite that, it is still beneficial to be able to evaluate the predictive performance of the model accurately. ## Moment matching correction for importance sampling To improve the accuracy of the `loo()` result above, we could perform leave-one-out cross-validation by explicitly leaving out single observations and refitting the model using MCMC repeatedly. However, the Pareto $k$ diagnostics indicate that there are 19 observations which are problematic. This would require 19 model refits which may require a lot of computation time. Instead of refitting with MCMC, we can perform a faster moment matching correction to the importance sampling for the problematic observations. This can be done with the `loo_moment_match()` function in the __loo__ package, which takes our existing `loo` object as input and modifies it. The moment matching requires some evaluations of the model posterior density. For models fitted with __rstan__, this can be conveniently done by using the existing `stanfit` object. First, we show how the moment matching can be used for a model fitted using __rstan__. It only requires setting the argument `moment_match` to `TRUE` in the `loo()` function. Optionally, you can also set the argument `k_threshold` which determines the Pareto $k$ threshold, above which moment matching is used. By default, it operates on all observations whose Pareto $k$ value is larger than 0.7. ```{r loo_moment_match} # available in rstan >= 2.21 loo2 <- loo(fit, moment_match = TRUE) loo2 ``` After the moment matching, all observations have the diagnostic Pareto $k$ less than 0.7, meaning that the estimates are now reliable. The total `elpd_loo` estimate also changed from `-5457.8` to `-5478.5`, showing that before moment matching, `loo()` overestimated the predictive performance of the model. The updated Pareto $k$ values stored in `loo2$diagnostics$pareto_k` are considered algorithmic diagnostic values that indicate the sampling accuracy. The original Pareto $k$ values are stored in `loo2$pointwise[,"influence_pareto_k"]` and these are not modified by the moment matching. These can be considered as diagnostics for how big influence each observation has on the posterior distribution. In addition to the Pareto $k$ diagnostics, moment matching also updates the effective sample size estimates. # Using `loo_moment_match()` directly The moment matching can also be performed by explicitly calling the function `loo_moment_match()`. This enables its use also for models that are not using __rstan__ or another package with built-in support for `loo_moment_match()`. To use `loo_moment_match()`, the user must give the model object `x`, the `loo` object, and 5 helper functions as arguments to `loo_moment_match()`. The helper functions are * `post_draws` + A function the takes `x` as the first argument and returns a matrix of posterior draws of the model parameters, `pars`. * `log_lik_i` + A function that takes `x` and `i` and returns a matrix (one column per chain) or a vector (all chains stacked) of log-likeliood draws of the ith observation based on the model `x`. If the draws are obtained using MCMC, the matrix with MCMC chains separated is preferred. * `unconstrain_pars` + A function that takes arguments `x` and `pars`, and returns posterior draws on the unconstrained space based on the posterior draws on the constrained space passed via `pars`. * `log_prob_upars` + A function that takes arguments `x` and `upars`, and returns a matrix of log-posterior density values of the unconstrained posterior draws passed via `upars`. * `log_lik_i_upars` + A function that takes arguments `x`, `upars`, and `i` and returns a vector of log-likelihood draws of the `i`th observation based on the unconstrained posterior draws passed via `upars`. Next, we show how the helper functions look like for RStan objects, and show an example of using `loo_moment_match()` directly. For stanfit objects from __rstan__ objects, the functions look like this: ```{r stanfitfuns} # create a named list of draws for use with rstan methods .rstan_relist <- function(x, skeleton) { out <- utils::relist(x, skeleton) for (i in seq_along(skeleton)) { dim(out[[i]]) <- dim(skeleton[[i]]) } out } # rstan helper function to get dims of parameters right .create_skeleton <- function(pars, dims) { out <- lapply(seq_along(pars), function(i) { len_dims <- length(dims[[i]]) if (len_dims < 1) return(0) return(array(0, dim = dims[[i]])) }) names(out) <- pars out } # extract original posterior draws post_draws_stanfit <- function(x, ...) { as.matrix(x) } # compute a matrix of log-likelihood values for the ith observation # matrix contains information about the number of MCMC chains log_lik_i_stanfit <- function(x, i, parameter_name = "log_lik", ...) { loo::extract_log_lik(x, parameter_name, merge_chains = FALSE)[, , i] } # transform parameters to the unconstraint space unconstrain_pars_stanfit <- function(x, pars, ...) { skeleton <- .create_skeleton(x@sim$pars_oi, x@par_dims[x@sim$pars_oi]) upars <- apply(pars, 1, FUN = function(theta) { rstan::unconstrain_pars(x, .rstan_relist(theta, skeleton)) }) # for one parameter models if (is.null(dim(upars))) { dim(upars) <- c(1, length(upars)) } t(upars) } # compute log_prob for each posterior draws on the unconstrained space log_prob_upars_stanfit <- function(x, upars, ...) { apply(upars, 1, rstan::log_prob, object = x, adjust_transform = TRUE, gradient = FALSE) } # compute log_lik values based on the unconstrained parameters log_lik_i_upars_stanfit <- function(x, upars, i, parameter_name = "log_lik", ...) { S <- nrow(upars) out <- numeric(S) for (s in seq_len(S)) { out[s] <- rstan::constrain_pars(x, upars = upars[s, ])[[parameter_name]][i] } out } ``` Using these function, we can call `loo_moment_match()` to update the existing `loo` object. ```{r loo_moment_match.default, message=FALSE} loo3 <- loo::loo_moment_match.default( x = fit, loo = loo1, post_draws = post_draws_stanfit, log_lik_i = log_lik_i_stanfit, unconstrain_pars = unconstrain_pars_stanfit, log_prob_upars = log_prob_upars_stanfit, log_lik_i_upars = log_lik_i_upars_stanfit ) loo3 ``` As expected, the result is identical to the previous result of `loo2 <- loo(fit, moment_match = TRUE)`. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2020) _RStan: the R interface to Stan, Version 2.21.1_ https://mc-stan.org Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2021). Implicitly adaptive importance sampling. _Statistics and Computing_, 31, 16. \doi:10.1007/s11222-020-09982-2. arXiv preprint arXiv:1906.08850. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/inst/doc/loo2-moment-matching.html0000644000176200001440000012225414411465147017430 0ustar liggesusers Avoiding model refits in leave-one-out cross-validation with moment matching

Avoiding model refits in leave-one-out cross-validation with moment matching

Topi Paananen, Paul Bürkner, Aki Vehtari and Jonah Gabry

2023-03-30

Introduction

This vignette demonstrates how to improve the Monte Carlo sampling accuracy of leave-one-out cross-validation with the loo package and Stan. The loo package automatically monitors the sampling accuracy using Pareto \(k\) diagnostics for each observation. Here, we present a method for quickly improving the accuracy when the Pareto diagnostics indicate problems. This is done by performing some additional computations using the existing posterior sample. If successful, this will decrease the Pareto \(k\) values, making the model assessment more reliable. loo also stores the original Pareto \(k\) values with the name influence_pareto_k which are not changed. They can be used as a diagnostic of how much each observation influences the posterior distribution.

The methodology presented is based on the paper

More information about the Pareto \(k\) diagnostics is given in the following papers

  • Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

  • Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

Example: Eradication of Roaches

We will use the same example as in the vignette Using the loo package (version >= 2.0.0). See the demo for a description of the problem and data. We will use the same Poisson regression model as in the case study.

Coding the Stan model

Here is the Stan code for fitting the Poisson regression model, which we will use for modeling the number of roaches.

stancode <- "
data {
  int<lower=1> K;
  int<lower=1> N;
  matrix[N,K] x;
  int y[N];
  vector[N] offset;

  real beta_prior_scale;
  real alpha_prior_scale;
}
parameters {
  vector[K] beta;
  real intercept;
}
model {
  y ~ poisson(exp(x * beta + intercept + offset));
  beta ~ normal(0,beta_prior_scale);
  intercept ~ normal(0,alpha_prior_scale);
}
generated quantities {
  vector[N] log_lik;
  for (n in 1:N)
    log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n]));
}
"

Following the usual approach recommended in Writing Stan programs for use with the loo package, we compute the log-likelihood for each observation in the generated quantities block of the Stan program.

Setup

In addition to loo, we load the rstan package for fitting the model, and the rstanarm package for the data.

library("rstan")
library("loo")
seed <- 9547
set.seed(seed)

Fitting the model with RStan

Next we fit the model in Stan using the rstan package:

# Prepare data
data(roaches, package = "rstanarm")
roaches$roach1 <- sqrt(roaches$roach1)
y <- roaches$y
x <- roaches[,c("roach1", "treatment", "senior")]
offset <- log(roaches[,"exposure2"])
n <- dim(x)[1]
k <- dim(x)[2]

standata <- list(N = n, K = k, x = as.matrix(x), y = y, offset = offset, beta_prior_scale = 2.5, alpha_prior_scale = 5.0)

# Compile
stanmodel <- stan_model(model_code = stancode)

# Fit model
fit <- sampling(stanmodel, data = standata, seed = seed, refresh = 0)
print(fit, pars = "beta")
Inference for Stan model: anon_model.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

         mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
beta[1]  0.16       0 0.00  0.16  0.16  0.16  0.16  0.16  2344    1
beta[2] -0.57       0 0.03 -0.62 -0.59 -0.57 -0.55 -0.52  2395    1
beta[3] -0.31       0 0.03 -0.38 -0.34 -0.31 -0.29 -0.25  2135    1

Samples were drawn using NUTS(diag_e) at Thu Mar 30 23:06:35 2023.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

Let us now evaluate the predictive performance of the model using loo().

loo1 <- loo(fit)
Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
loo1

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5459.4  694.1
p_loo       258.8   55.4
looic     10918.9 1388.2
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     241   92.0%   241       
 (0.5, 0.7]   (ok)         7    2.7%   53        
   (0.7, 1]   (bad)        7    2.7%   24        
   (1, Inf)   (very bad)   7    2.7%   2         
See help('pareto-k-diagnostic') for details.

The loo() function output warnings that there are some observations which are highly influential, and thus the accuracy of importance sampling is compromised as indicated by the large Pareto \(k\) diagnostic values (> 0.7). As discussed in the vignette Using the loo package (version >= 2.0.0), this may be an indication of model misspecification. Despite that, it is still beneficial to be able to evaluate the predictive performance of the model accurately.

Moment matching correction for importance sampling

To improve the accuracy of the loo() result above, we could perform leave-one-out cross-validation by explicitly leaving out single observations and refitting the model using MCMC repeatedly. However, the Pareto \(k\) diagnostics indicate that there are 19 observations which are problematic. This would require 19 model refits which may require a lot of computation time.

Instead of refitting with MCMC, we can perform a faster moment matching correction to the importance sampling for the problematic observations. This can be done with the loo_moment_match() function in the loo package, which takes our existing loo object as input and modifies it. The moment matching requires some evaluations of the model posterior density. For models fitted with rstan, this can be conveniently done by using the existing stanfit object.

First, we show how the moment matching can be used for a model fitted using rstan. It only requires setting the argument moment_match to TRUE in the loo() function. Optionally, you can also set the argument k_threshold which determines the Pareto \(k\) threshold, above which moment matching is used. By default, it operates on all observations whose Pareto \(k\) value is larger than 0.7.

# available in rstan >= 2.21
loo2 <- loo(fit, moment_match = TRUE)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
loo2

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5478.8  700.0
p_loo       269.8   61.5
looic     10957.7 1400.1
------
Monte Carlo SE of elpd_loo is 0.4.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     252   96.2%   236       
 (0.5, 0.7]   (ok)        10    3.8%   50        
   (0.7, 1]   (bad)        0    0.0%   <NA>      
   (1, Inf)   (very bad)   0    0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

After the moment matching, all observations have the diagnostic Pareto \(k\) less than 0.7, meaning that the estimates are now reliable. The total elpd_loo estimate also changed from -5457.8 to -5478.5, showing that before moment matching, loo() overestimated the predictive performance of the model.

The updated Pareto \(k\) values stored in loo2$diagnostics$pareto_k are considered algorithmic diagnostic values that indicate the sampling accuracy. The original Pareto \(k\) values are stored in loo2$pointwise[,"influence_pareto_k"] and these are not modified by the moment matching. These can be considered as diagnostics for how big influence each observation has on the posterior distribution. In addition to the Pareto \(k\) diagnostics, moment matching also updates the effective sample size estimates.

Using loo_moment_match() directly

The moment matching can also be performed by explicitly calling the function loo_moment_match(). This enables its use also for models that are not using rstan or another package with built-in support for loo_moment_match(). To use loo_moment_match(), the user must give the model object x, the loo object, and 5 helper functions as arguments to loo_moment_match(). The helper functions are

  • post_draws
    • A function the takes x as the first argument and returns a matrix of posterior draws of the model parameters, pars.
  • log_lik_i
    • A function that takes x and i and returns a matrix (one column per chain) or a vector (all chains stacked) of log-likeliood draws of the ith observation based on the model x. If the draws are obtained using MCMC, the matrix with MCMC chains separated is preferred.
  • unconstrain_pars
    • A function that takes arguments x and pars, and returns posterior draws on the unconstrained space based on the posterior draws on the constrained space passed via pars.
  • log_prob_upars
    • A function that takes arguments x and upars, and returns a matrix of log-posterior density values of the unconstrained posterior draws passed via upars.
  • log_lik_i_upars
    • A function that takes arguments x, upars, and i and returns a vector of log-likelihood draws of the ith observation based on the unconstrained posterior draws passed via upars.

Next, we show how the helper functions look like for RStan objects, and show an example of using loo_moment_match() directly. For stanfit objects from rstan objects, the functions look like this:

# create a named list of draws for use with rstan methods
.rstan_relist <- function(x, skeleton) {
  out <- utils::relist(x, skeleton)
  for (i in seq_along(skeleton)) {
    dim(out[[i]]) <- dim(skeleton[[i]])
  }
  out
}

# rstan helper function to get dims of parameters right
.create_skeleton <- function(pars, dims) {
  out <- lapply(seq_along(pars), function(i) {
    len_dims <- length(dims[[i]])
    if (len_dims < 1) return(0)
    return(array(0, dim = dims[[i]]))
  })
  names(out) <- pars
  out
}

# extract original posterior draws
post_draws_stanfit <- function(x, ...) {
  as.matrix(x)
}

# compute a matrix of log-likelihood values for the ith observation
# matrix contains information about the number of MCMC chains
log_lik_i_stanfit <- function(x, i, parameter_name = "log_lik", ...) {
  loo::extract_log_lik(x, parameter_name, merge_chains = FALSE)[, , i]
}

# transform parameters to the unconstraint space
unconstrain_pars_stanfit <- function(x, pars, ...) {
  skeleton <- .create_skeleton(x@sim$pars_oi, x@par_dims[x@sim$pars_oi])
  upars <- apply(pars, 1, FUN = function(theta) {
    rstan::unconstrain_pars(x, .rstan_relist(theta, skeleton))
  })
  # for one parameter models
  if (is.null(dim(upars))) {
    dim(upars) <- c(1, length(upars))
  }
  t(upars)
}

# compute log_prob for each posterior draws on the unconstrained space
log_prob_upars_stanfit <- function(x, upars, ...) {
  apply(upars, 1, rstan::log_prob, object = x,
        adjust_transform = TRUE, gradient = FALSE)
}

# compute log_lik values based on the unconstrained parameters
log_lik_i_upars_stanfit <- function(x, upars, i, parameter_name = "log_lik",
                                  ...) {
  S <- nrow(upars)
  out <- numeric(S)
  for (s in seq_len(S)) {
    out[s] <- rstan::constrain_pars(x, upars = upars[s, ])[[parameter_name]][i]
  }
  out
}

Using these function, we can call loo_moment_match() to update the existing loo object.

loo3 <- loo::loo_moment_match.default(
  x = fit,
  loo = loo1,
  post_draws = post_draws_stanfit,
  log_lik_i = log_lik_i_stanfit,
  unconstrain_pars = unconstrain_pars_stanfit,
  log_prob_upars = log_prob_upars_stanfit,
  log_lik_i_upars = log_lik_i_upars_stanfit
)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
loo3

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5478.8  700.0
p_loo       269.8   61.5
looic     10957.7 1400.1
------
Monte Carlo SE of elpd_loo is 0.4.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     252   96.2%   236       
 (0.5, 0.7]   (ok)        10    3.8%   50        
   (0.7, 1]   (bad)        0    0.0%   <NA>      
   (1, Inf)   (very bad)   0    0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

As expected, the result is identical to the previous result of loo2 <- loo(fit, moment_match = TRUE).

References

Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge University Press.

Stan Development Team (2020) RStan: the R interface to Stan, Version 2.21.1 https://mc-stan.org

Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2021). Implicitly adaptive importance sampling. Statistics and Computing, 31, 16. :10.1007/s11222-020-09982-2. arXiv preprint arXiv:1906.08850.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

loo/inst/doc/loo2-moment-matching.R0000644000176200001440000001014414411465147016657 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----stancode----------------------------------------------------------------- stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ## ----setup, message=FALSE----------------------------------------------------- library("rstan") library("loo") seed <- 9547 set.seed(seed) ## ----modelfit, message=FALSE-------------------------------------------------- # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) y <- roaches$y x <- roaches[,c("roach1", "treatment", "senior")] offset <- log(roaches[,"exposure2"]) n <- dim(x)[1] k <- dim(x)[2] standata <- list(N = n, K = k, x = as.matrix(x), y = y, offset = offset, beta_prior_scale = 2.5, alpha_prior_scale = 5.0) # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = standata, seed = seed, refresh = 0) print(fit, pars = "beta") ## ----loo1--------------------------------------------------------------------- loo1 <- loo(fit) loo1 ## ----loo_moment_match--------------------------------------------------------- # available in rstan >= 2.21 loo2 <- loo(fit, moment_match = TRUE) loo2 ## ----stanfitfuns-------------------------------------------------------------- # create a named list of draws for use with rstan methods .rstan_relist <- function(x, skeleton) { out <- utils::relist(x, skeleton) for (i in seq_along(skeleton)) { dim(out[[i]]) <- dim(skeleton[[i]]) } out } # rstan helper function to get dims of parameters right .create_skeleton <- function(pars, dims) { out <- lapply(seq_along(pars), function(i) { len_dims <- length(dims[[i]]) if (len_dims < 1) return(0) return(array(0, dim = dims[[i]])) }) names(out) <- pars out } # extract original posterior draws post_draws_stanfit <- function(x, ...) { as.matrix(x) } # compute a matrix of log-likelihood values for the ith observation # matrix contains information about the number of MCMC chains log_lik_i_stanfit <- function(x, i, parameter_name = "log_lik", ...) { loo::extract_log_lik(x, parameter_name, merge_chains = FALSE)[, , i] } # transform parameters to the unconstraint space unconstrain_pars_stanfit <- function(x, pars, ...) { skeleton <- .create_skeleton(x@sim$pars_oi, x@par_dims[x@sim$pars_oi]) upars <- apply(pars, 1, FUN = function(theta) { rstan::unconstrain_pars(x, .rstan_relist(theta, skeleton)) }) # for one parameter models if (is.null(dim(upars))) { dim(upars) <- c(1, length(upars)) } t(upars) } # compute log_prob for each posterior draws on the unconstrained space log_prob_upars_stanfit <- function(x, upars, ...) { apply(upars, 1, rstan::log_prob, object = x, adjust_transform = TRUE, gradient = FALSE) } # compute log_lik values based on the unconstrained parameters log_lik_i_upars_stanfit <- function(x, upars, i, parameter_name = "log_lik", ...) { S <- nrow(upars) out <- numeric(S) for (s in seq_len(S)) { out[s] <- rstan::constrain_pars(x, upars = upars[s, ])[[parameter_name]][i] } out } ## ----loo_moment_match.default, message=FALSE---------------------------------- loo3 <- loo::loo_moment_match.default( x = fit, loo = loo1, post_draws = post_draws_stanfit, log_lik_i = log_lik_i_stanfit, unconstrain_pars = unconstrain_pars_stanfit, log_prob_upars = log_prob_upars_stanfit, log_lik_i_upars = log_lik_i_upars_stanfit ) loo3 loo/inst/doc/loo2-non-factorized.Rmd0000644000176200001440000006631014407123455017037 0ustar liggesusers--- title: "Leave-one-out cross-validation for non-factorized models" author: "Aki Vehtari, Paul Bürkner and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes encoding: "UTF-8" params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r more-knitr-ops, include=FALSE} knitr::opts_chunk$set( cache=TRUE, message=FALSE, warning=FALSE ) ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction When computing ELPD-based LOO-CV for a Bayesian model we need to compute the log leave-one-out predictive densities $\log{p(y_i | y_{-i})}$ for every response value $y_i, \: i = 1, \ldots, N$, where $y_{-i}$ denotes all response values except observation $i$. To obtain $p(y_i | y_{-i})$, we need to have access to the pointwise likelihood $p(y_i\,|\, y_{-i}, \theta)$ and integrate over the model parameters $\theta$: $$ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\, y_{-i}, \theta) \, p(\theta\,|\, y_{-i}) \,d \theta $$ Here, $p(\theta\,|\, y_{-i})$ is the leave-one-out posterior distribution for $\theta$, that is, the posterior distribution for $\theta$ obtained by fitting the model while holding out the $i$th observation (we will later show how refitting the model to data $y_{-i}$ can be avoided). If the observation model is formulated directly as the product of the pointwise observation models, we call it a *factorized* model. In this case, the likelihood is also the product of the pointwise likelihood contributions $p(y_i\,|\, y_{-i}, \theta)$. To better illustrate possible structures of the observation models, we formally divide $\theta$ into two parts, observation-specific latent variables $f = (f_1, \ldots, f_N)$ and hyperparameters $\psi$, so that $p(y_i\,|\, y_{-i}, \theta) = p(y_i\,|\, y_{-i}, f_i, \psi)$. Depending on the model, one of the two parts of $\theta$ may also be empty. In very simple models, such as linear regression models, latent variables are not explicitly presented and response values are conditionally independent given $\psi$, so that $p(y_i\,|\, y_{-i}, f_i, \psi) = p(y_i \,|\, \psi)$. The full likelihood can then be written in the familiar form $$ p(y \,|\, \psi) = \prod_{i=1}^N p(y_i \,|\, \psi), $$ where $y = (y_1, \ldots, y_N)$ denotes the vector of all responses. When the likelihood factorizes this way, the conditional pointwise log-likelihood can be obtained easily by computing $p(y_i\,|\, \psi)$ for each $i$ with computational cost $O(n)$. Yet, there are several reasons why a *non-factorized* observation model may be necessary or preferred. In non-factorized models, the joint likelihood of the response values $p(y \,|\, \theta)$ is not factorized into observation-specific components, but rather given directly as one joint expression. For some models, an analytic factorized formulation is simply not available in which case we speak of a *non-factorizable* model. Even in models whose observation model can be factorized in principle, it may still be preferable to use a non-factorized form for reasons of efficiency and numerical stability (Bürkner et al. 2020). Whether a non-factorized model is used by necessity or for efficiency and stability, it comes at the cost of having no direct access to the leave-one-out predictive densities and thus to the overall leave-one-out predictive accuracy. In theory, we can express the observation-specific likelihoods in terms of the joint likelihood via $$ p(y_i \,|\, y_{i-1}, \theta) = \frac{p(y \,|\, \theta)}{p(y_{-i} \,|\, \theta)} = \frac{p(y \,|\, \theta)}{\int p(y \,|\, \theta) \, d y_i}, $$ but the expression on the right-hand side may not always have an analytical solution. Computing $\log p(y_i \,|\, y_{-i}, \theta)$ for non-factorized models is therefore often impossible, or at least inefficient and numerically unstable. However, there is a large class of multivariate normal and Student-$t$ models for which there are efficient analytical solutions available. More details can be found in our paper about LOO-CV for non-factorized models (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1810.10559). # LOO-CV for multivariate normal models In this vignette, we will focus on non-factorized multivariate normal models. Based on results of Sundararajan and Keerthi (2001), Bürkner et al. (2020) show that, for multivariate normal models with coriance matrix $C$, the LOO predictive mean and standard deviation can be computed as follows: \begin{align} \mu_{\tilde{y},-i} &= y_i-\bar{c}_{ii}^{-1} g_i \nonumber \\ \sigma_{\tilde{y},-i} &= \sqrt{\bar{c}_{ii}^{-1}}, \end{align} where $g_i$ and $\bar{c}_{ii}$ are \begin{align} g_i &= \left[C^{-1} y\right]_i \nonumber \\ \bar{c}_{ii} &= \left[C^{-1}\right]_{ii}. \end{align} Using these results, the log predictive density of the $i$th observation is then computed as $$ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{-i} - \frac{1}{2}\frac{(y_i-\mu_{-i})^2}{\sigma^2_{-i}}. $$ Expressing this same equation in terms of $g_i$ and $\bar{c}_{ii}$, the log predictive density becomes: $$ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) + \frac{1}{2}\log \bar{c}_{ii} - \frac{1}{2}\frac{g_i^2}{\bar{c}_{ii}}. $$ (Note that Vehtari et al. (2016) has a typo in the corresponding Equation 34.) From these equations we can now derive a recipe for obtaining the conditional pointwise log-likelihood for _all_ models that can be expressed conditionally in terms of a multivariate normal with invertible covariance matrix $C$. ## Approximate LOO-CV using integrated importance-sampling The above LOO equations for multivariate normal models are conditional on parameters $\theta$. Therefore, to obtain the leave-one-out predictive density $p(y_i \,|\, y_{-i})$ we need to integrate over $\theta$, $$ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\,y_{-i}, \theta) \, p(\theta\,|\,y_{-i}) \,d\theta. $$ Here, $p(\theta\,|\,y_{-i})$ is the leave-one-out posterior distribution for $\theta$, that is, the posterior distribution for $\theta$ obtained by fitting the model while holding out the $i$th observation. To avoid the cost of sampling from $N$ leave-one-out posteriors, it is possible to take the posterior draws $\theta^{(s)}, \, s=1,\ldots,S$, from the \emph{full} posterior $p(\theta\,|\,y)$, and then approximate the above integral using integrated importance sampling (Vehtari et al., 2016, Section 3.6.1): $$ p(y_i\,|\,y_{-i}) \approx \frac{ \sum_{s=1}^S p(y_i\,|\,y_{-i},\,\theta^{(s)}) \,w_i^{(s)}}{ \sum_{s=1}^S w_i^{(s)}}, $$ where $w_i^{(s)}$ are importance weights. First we compute the raw importance ratios $$ r_i^{(s)} \propto \frac{1}{p(y_i \,|\, y_{-i}, \,\theta^{(s)})}, $$ and then stabilize them using Pareto smoothed importance sampling (PSIS, Vehtari et al, 2019) to obtain the weights $w_i^{(s)}$. The resulting approximation is referred to as PSIS-LOO (Vehtari et al, 2017). ## Exact LOO-CV with re-fitting In order to validate the approximate LOO procedure, and also in order to allow exact computations to be made for a small number of leave-one-out folds for which the Pareto $k$ diagnostic (Vehtari et al, 2019) indicates an unstable approximation, we need to consider how we might to do _exact_ leave-one-out CV for a non-factorized model. In the case of a Gaussian process that has the marginalization property, we could just drop the one row and column of $C$ corresponding to the held out out observation. This does not hold in general for multivariate normal models, however, and to keep the original prior we may need to maintain the full covariance matrix $C$ even when one of the observations is left out. The solution is to model $y_i$ as a missing observation and estimate it along with all of the other model parameters. For a conditional multivariate normal model, $\log p(y_i\,|\,y_{-i})$ can be computed as follows. First, we model $y_i$ as missing and denote the corresponding parameter $y_i^{\mathrm{mis}}$. Then, we define $$ y_{\mathrm{mis}(i)} = (y_1, \ldots, y_{i-1}, y_i^{\mathrm{mis}}, y_{i+1}, \ldots, y_N). $$ to be the same as the full set of observations $y$, except replacing $y_i$ with the parameter $y_i^{\mathrm{mis}}$. Second, we compute the LOO predictive mean and standard deviations as above, but replace $y$ with $y_{\mathrm{mis}(i)}$ in the computation of $\mu_{\tilde{y},-i}$: $$ \mu_{\tilde{y},-i} = y_{{\mathrm{mis}}(i)}-\bar{c}_{ii}^{-1}g_i, $$ where in this case we have $$ g_i = \left[ C^{-1} y_{\mathrm{mis}(i)} \right]_i. $$ The conditional log predictive density is then computed with the above $\mu_{\tilde{y},-i}$ and the left out observation $y_i$: $$ \log p(y_i\,|\,y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{\tilde{y},-i} - \frac{1}{2}\frac{(y_i-\mu_{\tilde{y},-i})^2}{\sigma^2_{\tilde{y},-i}}. $$ Finally, the leave-one-out predictive distribution can then be estimated as $$ p(y_i\,|\,y_{-i}) \approx \sum_{s=1}^S p(y_i\,|\,y_{-i}, \theta_{-i}^{(s)}), $$ where $\theta_{-i}^{(s)}$ are draws from the posterior distribution $p(\theta\,|\,y_{\mathrm{mis}(i)})$. # Lagged SAR models A common non-factorized multivariate normal model is the simultaneously autoregressive (SAR) model, which is frequently used for spatially correlated data. The lagged SAR model is defined as $$ y = \rho Wy + \eta + \epsilon $$ or equivalently $$ (I - \rho W)y = \eta + \epsilon, $$ where $\rho$ is the spatial correlation parameter and $W$ is a user-defined weight matrix. The matrix $W$ has entries $w_{ii} = 0$ along the diagonal and the off-diagonal entries $w_{ij}$ are larger when areas $i$ and $j$ are closer to each other. In a linear model, the predictor term $\eta$ is given by $\eta = X \beta$ with design matrix $X$ and regression coefficients $\beta$. However, since the above equation holds for arbitrary $\eta$, these results are not restricted to linear models. If we have $\epsilon \sim {\mathrm N}(0, \,\sigma^2 I)$, it follows that $$ (I - \rho W)y \sim {\mathrm N}(\eta, \sigma^2 I), $$ which corresponds to the following log PDF coded in **Stan**: ```{r lpdf, eval=FALSE} /** * Normal log-pdf for spatially lagged responses * * @param y Vector of response values. * @param mu Mean parameter vector. * @param sigma Positive scalar residual standard deviation. * @param rho Positive scalar autoregressive parameter. * @param W Spatial weight matrix. * * @return A scalar to be added to the log posterior. */ real normal_lagsar_lpdf(vector y, vector mu, real sigma, real rho, matrix W) { int N = rows(y); real inv_sigma2 = 1 / square(sigma); matrix[N, N] W_tilde = -rho * W; vector[N] half_pred; for (n in 1:N) W_tilde[n,n] += 1; half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - 0.5 * dot_self(half_pred) * inv_sigma2; } ``` For the purpose of computing LOO-CV, it makes sense to rewrite the SAR model in slightly different form. Conditional on $\rho$, $\eta$, and $\sigma$, if we write \begin{align} y-(I-\rho W)^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(I-\rho W)^{-1}(I-\rho W)^{-T}), \end{align} or more compactly, with $\widetilde{W}=(I-\rho W)$, \begin{align} y-\widetilde{W}^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}), \end{align} then this has the same form as the zero mean Gaussian process from above. Accordingly, we can compute the leave-one-out predictive densities with the equations from Sundararajan and Keerthi (2001), replacing $y$ with $(y-\widetilde{W}^{-1}\eta)$ and taking the covariance matrix $C$ to be $\sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}$. ## Case Study: Neighborhood Crime in Columbus, Ohio In order to demonstrate how to carry out the computations implied by these equations, we will first fit a lagged SAR model to data on crime in 49 different neighborhoods of Columbus, Ohio during the year 1980. The data was originally described in Aneslin (1988) and ships with the **spdep** R package. In addition to the **loo** package, for this analysis we will use the **brms** interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting. ```{r setup, cache=FALSE} library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) SEED <- 10001 set.seed(SEED) # only sets seed for R (seed for Stan set later) # loads COL.OLD data frame and COL.nb neighbor list data(oldcol, package = "spdep") ``` The three variables in the data set relevant to this example are: * `CRIME`: the number of residential burglaries and vehicle thefts per thousand households in the neighbood * `HOVAL`: housing value in units of $1000 USD * `INC`: household income in units of $1000 USD ```{r data} str(COL.OLD[, c("CRIME", "HOVAL", "INC")]) ``` We will also use the object `COL.nb`, which is a list containing information about which neighborhoods border each other. From this list we will be able to construct the weight matrix to used to help account for the spatial dependency among the observations. ### Fit lagged SAR model A model predicting `CRIME` from `INC` and `HOVAL`, while accounting for the spatial dependency via an SAR structure, can be specified in **brms** as follows. ```{r fit, results="hide"} fit <- brm( CRIME ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 4, seed = SEED ) ``` The code above fits the model in **Stan** using a log PDF equivalent to the `normal_lagsar_lpdf` function we defined above. In the summary output below we see that both higher income and higher housing value predict lower crime rates in the neighborhood. Moreover, there seems to be substantial spatial correlation between adjacent neighborhoods, as indicated by the posterior distribution of the `lagsar` parameter. ```{r plot-lagsar, message=FALSE} lagsar <- as.matrix(fit, pars = "lagsar") estimates <- quantile(lagsar, probs = c(0.25, 0.5, 0.75)) mcmc_hist(lagsar) + vline_at(estimates, linetype = 2, size = 1) + ggtitle("lagsar: posterior median and 50% central interval") ``` ### Approximate LOO-CV After fitting the model, the next step is to compute the pointwise log-likelihood values needed for approximate LOO-CV. To do this we will use the recipe laid out in the previous sections. ```{r approx} posterior <- as.data.frame(fit) y <- fit$data$CRIME N <- length(y) S <- nrow(posterior) loglik <- yloo <- sdloo <- matrix(nrow = S, ncol = N) for (s in 1:S) { p <- posterior[s, ] eta <- p$b_Intercept + p$b_INC * fit$data$INC + p$b_HOVAL * fit$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y - solve(W_tilde, eta)) cbar <- diag(Cinv) yloo[s, ] <- y - g / cbar sdloo[s, ] <- sqrt(1 / cbar) loglik[s, ] <- dnorm(y, yloo[s, ], sdloo[s, ], log = TRUE) } # use loo for psis smoothing log_ratios <- -loglik psis_result <- psis(log_ratios) ``` The quality of the PSIS-LOO approximation can be investigated graphically by plotting the Pareto-k estimate for each observation. Ideally, they should not exceed $0.5$, but in practice the algorithm turns out to be robust up to values of $0.7$ (Vehtari et al, 2017, 2019). In the plot below, we see that the fourth observation is problematic and so may reduce the accuracy of the LOO-CV approximation. ```{r plot, cache = FALSE} plot(psis_result, label_points = TRUE) ``` We can also check that the conditional leave-one-out predictive distribution equations work correctly, for instance, using the last posterior draw: ```{r checklast, cache = FALSE} yloo_sub <- yloo[S, ] sdloo_sub <- sdloo[S, ] df <- data.frame( y = y, yloo = yloo_sub, ymin = yloo_sub - sdloo_sub * 2, ymax = yloo_sub + sdloo_sub * 2 ) ggplot(data=df, aes(x = y, y = yloo, ymin = ymin, ymax = ymax)) + geom_errorbar( width = 1, color = "skyblue3", position = position_jitter(width = 0.25) ) + geom_abline(color = "gray30", size = 1.2) + geom_point() ``` Finally, we use PSIS-LOO to approximate the expected log predictive density (ELPD) for new data, which we will validate using exact LOO-CV in the upcoming section. ```{r psisloo} (psis_loo <- loo(loglik)) ``` ### Exact LOO-CV Exact LOO-CV for the above example is somewhat more involved, as we need to re-fit the model $N$ times and each time model the held-out data point as a parameter. First, we create an empty dummy model that we will update below as we loop over the observations. ```{r fit_dummy, cache = TRUE} # see help("mi", "brms") for details on the mi() usage fit_dummy <- brm( CRIME | mi() ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 0 ) ``` Next, we fit the model $N$ times, each time leaving out a single observation and then computing the log predictive density for that observation. For obvious reasons, this takes much longer than the approximation we computed above, but it is necessary in order to validate the approximate LOO-CV method. Thanks to the PSIS-LOO approximation, in general doing these slow exact computations can be avoided. ```{r exact-loo-cv, results="hide", message=FALSE, warning=FALSE, cache = TRUE} S <- 500 res <- vector("list", N) loglik <- matrix(nrow = S, ncol = N) for (i in seq_len(N)) { dat_mi <- COL.OLD dat_mi$CRIME[i] <- NA fit_i <- update(fit_dummy, newdata = dat_mi, # just for vignette chains = 1, iter = S * 2) posterior <- as.data.frame(fit_i) yloo <- sdloo <- rep(NA, S) for (s in seq_len(S)) { p <- posterior[s, ] y_miss_i <- y y_miss_i[i] <- p$Ymi eta <- p$b_Intercept + p$b_INC * fit_i$data$INC + p$b_HOVAL * fit_i$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y_miss_i - solve(W_tilde, eta)) cbar <- diag(Cinv); yloo[s] <- y_miss_i[i] - g[i] / cbar[i] sdloo[s] <- sqrt(1 / cbar[i]) loglik[s, i] <- dnorm(y[i], yloo[s], sdloo[s], log = TRUE) } ypred <- rnorm(S, yloo, sdloo) res[[i]] <- data.frame(y = c(posterior$Ymi, ypred)) res[[i]]$type <- rep(c("pp", "loo"), each = S) res[[i]]$obs <- i } res <- do.call(rbind, res) ``` A first step in the validation of the pointwise predictive density is to compare the distribution of the implied response values for the left-out observation to the distribution of the $y_i^{\mathrm{mis}}$ posterior-predictive values estimated as part of the model. If the pointwise predictive density is correct, the two distributions should match very closely (up to sampling error). In the plot below, we overlay these two distributions for the first four observations and see that they match very closely (as is the case for all $49$ observations of in this example). ```{r yplots, cache = FALSE, fig.width=10, out.width="95%", fig.asp = 0.3} res_sub <- res[res$obs %in% 1:4, ] ggplot(res_sub, aes(y, fill = type)) + geom_density(alpha = 0.6) + facet_wrap("obs", scales = "fixed", ncol = 4) ``` In the final step, we compute the ELPD based on the exact LOO-CV and compare it to the approximate PSIS-LOO result computed earlier. ```{r loo_exact, cache=FALSE} log_mean_exp <- function(x) { # more stable than log(mean(exp(x))) max_x <- max(x) max_x + log(sum(exp(x - max_x))) - log(length(x)) } exact_elpds <- apply(loglik, 2, log_mean_exp) exact_elpd <- sum(exact_elpds) round(exact_elpd, 1) ``` The results of the approximate and exact LOO-CV are similar but not as close as we would expect if there were no problematic observations. We can investigate this issue more closely by plotting the approximate against the exact pointwise ELPD values. ```{r compare, fig.height=5} df <- data.frame( approx_elpd = psis_loo$pointwise[, "elpd_loo"], exact_elpd = exact_elpds ) ggplot(df, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + geom_point(data = df[4, ], size = 3, color = "red3") + xlab("Approximate elpds") + ylab("Exact elpds") + coord_fixed(xlim = c(-16, -3), ylim = c(-16, -3)) ``` In the plot above the fourth data point ---the observation flagged as problematic by the PSIS-LOO approximation--- is colored in red and is the clear outlier. Otherwise, the correspondence between the exact and approximate values is strong. In fact, summing over the pointwise ELPD values and leaving out the fourth observation yields practically equivalent results for approximate and exact LOO-CV: ````{r pt4} without_pt_4 <- c( approx = sum(psis_loo$pointwise[-4, "elpd_loo"]), exact = sum(exact_elpds[-4]) ) round(without_pt_4, 1) ``` From this we can conclude that the difference we found when including *all* observations does not indicate a bug in our implementation of the approximate LOO-CV but rather a violation of its assumptions. # Working with Stan directly So far, we have specified the models in brms and only used Stan implicitely behind the scenes. This allowed us to focus on the primary purpose of validating approximate LOO-CV for non-factorized models. However, we would also like to show how everything can be set up in Stan directly. The Stan code brms generates is human readable and so we can use it to learn some of the essential aspects of Stan and the particular model we are implementing. The Stan program below is a slightly modified version of the code extracted via `stancode(fit_dummy)`: ```{r brms-stan-code, eval=FALSE} // generated with brms 2.2.0 functions { /** * Normal log-pdf for spatially lagged responses * * @param y Vector of response values. * @param mu Mean parameter vector. * @param sigma Positive scalar residual standard deviation. * @param rho Positive scalar autoregressive parameter. * @param W Spatial weight matrix. * * @return A scalar to be added to the log posterior. */ real normal_lagsar_lpdf(vector y, vector mu, real sigma, real rho, matrix W) { int N = rows(y); real inv_sigma2 = 1 / square(sigma); matrix[N, N] W_tilde = -rho * W; vector[N] half_pred; for (n in 1:N) W_tilde[n, n] += 1; half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - 0.5 * dot_self(half_pred) * inv_sigma2; } } data { int N; // total number of observations vector[N] Y; // response variable int Nmi; // number of missings int Jmi[Nmi]; // positions of missings int K; // number of population-level effects matrix[N, K] X; // population-level design matrix matrix[N, N] W; // spatial weight matrix int prior_only; // should the likelihood be ignored? } transformed data { int Kc = K - 1; matrix[N, K - 1] Xc; // centered version of X vector[K - 1] means_X; // column means of X before centering for (i in 2:K) { means_X[i - 1] = mean(X[, i]); Xc[, i - 1] = X[, i] - means_X[i - 1]; } } parameters { vector[Nmi] Ymi; // estimated missings vector[Kc] b; // population-level effects real temp_Intercept; // temporary intercept real sigma; // residual SD real lagsar; // SAR parameter } transformed parameters { } model { vector[N] Yl = Y; vector[N] mu = Xc * b + temp_Intercept; Yl[Jmi] = Ymi; // priors including all constants target += student_t_lpdf(temp_Intercept | 3, 34, 17); target += student_t_lpdf(sigma | 3, 0, 17) - 1 * student_t_lccdf(0 | 3, 0, 17); // likelihood including all constants if (!prior_only) { target += normal_lagsar_lpdf(Yl | mu, sigma, lagsar, W); } } generated quantities { // actual population-level intercept real b_Intercept = temp_Intercept - dot_product(means_X, b); } ``` Here we want to focus on two aspects of the Stan code. First, because there is no built-in function in Stan that calculates the log-likelihood for the lag-SAR model, we define a new `normal_lagsar_lpdf` function in the `functions` block of the Stan program. This is the same function we showed earlier in the vignette and it can be used to compute the log-likelihood in an efficient and numerically stable way. The `_lpdf` suffix used in the function name informs Stan that this is a log probability density function. Second, this Stan program nicely illustrates how to set up missing value imputation. Instead of just computing the log-likelihood for the observed responses `Y`, we define a new variable `Yl` which is equal to `Y` if the reponse is observed and equal to `Ymi` if the response is missing. The latter is in turn defined as a parameter and thus estimated along with all other paramters of the model. More details about missing value imputation in Stan can be found in the *Missing Data & Partially Known Parameters* section of the [Stan manual](https://mc-stan.org/users/documentation/index.html). The Stan code extracted from brms is not only helpful when learning Stan, but can also drastically speed up the specification of models that are not support by brms. If brms can fit a model similar but not identical to the desired model, we can let brms generate the Stan program for the similar model and then mold it into the program that implements the model we actually want to fit. Rather than calling `stancode()`, which requires an existing fitted model object, we recommend using `make_stancode()` and specifying the `save_model` argument to write the Stan program to a file. The corresponding data can be prepared with `make_standata()` and then manually amended if needed. Once the code and data have been edited, they can be passed to RStan's `stan()` function via the `file` and `data` arguments. # Conclusion In summary, we have shown how to set up and validate approximate and exact LOO-CV for non-factorized multivariate normal models using Stan with the **brms** and **loo** packages. Although we focused on the particular example of a spatial SAR model, the presented recipe applies more generally to models that can be expressed in terms of a multivariate normal likelihood.
# References Anselin L. (1988). *Spatial econometrics: methods and models*. Dordrecht: Kluwer Academic. Bürkner P. C., Gabry J., & Vehtari A. (2020). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-t models. *Computational Statistics*, \doi:10.1007/s00180-020-01045-4. [ArXiv preprint](https://arxiv.org/abs/1810.10559). Sundararajan S. & Keerthi S. S. (2001). Predictive approaches for choosing hyperparameters in Gaussian processes. *Neural Computation*, 13(5), 1103--1118. Vehtari A., Mononen T., Tolvanen V., Sivula T., & Winther O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. *Journal of Machine Learning Research*, 17(103), 1--38. [Online](https://jmlr.org/papers/v17/14-540.html). Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/inst/doc/loo2-weights.html0000644000176200001440000014534114411465510016007 0ustar liggesusers Bayesian Stacking and Pseudo-BMA weights using the loo package

Bayesian Stacking and Pseudo-BMA weights using the loo package

Aki Vehtari and Jonah Gabry

2023-03-30

Introduction

This vignette demonstrates the new functionality in loo v2.0.0 for Bayesian stacking and Pseudo-BMA weighting. In this vignette we can’t provide all of the necessary background on this topic, so we encourage readers to refer to the paper

  • Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, :10.1214/17-BA1091. Online

which provides important details on the methods demonstrated in this vignette. Here we just quote from the abstract of the paper:

Abstract: Bayesian model averaging is flawed in the \(\mathcal{M}\)-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions. We extend the utility function to any proper scoring rule and use Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions. We compare stacking of predictive distributions to several alternatives: stacking of means, Bayesian model averaging (BMA), Pseudo-BMA, and a variant of Pseudo-BMA that is stabilized using the Bayesian bootstrap. Based on simulations and real-data applications, we recommend stacking of predictive distributions, with bootstrapped-Pseudo-BMA as an approximate alternative when computation cost is an issue.

Ideally, we would avoid the Bayesian model combination problem by extending the model to include the separate models as special cases, and preferably as a continuous expansion of the model space. For example, instead of model averaging over different covariate combinations, all potentially relevant covariates should be included in a predictive model (for causal analysis more care is needed) and a prior assumption that only some of the covariates are relevant can be presented with regularized horseshoe prior (Piironen and Vehtari, 2017a). For variable selection we recommend projective predictive variable selection (Piironen and Vehtari, 2017a; projpred package).

To demonstrate how to use loo package to compute Bayesian stacking and Pseudo-BMA weights, we repeat two simple model averaging examples from Chapters 6 and 10 of Statistical Rethinking by Richard McElreath. In Statistical Rethinking WAIC is used to form weights which are similar to classical “Akaike weights”. Pseudo-BMA weighting using PSIS-LOO for computation is close to these WAIC weights, but named after the Pseudo Bayes Factor by Geisser and Eddy (1979). As discussed below, in general we prefer using stacking rather than WAIC weights or the similar pseudo-BMA weights.

Setup

In addition to the loo package we will also load the rstanarm package for fitting the models.

library(rstanarm)
library(loo)

Example: Primate milk

In Statistical Rethinking, McElreath describes the data for the primate milk example as follows:

A popular hypothesis has it that primates with larger brains produce more energetic milk, so that brains can grow quickly. … The question here is to what extent energy content of milk, measured here by kilocalories, is related to the percent of the brain mass that is neocortex. … We’ll end up needing female body mass as well, to see the masking that hides the relationships among the variables.

data(milk)
d <- milk[complete.cases(milk),]
d$neocortex <- d$neocortex.perc /100
str(d)
'data.frame':   17 obs. of  9 variables:
 $ clade         : Factor w/ 4 levels "Ape","New World Monkey",..: 4 2 2 2 2 2 2 2 3 3 ...
 $ species       : Factor w/ 29 levels "A palliata","Alouatta seniculus",..: 11 2 1 6 27 5 3 4 21 19 ...
 $ kcal.per.g    : num  0.49 0.47 0.56 0.89 0.92 0.8 0.46 0.71 0.68 0.97 ...
 $ perc.fat      : num  16.6 21.2 29.7 53.4 50.6 ...
 $ perc.protein  : num  15.4 23.6 23.5 15.8 22.3 ...
 $ perc.lactose  : num  68 55.2 46.9 30.8 27.1 ...
 $ mass          : num  1.95 5.25 5.37 2.51 0.68 0.12 0.47 0.32 1.55 3.24 ...
 $ neocortex.perc: num  55.2 64.5 64.5 67.6 68.8 ...
 $ neocortex     : num  0.552 0.645 0.645 0.676 0.688 ...

We repeat the analysis in Chapter 6 of Statistical Rethinking using the following four models (here we use the default weakly informative priors in rstanarm, while flat priors were used in Statistical Rethinking).

fit1 <- stan_glm(kcal.per.g ~ 1, data = d, seed = 2030)
fit2 <- update(fit1, formula = kcal.per.g ~ neocortex)
fit3 <- update(fit1, formula = kcal.per.g ~ log(mass))
fit4 <- update(fit1, formula = kcal.per.g ~ neocortex + log(mass))

McElreath uses WAIC for model comparison and averaging, so we’ll start by also computing WAIC for these models so we can compare the results to the other options presented later in the vignette. The loo package provides waic methods for log-likelihood arrays, matrices and functions. Since we fit our model with rstanarm we can use the waic method provided by the rstanarm package (a wrapper around waic from the loo package), which allows us to just pass in our fitted model objects instead of first extracting the log-likelihood values.

waic1 <- waic(fit1)
waic2 <- waic(fit2)
waic3 <- waic(fit3)
Warning: 
1 (5.9%) p_waic estimates greater than 0.4. We recommend trying loo instead.
waic4 <- waic(fit4)
Warning: 
2 (11.8%) p_waic estimates greater than 0.4. We recommend trying loo instead.
waics <- c(
  waic1$estimates["elpd_waic", 1],
  waic2$estimates["elpd_waic", 1],
  waic3$estimates["elpd_waic", 1],
  waic4$estimates["elpd_waic", 1]
)

We get some warnings when computing WAIC for models 3 and 4, indicating that we shouldn’t trust the WAIC weights we will compute later. Following the recommendation in the warning, we next use the loo methods to compute PSIS-LOO instead. The loo package provides loo methods for log-likelihood arrays, matrices, and functions, but since we fit our model with rstanarm we can just pass the fitted model objects directly and rstanarm will extract the needed values to pass to the loo package. (Like rstanarm, some other R packages for fitting Stan models, e.g. brms, also provide similar methods for interfacing with the loo package.)

# note: the loo function accepts a 'cores' argument that we recommend specifying
# when working with bigger datasets

loo1 <- loo(fit1)
loo2 <- loo(fit2)
loo3 <- loo(fit3)
loo4 <- loo(fit4)
lpd_point <- cbind(
  loo1$pointwise[,"elpd_loo"], 
  loo2$pointwise[,"elpd_loo"],
  loo3$pointwise[,"elpd_loo"], 
  loo4$pointwise[,"elpd_loo"]
)

With loo we don’t get any warnings for models 3 and 4, but for illustration of good results, we display the diagnostic details for these models anyway.

print(loo3)

Computed from 4000 by 17 log-likelihood matrix

         Estimate  SE
elpd_loo      4.5 2.3
p_loo         2.1 0.5
looic        -9.1 4.6
------
Monte Carlo SE of elpd_loo is 0.0.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     15    88.2%   1658      
 (0.5, 0.7]   (ok)        2    11.8%   768       
   (0.7, 1]   (bad)       0     0.0%   <NA>      
   (1, Inf)   (very bad)  0     0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.
print(loo4)

Computed from 4000 by 17 log-likelihood matrix

         Estimate  SE
elpd_loo      8.4 2.8
p_loo         3.3 0.9
looic       -16.8 5.5
------
Monte Carlo SE of elpd_loo is 0.1.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     16    94.1%   532       
 (0.5, 0.7]   (ok)        1     5.9%   404       
   (0.7, 1]   (bad)       0     0.0%   <NA>      
   (1, Inf)   (very bad)  0     0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

One benefit of PSIS-LOO over WAIC is better diagnostics. Here for both models 3 and 4 all \(k<0.7\) and the Monte Carlo SE of elpd_loo is 0.1 or less, and we can expect the model comparison to be reliable.

Next we compute and compare 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights.

waic_wts <- exp(waics) / sum(exp(waics))
pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE)
pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE
stacking_wts <- stacking_weights(lpd_point)
round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2)
       waic_wts pbma_wts pbma_BB_wts stacking_wts
model1     0.01     0.02        0.07         0.01
model2     0.01     0.01        0.04         0.00
model3     0.02     0.02        0.04         0.00
model4     0.96     0.95        0.85         0.99

With all approaches Model 4 with neocortex and log(mass) gets most of the weight. Based on theory, Pseudo-BMA weights without Bayesian bootstrap should be close to WAIC weights, and we can also see that here. Pseudo-BMA+ weights with Bayesian bootstrap provide more cautious weights further away from 0 and 1 (see Yao et al. (2018) for a discussion of why this can be beneficial and results from related experiments). In this particular example, the Bayesian stacking weights are not much different from the other weights.

One of the benefits of stacking is that it manages well if there are many similar models. Consider for example that there could be many irrelevant covariates that when included would produce a similar model to one of the existing models. To emulate this situation here we simply copy the first model a bunch of times, but you can imagine that instead we would have ten alternative models with about the same predictive performance. WAIC weights for such a scenario would be close to the following:

waic_wts_demo <- 
  exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)]) /
  sum(exp(waics[c(1,1,1,1,1,1,1,1,1,1,2,3,4)]))
round(waic_wts_demo, 3)
 [1] 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.006 0.016
[13] 0.847

Notice how much the weight for model 4 is lowered now that more models similar to model 1 (or in this case identical) have been added. Both WAIC weights and Pseudo-BMA approaches first estimate the predictive performance separately for each model and then compute weights based on estimated relative predictive performances. Similar models share similar weights so the weights of other models must be reduced for the total sum of the weights to remain the same.

On the other hand, stacking optimizes the weights jointly, allowing for the very similar models (in this toy example repeated models) to share their weight while more unique models keep their original weights. In our example we can see this difference clearly:

stacking_weights(lpd_point[,c(1,1,1,1,1,1,1,1,1,1,2,3,4)])
Method: stacking
------
        weight
model1  0.001 
model2  0.001 
model3  0.001 
model4  0.001 
model5  0.001 
model6  0.001 
model7  0.001 
model8  0.001 
model9  0.001 
model10 0.001 
model11 0.000 
model12 0.000 
model13 0.987

Using stacking, the weight for the best model stays essentially unchanged.

Example: Oceanic tool complexity

Another example we consider is the Kline oceanic tool complexity data, which McElreath describes as follows:

Different historical island populations possessed tool kits of different size. These kits include fish hooks, axes, boats, hand plows, and many other types of tools. A number of theories predict that larger populations will both develop and sustain more complex tool kits. … It’s also suggested that contact rates among populations effectively increases population [sic, probably should be tool kit] size, as it’s relevant to technological evolution.

We build models predicting the total number of tools given the log population size and the contact rate (high vs. low).

data(Kline)
d <- Kline
d$log_pop <- log(d$population)
d$contact_high <- ifelse(d$contact=="high", 1, 0)
str(d)
'data.frame':   10 obs. of  7 variables:
 $ culture     : Factor w/ 10 levels "Chuuk","Hawaii",..: 4 7 6 10 3 9 1 5 8 2
 $ population  : int  1100 1500 3600 4791 7400 8000 9200 13000 17500 275000
 $ contact     : Factor w/ 2 levels "high","low": 2 2 2 1 1 1 1 2 1 2
 $ total_tools : int  13 22 24 43 33 19 40 28 55 71
 $ mean_TU     : num  3.2 4.7 4 5 5 4 3.8 6.6 5.4 6.6
 $ log_pop     : num  7 7.31 8.19 8.47 8.91 ...
 $ contact_high: num  0 0 0 1 1 1 1 0 1 0

We start with a Poisson regression model with the log population size, the contact rate, and an interaction term between them (priors are informative priors as in Statistical Rethinking).

fit10 <-
  stan_glm(
    total_tools ~ log_pop + contact_high + log_pop * contact_high,
    family = poisson(link = "log"),
    data = d,
    prior = normal(0, 1, autoscale = FALSE),
    prior_intercept = normal(0, 100, autoscale = FALSE),
    seed = 2030
  )

Before running other models, we check whether Poisson is good choice as the conditional observation model.

loo10 <- loo(fit10)
Warning: Found 2 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 2 times to compute the ELPDs for the problematic observations directly.
print(loo10)

Computed from 4000 by 10 log-likelihood matrix

         Estimate   SE
elpd_loo    -40.9  6.0
p_loo         5.8  2.0
looic        81.8 12.0
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     5     50.0%   1116      
 (0.5, 0.7]   (ok)       3     30.0%   92        
   (0.7, 1]   (bad)      2     20.0%   62        
   (1, Inf)   (very bad) 0      0.0%   <NA>      
See help('pareto-k-diagnostic') for details.

We get at least one observation with \(k>0.7\) and the estimated effective number of parameters p_loo is larger than the total number of parameters in the model. This indicates that Poisson might be too narrow. A negative binomial model might be better, but with so few observations it is not so clear.

We can compute LOO more accurately by running Stan again for the leave-one-out folds with high \(k\) estimates. When using rstanarm this can be done by specifying the k_threshold argument:

loo10 <- loo(fit10, k_threshold=0.7)
2 problematic observation(s) found.
Model will be refit 2 times.

Fitting model 1 out of 2 (leaving out observation 9)

Fitting model 2 out of 2 (leaving out observation 10)
print(loo10)

Computed from 4000 by 10 log-likelihood matrix

         Estimate   SE
elpd_loo    -41.2  6.0
p_loo         6.1  2.0
looic        82.4 12.0
------
Monte Carlo SE of elpd_loo is 0.2.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     5     62.5%   1116      
 (0.5, 0.7]   (ok)       3     37.5%   92        
   (0.7, 1]   (bad)      0      0.0%   <NA>      
   (1, Inf)   (very bad) 0      0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

In this case we see that there is not much difference, and thus it is relatively safe to continue.

As a comparison we also compute WAIC:

waic10 <- waic(fit10)
Warning: 
4 (40.0%) p_waic estimates greater than 0.4. We recommend trying loo instead.
print(waic10)

Computed from 4000 by 10 log-likelihood matrix

          Estimate   SE
elpd_waic    -40.2  5.9
p_waic         5.0  1.8
waic          80.3 11.8

4 (40.0%) p_waic estimates greater than 0.4. We recommend trying loo instead.

The WAIC computation is giving warnings and the estimated ELPD is slightly more optimistic. We recommend using the PSIS-LOO results instead.

To assess whether the contact rate and interaction term are useful, we can make a comparison to models without these terms.

fit11 <- update(fit10, formula = total_tools ~ log_pop + contact_high)
fit12 <- update(fit10, formula = total_tools ~ log_pop)
(loo11 <- loo(fit11))

Computed from 4000 by 10 log-likelihood matrix

         Estimate   SE
elpd_loo    -39.7  5.8
p_loo         4.4  1.6
looic        79.4 11.6
------
Monte Carlo SE of elpd_loo is 0.1.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     8     80.0%   351       
 (0.5, 0.7]   (ok)       2     20.0%   308       
   (0.7, 1]   (bad)      0      0.0%   <NA>      
   (1, Inf)   (very bad) 0      0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.
(loo12 <- loo(fit12))
Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly.

Computed from 4000 by 10 log-likelihood matrix

         Estimate  SE
elpd_loo    -42.5 4.7
p_loo         4.1 1.1
looic        85.0 9.4
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     9     90.0%   649       
 (0.5, 0.7]   (ok)       0      0.0%   <NA>      
   (0.7, 1]   (bad)      1     10.0%   68        
   (1, Inf)   (very bad) 0      0.0%   <NA>      
See help('pareto-k-diagnostic') for details.
loo11 <- loo(fit11, k_threshold=0.7)
All pareto_k estimates below user-specified threshold of 0.7. 
Returning loo object.
loo12 <- loo(fit12, k_threshold=0.7)
1 problematic observation(s) found.
Model will be refit 1 times.

Fitting model 1 out of 1 (leaving out observation 10)
lpd_point <- cbind(
  loo10$pointwise[, "elpd_loo"], 
  loo11$pointwise[, "elpd_loo"], 
  loo12$pointwise[, "elpd_loo"]
)

For comparison we’ll also compute WAIC values for these additional models:

waic11 <- waic(fit11)
Warning: 
3 (30.0%) p_waic estimates greater than 0.4. We recommend trying loo instead.
waic12 <- waic(fit12)
Warning: 
5 (50.0%) p_waic estimates greater than 0.4. We recommend trying loo instead.
waics <- c(
  waic10$estimates["elpd_waic", 1], 
  waic11$estimates["elpd_waic", 1], 
  waic12$estimates["elpd_waic", 1]
)

The WAIC computation again gives warnings, and we recommend using PSIS-LOO instead.

Finally, we compute 1) WAIC weights, 2) Pseudo-BMA weights without Bayesian bootstrap, 3) Pseudo-BMA+ weights with Bayesian bootstrap, and 4) Bayesian stacking weights.

waic_wts <- exp(waics) / sum(exp(waics))
pbma_wts <- pseudobma_weights(lpd_point, BB=FALSE)
pbma_BB_wts <- pseudobma_weights(lpd_point) # default is BB=TRUE
stacking_wts <- stacking_weights(lpd_point)
round(cbind(waic_wts, pbma_wts, pbma_BB_wts, stacking_wts), 2)
       waic_wts pbma_wts pbma_BB_wts stacking_wts
model1     0.33     0.18        0.16          0.0
model2     0.63     0.81        0.67          0.8
model3     0.04     0.02        0.17          0.2

All weights favor the second model with the log population and the contact rate. WAIC weights and Pseudo-BMA weights (without Bayesian bootstrap) are similar, while Pseudo-BMA+ is more cautious and closer to stacking weights.

It may seem surprising that Bayesian stacking is giving zero weight to the first model, but this is likely due to the fact that the estimated effect for the interaction term is close to zero and thus models 1 and 2 give very similar predictions. In other words, incorporating the model with the interaction (model 1) into the model average doesn’t improve the predictions at all and so model 1 is given a weight of 0. On the other hand, models 2 and 3 are giving slightly different predictions and thus their combination may be slightly better than either alone. This behavior is related to the repeated similar model illustration in the milk example above.

Simpler coding using loo_model_weights function

Although in the examples above we called the stacking_weights and pseudobma_weights functions directly, we can also use the loo_model_weights wrapper, which takes as its input either a list of pointwise log-likelihood matrices or a list of precomputed loo objects. There are also loo_model_weights methods for stanreg objects (fitted model objects from rstanarm) as well as fitted model objects from other packages (e.g. brms) that do the preparation work for the user (see, e.g., the examples at help("loo_model_weights", package = "rstanarm")).

# using list of loo objects
loo_list <- list(loo10, loo11, loo12)
loo_model_weights(loo_list)
Method: stacking
------
      weight
fit10 0.000 
fit11 0.802 
fit12 0.198 
loo_model_weights(loo_list, method = "pseudobma")
Method: pseudo-BMA+ with Bayesian bootstrap
------
      weight
fit10 0.159 
fit11 0.679 
fit12 0.162 
loo_model_weights(loo_list, method = "pseudobma", BB = FALSE)
Method: pseudo-BMA
------
      weight
fit10 0.175 
fit11 0.805 
fit12 0.020

References

McElreath, R. (2016). Statistical rethinking: A Bayesian course with examples in R and Stan. Chapman & Hall/CRC. http://xcelab.net/rm/statistical-rethinking/

Piironen, J. and Vehtari, A. (2017a). Sparsity information and regularization in the horseshoe and other shrinkage priors. In Electronic Journal of Statistics, 11(2):5018-5051. Online.

Piironen, J. and Vehtari, A. (2017b). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3):711-735. :10.1007/s11222-016-9649-y. Online.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. online, arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian Analysis, :10.1214/17-BA1091. Online.

loo/inst/doc/loo2-mixis.Rmd0000644000176200001440000002204314407123455015241 0ustar liggesusers--- title: "Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models" author: "Luca Silva and Giacomo Zanella" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette shows how to perform Bayesian leave-one-out cross-validation (LOO-CV) using the mixture estimators proposed in the paper [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-$k$ diagnostics and potential instability of PSIS-LOO estimators. For this illustration we consider a high-dimensional Bayesian Logistic regression model applied to the _Voice_ dataset. ## Setup: load packages and set seed ```{r, warnings=FALSE, message=FALSE} library("rstan") library("loo") library("matrixStats") options(mc.cores = parallel::detectCores(), parallel=FALSE) set.seed(24877) ``` ## Model This is the Stan code for a logistic regression model with regularized horseshoe prior. The code includes an if statement to include a code line needed later for the MixIS approach. ```{r stancode_horseshoe} stancode_horseshoe <- " data { int n; int p; int y[n]; matrix [n,p] X; real scale_global; int mixis; } transformed data { real nu_global=1; // degrees of freedom for the half-t priors for tau real nu_local=1; // degrees of freedom for the half-t priors for lambdas // (nu_local = 1 corresponds to the horseshoe) real slab_scale=2;// for the regularized horseshoe real slab_df=100; // for the regularized horseshoe } parameters { vector[p] z; // for non-centered parameterization real tau; // global shrinkage parameter vector [p] lambda; // local shrinkage parameter real caux; } transformed parameters { vector[p] beta; { vector[p] lambda_tilde; // 'truncated' local shrinkage parameter real c = slab_scale * sqrt(caux); // slab scale lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda))); beta = z .* lambda_tilde*tau; } } model { vector[n] means=X*beta; vector[n] log_lik; target += std_normal_lpdf(z); target += student_t_lpdf(lambda | nu_local, 0, 1); target += student_t_lpdf(tau | nu_global, 0, scale_global); target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df); for (index in 1:n) { log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]); } target += sum(log_lik); if (mixis) { target += log_sum_exp(-log_lik); } } generated quantities { vector[n] means=X*beta; vector[n] log_lik; for (index in 1:n) { log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]); } } " ``` ## Dataset The _LSVT Voice Rehabilitation Data Set_ (see [link](https://archive.ics.uci.edu/ml/datasets/LSVT+Voice+Rehabilitation) for details) has $p=312$ covariates and $n=126$ observations with binary response. We construct data list for Stan. ```{r, results='hide', warning=FALSE, message=FALSE, error=FALSE} data(voice) y <- voice$y X <- voice[2:length(voice)] n <- dim(X)[1] p <- dim(X)[2] p0 <- 10 scale_global <- 2*p0/(p-p0)/sqrt(n-1) standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) ``` Note that in our prior specification we divide the prior variance by the number of covariates $p$. This is often done in high-dimensional contexts to have a prior variance for the linear predictors $X\beta$ that remains bounded as $p$ increases. ## PSIS estimators and Pareto-$k$ diagnostics LOO-CV computations are challenging in this context due to high-dimensionality of the parameter space. To show that, we compute PSIS-LOO estimators, which require sampling from the posterior distribution, and inspect the associated Pareto-$k$ diagnostics. ```{r, results='hide', warning=FALSE} chains <- 4 n_iter <- 2000 warm_iter <- 1000 stanmodel <- stan_model(model_code = stancode_horseshoe) fit_post <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0) loo_post <-loo(fit_post) ``` ```{r} print(loo_post) ``` As we can see the diagnostics signal either "bad" or "very bad" Pareto-$k$ values for roughly $15-30\%$ of the observations which is a significant portion of the dataset. ## Mixture estimators We now compute the mixture estimators proposed in Silva and Zanella (2022). These require to sample from the following mixture of leave-one-out posteriors \begin{equation} q_{mix}(\theta) = \frac{\sum_{i=1}^n p(y_{-i}|\theta)p(\theta)}{\sum_{i=1}^np(y_{-i})}\propto p(\theta|y)\cdot \left(\sum_{i=1}^np(y_i|\theta)^{-1}\right). \end{equation} The code to generate a Stan model for the above mixture distribution is the same to the one for the posterior, just enabling one line of code with a _LogSumExp_ contribution to account for the last term in the equation above. ``` if (mixis) { target += log_sum_exp(-log_lik); } ``` We sample from the mixture and collect the log-likelihoods term. ```{r, results='hide', warnings=FALSE} standata$mixis <- 1 fit_mix <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0, pars = "log_lik") log_lik_mix <- extract(fit_mix)$log_lik ``` We now compute the mixture estimators, following the numerically stable implementation in Appendix A.2 of [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). The code below makes use of the package "matrixStats". ```{r} l_common_mix <- rowLogSumExps(-log_lik_mix) log_weights <- -log_lik_mix - l_common_mix elpd_mixis <- logSumExp(-l_common_mix) - rowLogSumExps(t(log_weights)) ``` ## Comparison with benchmark values obtained with long simulations To evaluate the performance of mixture estimators (MixIS) we also generate _benchmark values_, i.e.\ accurate approximations of the LOO predictives $\{p(y_i|y_{-i})\}_{i=1,\dots,n}$, obtained by brute-force sampling from the leave-one-out posteriors directly, getting $90k$ samples from each and discarding the first $10k$ as warmup. This is computationally heavy, hence we have saved the results and we just load them in the current vignette. ```{r} data(voice_loo) elpd_loo <- voice_loo$elpd_loo ``` We can then compute the root mean squared error (RMSE) of the PSIS and mixture estimators relative to such benchmark values. ```{r} elpd_psis <- loo_post$pointwise[,1] print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2))) print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2))) ``` Here mixture estimator provides a reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results). We then compare the overall ELPD estimates with the brute force one. ```{r} elpd_psis <- loo_post$pointwise[,1] print(paste("ELPD (PSIS)=",round(sum(elpd_psis),2))) print(paste("ELPD (MixIS)=",round(sum(elpd_mixis),2))) print(paste("ELPD (brute force)=",round(sum(elpd_loo),2))) ``` In this example, MixIS provides a more accurate ELPD estimate closer to the brute force estimate, while PSIS severely overestimates the ELPD. Note that low accuracy of the PSIS ELPD estimate is expected in this example given the large number of large Pareto-$k$ values. In this example, the accuracy of MixIS estimate will also improve with bigger MCMC sample size. More generally, mixture estimators can be useful in situations where standard PSIS estimators struggle and return many large Pareto-$k$ values. In these contexts MixIS often provides more accurate LOO-CV and ELPD estimates with a single sampling routine (i.e. with a cost comparable to sampling from the original posterior). ## References Silva L. and Zanella G. (2022). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Preprint at [arXiv:2209.09190](https://arxiv.org/abs/2209.09190) Vehtari A., Gelman A., and Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. Preprint at [arXiv:1507.04544](https://arxiv.org/abs/1507.04544) Vehtari A., Simpson D., Gelman A., Yao Y., and Gabry J. (2022). Pareto smoothed importance sampling. Preprint at [arXiv:1507.02646](https://arxiv.org/abs/1507.02646) loo/inst/doc/loo2-example.R0000644000176200001440000000463614411455345015233 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----setup, message=FALSE----------------------------------------------------- library("rstanarm") library("bayesplot") library("loo") ## ----data--------------------------------------------------------------------- # the 'roaches' data frame is included with the rstanarm package data(roaches) str(roaches) # rescale to units of hundreds of roaches roaches$roach1 <- roaches$roach1 / 100 ## ----count-roaches-mcmc, results="hide"--------------------------------------- fit1 <- stan_glm( formula = y ~ roach1 + treatment + senior, offset = log(exposure2), data = roaches, family = poisson(link = "log"), prior = normal(0, 2.5, autoscale = TRUE), prior_intercept = normal(0, 5, autoscale = TRUE), seed = 12345 ) ## ----loo1--------------------------------------------------------------------- loo1 <- loo(fit1, save_psis = TRUE) ## ----print-loo1--------------------------------------------------------------- print(loo1) ## ----plot-loo1, out.width = "70%"--------------------------------------------- plot(loo1) ## ----ppc_loo_pit_overlay------------------------------------------------------ yrep <- posterior_predict(fit1) ppc_loo_pit_overlay( y = roaches$y, yrep = yrep, lw = weights(loo1$psis_object) ) ## ----count-roaches-negbin, results="hide"------------------------------------- fit2 <- update(fit1, family = neg_binomial_2) ## ----loo2--------------------------------------------------------------------- loo2 <- loo(fit2, save_psis = TRUE, cores = 2) print(loo2) ## ----plot-loo2---------------------------------------------------------------- plot(loo2, label_points = TRUE) ## ----reloo-------------------------------------------------------------------- if (any(pareto_k_values(loo2) > 0.7)) { loo2 <- loo(fit2, save_psis = TRUE, k_threshold = 0.7) } print(loo2) ## ----ppc_loo_pit_overlay-negbin----------------------------------------------- yrep <- posterior_predict(fit2) ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2$psis_object)) ## ----loo_compare-------------------------------------------------------------- loo_compare(loo1, loo2) loo/inst/doc/loo2-large-data.Rmd0000644000176200001440000005036613764522153016125 0ustar liggesusers--- title: "Using Leave-one-out cross-validation for large data" author: "Mans Magnusson, Paul Bürkner, Aki Vehtari and Jonah Gabry" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r settings, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to do leave-one-out cross-validation for large data using the __loo__ package and Stan. There are two approaches covered: LOO with subsampling and LOO using approximations to posterior distributions. Some sections from this vignette are excerpted from the papers * Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. [arXiv preprint arXiv:2001.00980](https://arxiv.org/abs/2001.00980). * Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 [online](http://proceedings.mlr.press/v97/magnusson19a.html), [arXiv preprint arXiv:1904.10679](https://arxiv.org/abs/1904.10679). * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). * Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). which provide important background for understanding the methods implemented in the package. # Setup In addition to the __loo__ package, we'll also be using __rstan__: ```{r setup, message=FALSE} library("rstan") library("loo") set.seed(4711) ``` # Example: Well water in Bangladesh We will use the same example as in the vignette [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html). See that vignette for a description of the problem and data. The sample size in this example is only $N=3020$, which is not large enough to _require_ the special methods for large data described in this vignette, but is sufficient for demonstration purposes in this tutorial. ## Coding the Stan model Here is the Stan code for fitting the logistic regression model, which we save in a file called `logistic.stan`: ``` // save in `logistic.stan` data { int N; // number of data points int P; // number of predictors (including intercept) matrix[N,P] X; // predictors (including 1s for intercept) int y[N]; // binary outcome } parameters { vector[P] beta; } model { beta ~ normal(0, 1); y ~ bernoulli_logit(X * beta); } ``` Importantly, unlike the general approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we do _not_ compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. Here we are assuming we have a large data set (larger than the one we're actually using in this demonstration) and so it is preferable to instead define a function in R to compute the log-likelihood for each data point when needed rather than storing all of the log-likelihood values in memory. The log-likelihood in R can be coded as follows: ```{r llfun_logistic} # we'll add an argument log to toggle whether this is a log-likelihood or # likelihood function. this will be useful later in the vignette. llfun_logistic <- function(data_i, draws, log = TRUE) { x_i <- as.matrix(data_i[, which(grepl(colnames(data_i), pattern = "X")), drop=FALSE]) logit_pred <- draws %*% t(x_i) dbinom(x = data_i$y, size = 1, prob = 1/(1 + exp(-logit_pred)), log = log) } ``` The function `llfun_logistic()` needs to have arguments `data_i` and `draws`. Below we will test that the function is working by using the `loo_i()` function. ## Fitting the model with RStan Next we fit the model in Stan using the **rstan** package: ```{r, eval=FALSE} # Prepare data url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" wells <- read.table(url) wells$dist100 <- with(wells, dist / 100) X <- model.matrix(~ dist100 + arsenic, wells) standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # Compile stan_mod <- stan_model("logistic.stan") # Fit model fit_1 <- sampling(stan_mod, data = standata, seed = 4711) print(fit_1, pars = "beta") ``` ``` mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat beta[1] 0.00 0 0.08 -0.15 -0.05 0.00 0.06 0.16 1933 1 beta[2] -0.89 0 0.10 -1.09 -0.96 -0.89 -0.82 -0.69 2332 1 beta[3] 0.46 0 0.04 0.38 0.43 0.46 0.49 0.54 2051 1 ``` Before we move on to computing LOO we can now test that the log-likelihood function we wrote is working as it should. The `loo_i()` function is a helper function that can be used to test a log-likelihood function on a single observation. ```{r, eval=FALSE} # used for draws argument to loo_i parameter_draws_1 <- extract(fit_1)$beta # used for data argument to loo_i stan_df_1 <- as.data.frame(standata) # compute relative efficiency (this is slow and optional but is recommended to allow # for adjusting PSIS effective sample size based on MCMC effective sample size) r_eff <- relative_eff(llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_1, draws = parameter_draws_1, cores = 2) loo_i(i = 1, llfun_logistic, r_eff = r_eff, data = stan_df_1, draws = parameter_draws_1) ``` ``` $pointwise elpd_loo mcse_elpd_loo p_loo looic influence_pareto_k 1 -0.3314552 0.0002887608 0.0003361772 0.6629103 -0.05679886 ... ``` # Approximate LOO-CV using PSIS-LOO and subsampling We can then use the `loo_subsample()` function to compute the efficient PSIS-LOO approximation to exact LOO-CV using subsampling: ```{r, eval=FALSE} set.seed(4711) loo_ss_1 <- loo_subsample( llfun_logistic, observations = 100, # take a subsample of size 100 cores = 2, # these next objects were computed above r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1 ) print(loo_ss_1) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.5 15.6 0.3 p_loo 3.1 0.1 0.4 looic 3936.9 31.2 0.6 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` The `loo_subsample()` function creates an object of class `psis_loo_ss`, that inherits from `psis_loo, loo` (the classes of regular `loo` objects). The printed output above shows the estimates $\widehat{\mbox{elpd}}_{\rm loo}$ (expected log predictive density), $\widehat{p}_{\rm loo}$ (effective number of parameters), and ${\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}$ (the LOO information criterion). Unlike when using `loo()`, when using `loo_subsample()` there is an additional column giving the "subsampling SE", which reflects the additional uncertainty due to the subsampling used. The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the $k$ parameter is explained in `help('pareto-k-diagnostic')` and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case, the message tells us that all of the estimates for $k$ are fine _for this given subsample_. ## Adding additional subsamples If we are not satisfied with the subsample size (i.e., the accuracy) we can simply add more samples until we are satisfied using the `update()` method. ```{r, eval=FALSE} set.seed(4711) loo_ss_1b <- update( loo_ss_1, observations = 200, # subsample 200 instead of 100 r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1 ) print(loo_ss_1b) ``` ``` Computed from 4000 by 200 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.3 15.6 0.2 p_loo 3.2 0.1 0.4 looic 3936.7 31.2 0.5 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` ## Specifying estimator and sampling method The performance relies on two components: the estimation method and the approximation used for the elpd. See the documentation for `loo_subsample()` more information on which estimators and approximations are implemented. The default implementation is using the point log predictive density evaluated at the mean of the posterior (`loo_approximation="plpd"`) and the difference estimator (`estimator="diff_srs"`). This combination has a focus on fast inference. But we can easily use other estimators as well as other elpd approximations, for example: ```{r, eval=FALSE} set.seed(4711) loo_ss_1c <- loo_subsample( x = llfun_logistic, r_eff = r_eff, draws = parameter_draws_1, data = stan_df_1, observations = 100, estimator = "hh_pps", # use Hansen-Hurwitz loo_approximation = "lpd", # use lpd instead of plpd loo_approximation_draws = 100, cores = 2 ) print(loo_ss_1c) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.9 15.4 0.5 p_loo 3.5 0.2 0.5 looic 3937.9 30.7 1.1 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` See the documentation and references for `loo_subsample()` for details on the implemented approximations. # Approximate LOO-CV using PSIS-LOO with posterior approximations Using posterior approximations, such as variational inference and Laplace approximations, can further speed-up LOO-CV for large data. Here we demonstrate using a Laplace approximation in Stan. ```{r, eval=FALSE} fit_laplace <- optimizing(stan_mod, data = standata, draws = 2000, importance_resampling = TRUE) parameter_draws_laplace <- fit_laplace$theta_tilde # draws from approximate posterior log_p <- fit_laplace$log_p # log density of the posterior log_g <- fit_laplace$log_g # log density of the approximation ``` Using the posterior approximation we can then do LOO-CV by correcting for the posterior approximation when we compute the elpd. To do this we use the `loo_approximate_posterior()` function. ```{r, eval=FALSE} set.seed(4711) loo_ap_1 <- loo_approximate_posterior( x = llfun_logistic, draws = parameter_draws_laplace, data = stan_df_1, log_p = log_p, log_g = log_g, cores = 2 ) print(loo_ap_1) ``` The function creates a class, `psis_loo_ap` that inherits from `psis_loo, loo`. ``` Computed from 2000 by 3020 log-likelihood matrix Estimate SE elpd_loo -1968.4 15.6 p_loo 3.2 0.2 looic 3936.8 31.2 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. Pareto k diagnostic values: Count Pct. Min. n_eff (-Inf, 0.5] (good) 2989 99.0% 1827 (0.5, 0.7] (ok) 31 1.0% 1996 (0.7, 1] (bad) 0 0.0% (1, Inf) (very bad) 0 0.0% All Pareto k estimates are ok (k < 0.7). See help('pareto-k-diagnostic') for details. ``` ## Combining the posterior approximation method with subsampling The posterior approximation correction can also be used together with subsampling: ```{r, eval=FALSE} set.seed(4711) loo_ap_ss_1 <- loo_subsample( x = llfun_logistic, draws = parameter_draws_laplace, data = stan_df_1, log_p = log_p, log_g = log_g, observations = 100, cores = 2 ) print(loo_ap_ss_1) ``` ``` Computed from 2000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1968.2 15.6 0.4 p_loo 2.9 0.1 0.5 looic 3936.4 31.1 0.8 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. Pareto k diagnostic values: Count Pct. Min. n_eff (-Inf, 0.5] (good) 97 97.0% 1971 (0.5, 0.7] (ok) 3 3.0% 1997 (0.7, 1] (bad) 0 0.0% (1, Inf) (very bad) 0 0.0% All Pareto k estimates are ok (k < 0.7). See help('pareto-k-diagnostic') for details. ``` The object created is of class `psis_loo_ss`, which inherits from the `psis_loo_ap` class previously described. ## Comparing models To compare this model to an alternative model for the same data we can use the `loo_compare()` function just as we would if using `loo()` instead of `loo_subsample()` or `loo_approximate_posterior()`. First we'll fit a second model to the well-switching data, using `log(arsenic)` instead of `arsenic` as a predictor: ```{r, eval=FALSE} standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) fit_2 <- sampling(stan_mod, data = standata) parameter_draws_2 <- extract(fit_2)$beta stan_df_2 <- as.data.frame(standata) # recompute subsampling loo for first model for demonstration purposes # compute relative efficiency (this is slow and optional but is recommended to allow # for adjusting PSIS effective sample size based on MCMC effective sample size) r_eff_1 <- relative_eff( llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_1, draws = parameter_draws_1, cores = 2 ) set.seed(4711) loo_ss_1 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_1, draws = parameter_draws_1, data = stan_df_1, observations = 200, cores = 2 ) # compute subsampling loo for a second model (with log-arsenic) r_eff_2 <- relative_eff( llfun_logistic, log = FALSE, # relative_eff wants likelihood not log-likelihood values chain_id = rep(1:4, each = 1000), data = stan_df_2, draws = parameter_draws_2, cores = 2 ) loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = 200, cores = 2 ) print(loo_ss_2) ``` ``` Computed from 4000 by 100 subsampled log-likelihood values from 3020 total observations. Estimate SE subsampling SE elpd_loo -1952.0 16.2 0.2 p_loo 2.6 0.1 0.3 looic 3903.9 32.4 0.4 ------ Monte Carlo SE of elpd_loo is 0.0. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. ``` We can now compare the models on LOO using the `loo_compare` function: ```{r, eval=FALSE} # Compare comp <- loo_compare(loo_ss_1, loo_ss_2) print(comp) ``` ``` Warning: Different subsamples in 'model2' and 'model1'. Naive diff SE is used. elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.5 22.5 0.4 ``` This new object `comp` contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error. As the warning indicates, because different subsamples were used the comparison will not take the correlations between different observations into account. Here we see that the naive SE is 22.5 and we cannot see any difference in performance between the models. To force subsampling to use the same observations for each of the models we can simply extract the observations used in `loo_ss_1` and use them in `loo_ss_2` by supplying the `loo_ss_1` object to the `observations` argument. ```{r, eval=FALSE} loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = loo_ss_1, cores = 2 ) ``` We could also supply the subsampling indices using the `obs_idx()` helper function: ```{r, eval=FALSE} idx <- obs_idx(loo_ss_1) loo_ss_2 <- loo_subsample( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, observations = idx, cores = 2 ) ``` ``` Simple random sampling with replacement assumed. ``` This results in a message indicating that we assume these observations to have been sampled with simple random sampling, which is true because we had used the default `"diff_srs"` estimator for `loo_ss_1`. We can now compare the models and estimate the difference based on the same subsampled observations. ```{r, eval=FALSE} comp <- loo_compare(loo_ss_1, loo_ss_2) print(comp) ``` ``` elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.1 4.4 0.1 ``` First, notice that now the `se_diff` is now around 4 (as opposed to 20 when using different subsamples). The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in `elpd` and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (`model2`). Since the subsampling uncertainty is so small in this case it can effectively be ignored. If we need larger subsamples we can simply add samples using the `update()` method demonstrated earlier. It is also possible to compare a subsampled loo computation with a full loo object. ```{r, eval=FALSE} # use loo() instead of loo_subsample() to compute full PSIS-LOO for model 2 loo_full_2 <- loo( x = llfun_logistic, r_eff = r_eff_2, draws = parameter_draws_2, data = stan_df_2, cores = 2 ) loo_compare(loo_ss_1, loo_full_2) ``` ``` Estimated elpd_diff using observations included in loo calculations for all models. ``` Because we are comparing a non-subsampled loo calculation to a subsampled calculation we get the message that only the observations that are included in the loo calculations for both `model1` and `model2` are included in the computations for the comparison. ``` elpd_diff se_diff subsampling_se_diff model2 0.0 0.0 0.0 model1 16.3 4.4 0.3 ``` Here we actually see an increase in `subsampling_se_diff`, but this is due to a technical detail not elaborated here. In general, the difference should be better or negligible. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2017). _The Stan C++ Library, Version 2.17.0._ https://mc-stan.org/ Stan Development Team (2018) _RStan: the R interface to Stan, Version 2.17.3._ https://mc-stan.org/ Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. [arXiv preprint arXiv:2001.00980](https://arxiv.org/abs/2001.00980). Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 [online](http://proceedings.mlr.press/v97/magnusson19a.html), [arXiv preprint arXiv:1904.10679](https://arxiv.org/abs/1904.10679). Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). loo/inst/doc/loo2-non-factorized.html0000644000176200001440000130507714411465466017276 0ustar liggesusers Leave-one-out cross-validation for non-factorized models

Leave-one-out cross-validation for non-factorized models

Aki Vehtari, Paul Bürkner and Jonah Gabry

2023-03-30

Introduction

When computing ELPD-based LOO-CV for a Bayesian model we need to compute the log leave-one-out predictive densities \(\log{p(y_i | y_{-i})}\) for every response value \(y_i, \: i = 1, \ldots, N\), where \(y_{-i}\) denotes all response values except observation \(i\). To obtain \(p(y_i | y_{-i})\), we need to have access to the pointwise likelihood \(p(y_i\,|\, y_{-i}, \theta)\) and integrate over the model parameters \(\theta\):

\[ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\, y_{-i}, \theta) \, p(\theta\,|\, y_{-i}) \,d \theta \]

Here, \(p(\theta\,|\, y_{-i})\) is the leave-one-out posterior distribution for \(\theta\), that is, the posterior distribution for \(\theta\) obtained by fitting the model while holding out the \(i\)th observation (we will later show how refitting the model to data \(y_{-i}\) can be avoided).

If the observation model is formulated directly as the product of the pointwise observation models, we call it a factorized model. In this case, the likelihood is also the product of the pointwise likelihood contributions \(p(y_i\,|\, y_{-i}, \theta)\). To better illustrate possible structures of the observation models, we formally divide \(\theta\) into two parts, observation-specific latent variables \(f = (f_1, \ldots, f_N)\) and hyperparameters \(\psi\), so that \(p(y_i\,|\, y_{-i}, \theta) = p(y_i\,|\, y_{-i}, f_i, \psi)\). Depending on the model, one of the two parts of \(\theta\) may also be empty. In very simple models, such as linear regression models, latent variables are not explicitly presented and response values are conditionally independent given \(\psi\), so that \(p(y_i\,|\, y_{-i}, f_i, \psi) = p(y_i \,|\, \psi)\). The full likelihood can then be written in the familiar form

\[ p(y \,|\, \psi) = \prod_{i=1}^N p(y_i \,|\, \psi), \]

where \(y = (y_1, \ldots, y_N)\) denotes the vector of all responses. When the likelihood factorizes this way, the conditional pointwise log-likelihood can be obtained easily by computing \(p(y_i\,|\, \psi)\) for each \(i\) with computational cost \(O(n)\).

Yet, there are several reasons why a non-factorized observation model may be necessary or preferred. In non-factorized models, the joint likelihood of the response values \(p(y \,|\, \theta)\) is not factorized into observation-specific components, but rather given directly as one joint expression. For some models, an analytic factorized formulation is simply not available in which case we speak of a non-factorizable model. Even in models whose observation model can be factorized in principle, it may still be preferable to use a non-factorized form for reasons of efficiency and numerical stability (Bürkner et al. 2020).

Whether a non-factorized model is used by necessity or for efficiency and stability, it comes at the cost of having no direct access to the leave-one-out predictive densities and thus to the overall leave-one-out predictive accuracy. In theory, we can express the observation-specific likelihoods in terms of the joint likelihood via

\[ p(y_i \,|\, y_{i-1}, \theta) = \frac{p(y \,|\, \theta)}{p(y_{-i} \,|\, \theta)} = \frac{p(y \,|\, \theta)}{\int p(y \,|\, \theta) \, d y_i}, \]

but the expression on the right-hand side may not always have an analytical solution. Computing \(\log p(y_i \,|\, y_{-i}, \theta)\) for non-factorized models is therefore often impossible, or at least inefficient and numerically unstable. However, there is a large class of multivariate normal and Student-\(t\) models for which there are efficient analytical solutions available.

More details can be found in our paper about LOO-CV for non-factorized models (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1810.10559).

LOO-CV for multivariate normal models

In this vignette, we will focus on non-factorized multivariate normal models. Based on results of Sundararajan and Keerthi (2001), Bürkner et al. (2020) show that, for multivariate normal models with coriance matrix \(C\), the LOO predictive mean and standard deviation can be computed as follows:

\[\begin{align} \mu_{\tilde{y},-i} &= y_i-\bar{c}_{ii}^{-1} g_i \nonumber \\ \sigma_{\tilde{y},-i} &= \sqrt{\bar{c}_{ii}^{-1}}, \end{align}\] where \(g_i\) and \(\bar{c}_{ii}\) are \[\begin{align} g_i &= \left[C^{-1} y\right]_i \nonumber \\ \bar{c}_{ii} &= \left[C^{-1}\right]_{ii}. \end{align}\]

Using these results, the log predictive density of the \(i\)th observation is then computed as

\[ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{-i} - \frac{1}{2}\frac{(y_i-\mu_{-i})^2}{\sigma^2_{-i}}. \]

Expressing this same equation in terms of \(g_i\) and \(\bar{c}_{ii}\), the log predictive density becomes:

\[ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) + \frac{1}{2}\log \bar{c}_{ii} - \frac{1}{2}\frac{g_i^2}{\bar{c}_{ii}}. \] (Note that Vehtari et al. (2016) has a typo in the corresponding Equation 34.)

From these equations we can now derive a recipe for obtaining the conditional pointwise log-likelihood for all models that can be expressed conditionally in terms of a multivariate normal with invertible covariance matrix \(C\).

Approximate LOO-CV using integrated importance-sampling

The above LOO equations for multivariate normal models are conditional on parameters \(\theta\). Therefore, to obtain the leave-one-out predictive density \(p(y_i \,|\, y_{-i})\) we need to integrate over \(\theta\),

\[ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\,y_{-i}, \theta) \, p(\theta\,|\,y_{-i}) \,d\theta. \]

Here, \(p(\theta\,|\,y_{-i})\) is the leave-one-out posterior distribution for \(\theta\), that is, the posterior distribution for \(\theta\) obtained by fitting the model while holding out the \(i\)th observation.

To avoid the cost of sampling from \(N\) leave-one-out posteriors, it is possible to take the posterior draws \(\theta^{(s)}, \, s=1,\ldots,S\), from the posterior \(p(\theta\,|\,y)\), and then approximate the above integral using integrated importance sampling (Vehtari et al., 2016, Section 3.6.1):

\[ p(y_i\,|\,y_{-i}) \approx \frac{ \sum_{s=1}^S p(y_i\,|\,y_{-i},\,\theta^{(s)}) \,w_i^{(s)}}{ \sum_{s=1}^S w_i^{(s)}}, \]

where \(w_i^{(s)}\) are importance weights. First we compute the raw importance ratios

\[ r_i^{(s)} \propto \frac{1}{p(y_i \,|\, y_{-i}, \,\theta^{(s)})}, \]

and then stabilize them using Pareto smoothed importance sampling (PSIS, Vehtari et al, 2019) to obtain the weights \(w_i^{(s)}\). The resulting approximation is referred to as PSIS-LOO (Vehtari et al, 2017).

Exact LOO-CV with re-fitting

In order to validate the approximate LOO procedure, and also in order to allow exact computations to be made for a small number of leave-one-out folds for which the Pareto \(k\) diagnostic (Vehtari et al, 2019) indicates an unstable approximation, we need to consider how we might to do exact leave-one-out CV for a non-factorized model. In the case of a Gaussian process that has the marginalization property, we could just drop the one row and column of \(C\) corresponding to the held out out observation. This does not hold in general for multivariate normal models, however, and to keep the original prior we may need to maintain the full covariance matrix \(C\) even when one of the observations is left out.

The solution is to model \(y_i\) as a missing observation and estimate it along with all of the other model parameters. For a conditional multivariate normal model, \(\log p(y_i\,|\,y_{-i})\) can be computed as follows. First, we model \(y_i\) as missing and denote the corresponding parameter \(y_i^{\mathrm{mis}}\). Then, we define

\[ y_{\mathrm{mis}(i)} = (y_1, \ldots, y_{i-1}, y_i^{\mathrm{mis}}, y_{i+1}, \ldots, y_N). \] to be the same as the full set of observations \(y\), except replacing \(y_i\) with the parameter \(y_i^{\mathrm{mis}}\).

Second, we compute the LOO predictive mean and standard deviations as above, but replace \(y\) with \(y_{\mathrm{mis}(i)}\) in the computation of \(\mu_{\tilde{y},-i}\):

\[ \mu_{\tilde{y},-i} = y_{{\mathrm{mis}}(i)}-\bar{c}_{ii}^{-1}g_i, \]

where in this case we have

\[ g_i = \left[ C^{-1} y_{\mathrm{mis}(i)} \right]_i. \]

The conditional log predictive density is then computed with the above \(\mu_{\tilde{y},-i}\) and the left out observation \(y_i\):

\[ \log p(y_i\,|\,y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{\tilde{y},-i} - \frac{1}{2}\frac{(y_i-\mu_{\tilde{y},-i})^2}{\sigma^2_{\tilde{y},-i}}. \]

Finally, the leave-one-out predictive distribution can then be estimated as

\[ p(y_i\,|\,y_{-i}) \approx \sum_{s=1}^S p(y_i\,|\,y_{-i}, \theta_{-i}^{(s)}), \]

where \(\theta_{-i}^{(s)}\) are draws from the posterior distribution \(p(\theta\,|\,y_{\mathrm{mis}(i)})\).

Lagged SAR models

A common non-factorized multivariate normal model is the simultaneously autoregressive (SAR) model, which is frequently used for spatially correlated data. The lagged SAR model is defined as

\[ y = \rho Wy + \eta + \epsilon \] or equivalently \[ (I - \rho W)y = \eta + \epsilon, \] where \(\rho\) is the spatial correlation parameter and \(W\) is a user-defined weight matrix. The matrix \(W\) has entries \(w_{ii} = 0\) along the diagonal and the off-diagonal entries \(w_{ij}\) are larger when areas \(i\) and \(j\) are closer to each other. In a linear model, the predictor term \(\eta\) is given by \(\eta = X \beta\) with design matrix \(X\) and regression coefficients \(\beta\). However, since the above equation holds for arbitrary \(\eta\), these results are not restricted to linear models.

If we have \(\epsilon \sim {\mathrm N}(0, \,\sigma^2 I)\), it follows that \[ (I - \rho W)y \sim {\mathrm N}(\eta, \sigma^2 I), \] which corresponds to the following log PDF coded in Stan:

/** 
 * Normal log-pdf for spatially lagged responses
 * 
 * @param y Vector of response values.
 * @param mu Mean parameter vector.
 * @param sigma Positive scalar residual standard deviation.
 * @param rho Positive scalar autoregressive parameter.
 * @param W Spatial weight matrix.
 *
 * @return A scalar to be added to the log posterior.
 */
real normal_lagsar_lpdf(vector y, vector mu, real sigma, 
                        real rho, matrix W) {
  int N = rows(y);
  real inv_sigma2 = 1 / square(sigma);
  matrix[N, N] W_tilde = -rho * W;
  vector[N] half_pred;
  
  for (n in 1:N) W_tilde[n,n] += 1;
  
  half_pred = W_tilde * (y - mdivide_left(W_tilde, mu));
  
  return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) -
         0.5 * dot_self(half_pred) * inv_sigma2;
}

For the purpose of computing LOO-CV, it makes sense to rewrite the SAR model in slightly different form. Conditional on \(\rho\), \(\eta\), and \(\sigma\), if we write

\[\begin{align} y-(I-\rho W)^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(I-\rho W)^{-1}(I-\rho W)^{-T}), \end{align}\] or more compactly, with \(\widetilde{W}=(I-\rho W)\), \[\begin{align} y-\widetilde{W}^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}), \end{align}\]

then this has the same form as the zero mean Gaussian process from above. Accordingly, we can compute the leave-one-out predictive densities with the equations from Sundararajan and Keerthi (2001), replacing \(y\) with \((y-\widetilde{W}^{-1}\eta)\) and taking the covariance matrix \(C\) to be \(\sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}\).

Case Study: Neighborhood Crime in Columbus, Ohio

In order to demonstrate how to carry out the computations implied by these equations, we will first fit a lagged SAR model to data on crime in 49 different neighborhoods of Columbus, Ohio during the year 1980. The data was originally described in Aneslin (1988) and ships with the spdep R package.

In addition to the loo package, for this analysis we will use the brms interface to Stan to generate a Stan program and fit the model, and also the bayesplot and ggplot2 packages for plotting.

library("loo")
library("brms")
library("bayesplot")
library("ggplot2")
color_scheme_set("brightblue")
theme_set(theme_default())


SEED <- 10001 
set.seed(SEED) # only sets seed for R (seed for Stan set later)

# loads COL.OLD data frame and COL.nb neighbor list
data(oldcol, package = "spdep")

The three variables in the data set relevant to this example are:

  • CRIME: the number of residential burglaries and vehicle thefts per thousand households in the neighbood
  • HOVAL: housing value in units of $1000 USD
  • INC: household income in units of $1000 USD
str(COL.OLD[, c("CRIME", "HOVAL", "INC")])
'data.frame':   49 obs. of  3 variables:
 $ CRIME: num  18.802 32.388 38.426 0.178 15.726 ...
 $ HOVAL: num  44.6 33.2 37.1 75 80.5 ...
 $ INC  : num  21.23 4.48 11.34 8.44 19.53 ...

We will also use the object COL.nb, which is a list containing information about which neighborhoods border each other. From this list we will be able to construct the weight matrix to used to help account for the spatial dependency among the observations.

Fit lagged SAR model

A model predicting CRIME from INC and HOVAL, while accounting for the spatial dependency via an SAR structure, can be specified in brms as follows.

fit <- brm(
  CRIME ~ INC + HOVAL + sar(COL.nb, type = "lag"), 
  data = COL.OLD,
  data2 = list(COL.nb = COL.nb),
  chains = 4,
  seed = SEED
)

The code above fits the model in Stan using a log PDF equivalent to the normal_lagsar_lpdf function we defined above. In the summary output below we see that both higher income and higher housing value predict lower crime rates in the neighborhood. Moreover, there seems to be substantial spatial correlation between adjacent neighborhoods, as indicated by the posterior distribution of the lagsar parameter.

lagsar <- as.matrix(fit, pars = "lagsar")
estimates <- quantile(lagsar, probs = c(0.25, 0.5, 0.75))
mcmc_hist(lagsar) + 
  vline_at(estimates, linetype = 2, size = 1) +
  ggtitle("lagsar: posterior median and 50% central interval")

Approximate LOO-CV

After fitting the model, the next step is to compute the pointwise log-likelihood values needed for approximate LOO-CV. To do this we will use the recipe laid out in the previous sections.

posterior <- as.data.frame(fit)
y <- fit$data$CRIME
N <- length(y)
S <- nrow(posterior)
loglik <- yloo <- sdloo <- matrix(nrow = S, ncol = N)

for (s in 1:S) {
  p <- posterior[s, ]
  eta <- p$b_Intercept + p$b_INC * fit$data$INC + p$b_HOVAL * fit$data$HOVAL
  W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb)
  Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2
  g <- Cinv %*% (y - solve(W_tilde, eta))
  cbar <- diag(Cinv)
  yloo[s, ] <- y - g / cbar
  sdloo[s, ] <- sqrt(1 / cbar)
  loglik[s, ] <- dnorm(y, yloo[s, ], sdloo[s, ], log = TRUE)
}

# use loo for psis smoothing
log_ratios <- -loglik
psis_result <- psis(log_ratios)

The quality of the PSIS-LOO approximation can be investigated graphically by plotting the Pareto-k estimate for each observation. Ideally, they should not exceed \(0.5\), but in practice the algorithm turns out to be robust up to values of \(0.7\) (Vehtari et al, 2017, 2019). In the plot below, we see that the fourth observation is problematic and so may reduce the accuracy of the LOO-CV approximation.

plot(psis_result, label_points = TRUE)

We can also check that the conditional leave-one-out predictive distribution equations work correctly, for instance, using the last posterior draw:

yloo_sub <- yloo[S, ]
sdloo_sub <- sdloo[S, ]
df <- data.frame(
  y = y, 
  yloo = yloo_sub,
  ymin = yloo_sub - sdloo_sub * 2,
  ymax = yloo_sub + sdloo_sub * 2
)
ggplot(data=df, aes(x = y, y = yloo, ymin = ymin, ymax = ymax)) +
  geom_errorbar(
    width = 1, 
    color = "skyblue3", 
    position = position_jitter(width = 0.25)
  ) +
  geom_abline(color = "gray30", size = 1.2) +
  geom_point()

Finally, we use PSIS-LOO to approximate the expected log predictive density (ELPD) for new data, which we will validate using exact LOO-CV in the upcoming section.

(psis_loo <- loo(loglik))

Computed from 4000 by 49 log-likelihood matrix

         Estimate   SE
elpd_loo   -186.7 10.4
p_loo         7.8  4.9
looic       373.5 20.8
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     47    95.9%   2252      
 (0.5, 0.7]   (ok)        0     0.0%   <NA>      
   (0.7, 1]   (bad)       2     4.1%   57        
   (1, Inf)   (very bad)  0     0.0%   <NA>      
See help('pareto-k-diagnostic') for details.

Exact LOO-CV

Exact LOO-CV for the above example is somewhat more involved, as we need to re-fit the model \(N\) times and each time model the held-out data point as a parameter. First, we create an empty dummy model that we will update below as we loop over the observations.

# see help("mi", "brms") for details on the mi() usage
fit_dummy <- brm(
  CRIME | mi() ~ INC + HOVAL + sar(COL.nb, type = "lag"), 
  data = COL.OLD,
  data2 = list(COL.nb = COL.nb),
  chains = 0
)

Next, we fit the model \(N\) times, each time leaving out a single observation and then computing the log predictive density for that observation. For obvious reasons, this takes much longer than the approximation we computed above, but it is necessary in order to validate the approximate LOO-CV method. Thanks to the PSIS-LOO approximation, in general doing these slow exact computations can be avoided.

S <- 500
res <- vector("list", N)
loglik <- matrix(nrow = S, ncol = N)
for (i in seq_len(N)) {
  dat_mi <- COL.OLD
  dat_mi$CRIME[i] <- NA
  fit_i <- update(fit_dummy, newdata = dat_mi, 
                  # just for vignette
                  chains = 1, iter = S * 2)
  posterior <- as.data.frame(fit_i)
  yloo <- sdloo <- rep(NA, S)
  for (s in seq_len(S)) {
    p <- posterior[s, ]
    y_miss_i <- y
    y_miss_i[i] <- p$Ymi
    eta <- p$b_Intercept + p$b_INC * fit_i$data$INC + p$b_HOVAL * fit_i$data$HOVAL
    W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb)
    Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2
    g <- Cinv %*% (y_miss_i - solve(W_tilde, eta))
    cbar <- diag(Cinv);
    yloo[s] <- y_miss_i[i] - g[i] / cbar[i]
    sdloo[s] <- sqrt(1 / cbar[i])
    loglik[s, i] <- dnorm(y[i], yloo[s], sdloo[s], log = TRUE)
  }
  ypred <- rnorm(S, yloo, sdloo)
  res[[i]] <- data.frame(y = c(posterior$Ymi, ypred))
  res[[i]]$type <- rep(c("pp", "loo"), each = S)
  res[[i]]$obs <- i
}
res <- do.call(rbind, res)

A first step in the validation of the pointwise predictive density is to compare the distribution of the implied response values for the left-out observation to the distribution of the \(y_i^{\mathrm{mis}}\) posterior-predictive values estimated as part of the model. If the pointwise predictive density is correct, the two distributions should match very closely (up to sampling error). In the plot below, we overlay these two distributions for the first four observations and see that they match very closely (as is the case for all \(49\) observations of in this example).

res_sub <- res[res$obs %in% 1:4, ]
ggplot(res_sub, aes(y, fill = type)) +
  geom_density(alpha = 0.6) +
  facet_wrap("obs", scales = "fixed", ncol = 4)

In the final step, we compute the ELPD based on the exact LOO-CV and compare it to the approximate PSIS-LOO result computed earlier.

log_mean_exp <- function(x) {
  # more stable than log(mean(exp(x)))
  max_x <- max(x)
  max_x + log(sum(exp(x - max_x))) - log(length(x))
}
exact_elpds <- apply(loglik, 2, log_mean_exp)
exact_elpd <- sum(exact_elpds)
round(exact_elpd, 1)
[1] -189

The results of the approximate and exact LOO-CV are similar but not as close as we would expect if there were no problematic observations. We can investigate this issue more closely by plotting the approximate against the exact pointwise ELPD values.

df <- data.frame(
  approx_elpd = psis_loo$pointwise[, "elpd_loo"],
  exact_elpd = exact_elpds
)
ggplot(df, aes(x = approx_elpd, y = exact_elpd)) +
  geom_abline(color = "gray30") +
  geom_point(size = 2) +
  geom_point(data = df[4, ], size = 3, color = "red3") +
  xlab("Approximate elpds") +
  ylab("Exact elpds") +
  coord_fixed(xlim = c(-16, -3), ylim = c(-16, -3))

In the plot above the fourth data point —the observation flagged as problematic by the PSIS-LOO approximation— is colored in red and is the clear outlier. Otherwise, the correspondence between the exact and approximate values is strong. In fact, summing over the pointwise ELPD values and leaving out the fourth observation yields practically equivalent results for approximate and exact LOO-CV:

without_pt_4 <- c(
  approx = sum(psis_loo$pointwise[-4, "elpd_loo"]),
  exact = sum(exact_elpds[-4])  
)
round(without_pt_4, 1)
approx  exact 
-173.3 -173.0

From this we can conclude that the difference we found when including all observations does not indicate a bug in our implementation of the approximate LOO-CV but rather a violation of its assumptions.

Working with Stan directly

So far, we have specified the models in brms and only used Stan implicitely behind the scenes. This allowed us to focus on the primary purpose of validating approximate LOO-CV for non-factorized models. However, we would also like to show how everything can be set up in Stan directly. The Stan code brms generates is human readable and so we can use it to learn some of the essential aspects of Stan and the particular model we are implementing. The Stan program below is a slightly modified version of the code extracted via stancode(fit_dummy):

// generated with brms 2.2.0
functions {
/** 
 * Normal log-pdf for spatially lagged responses
 * 
 * @param y Vector of response values.
 * @param mu Mean parameter vector.
 * @param sigma Positive scalar residual standard deviation.
 * @param rho Positive scalar autoregressive parameter.
 * @param W Spatial weight matrix.
 *
 * @return A scalar to be added to the log posterior.
 */
  real normal_lagsar_lpdf(vector y, vector mu, real sigma,
                          real rho, matrix W) {
    int N = rows(y);
    real inv_sigma2 = 1 / square(sigma);
    matrix[N, N] W_tilde = -rho * W;
    vector[N] half_pred;
    for (n in 1:N) W_tilde[n, n] += 1;
    half_pred = W_tilde * (y - mdivide_left(W_tilde, mu));
    return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) -
           0.5 * dot_self(half_pred) * inv_sigma2;
  }
}
data {
  int<lower=1> N;  // total number of observations
  vector[N] Y;  // response variable
  int<lower=0> Nmi;  // number of missings
  int<lower=1> Jmi[Nmi];  // positions of missings
  int<lower=1> K;  // number of population-level effects
  matrix[N, K] X;  // population-level design matrix
  matrix[N, N] W;  // spatial weight matrix
  int prior_only;  // should the likelihood be ignored?
}
transformed data {
  int Kc = K - 1;
  matrix[N, K - 1] Xc;  // centered version of X
  vector[K - 1] means_X;  // column means of X before centering
  for (i in 2:K) {
    means_X[i - 1] = mean(X[, i]);
    Xc[, i - 1] = X[, i] - means_X[i - 1];
  }
}
parameters {
  vector[Nmi] Ymi;  // estimated missings
  vector[Kc] b;  // population-level effects
  real temp_Intercept;  // temporary intercept
  real<lower=0> sigma;  // residual SD
  real<lower=0,upper=1> lagsar;  // SAR parameter
}
transformed parameters {
}
model {
  vector[N] Yl = Y;
  vector[N] mu = Xc * b + temp_Intercept;
  Yl[Jmi] = Ymi;
  // priors including all constants
  target += student_t_lpdf(temp_Intercept | 3, 34, 17);
  target += student_t_lpdf(sigma | 3, 0, 17)
    - 1 * student_t_lccdf(0 | 3, 0, 17);
  // likelihood including all constants
  if (!prior_only) {
    target += normal_lagsar_lpdf(Yl | mu, sigma, lagsar, W);
  }
}
generated quantities {
  // actual population-level intercept
  real b_Intercept = temp_Intercept - dot_product(means_X, b);
}

Here we want to focus on two aspects of the Stan code. First, because there is no built-in function in Stan that calculates the log-likelihood for the lag-SAR model, we define a new normal_lagsar_lpdf function in the functions block of the Stan program. This is the same function we showed earlier in the vignette and it can be used to compute the log-likelihood in an efficient and numerically stable way. The _lpdf suffix used in the function name informs Stan that this is a log probability density function.

Second, this Stan program nicely illustrates how to set up missing value imputation. Instead of just computing the log-likelihood for the observed responses Y, we define a new variable Yl which is equal to Y if the reponse is observed and equal to Ymi if the response is missing. The latter is in turn defined as a parameter and thus estimated along with all other paramters of the model. More details about missing value imputation in Stan can be found in the Missing Data & Partially Known Parameters section of the Stan manual.

The Stan code extracted from brms is not only helpful when learning Stan, but can also drastically speed up the specification of models that are not support by brms. If brms can fit a model similar but not identical to the desired model, we can let brms generate the Stan program for the similar model and then mold it into the program that implements the model we actually want to fit. Rather than calling stancode(), which requires an existing fitted model object, we recommend using make_stancode() and specifying the save_model argument to write the Stan program to a file. The corresponding data can be prepared with make_standata() and then manually amended if needed. Once the code and data have been edited, they can be passed to RStan’s stan() function via the file and data arguments.

Conclusion

In summary, we have shown how to set up and validate approximate and exact LOO-CV for non-factorized multivariate normal models using Stan with the brms and loo packages. Although we focused on the particular example of a spatial SAR model, the presented recipe applies more generally to models that can be expressed in terms of a multivariate normal likelihood.


References

Anselin L. (1988). Spatial econometrics: methods and models. Dordrecht: Kluwer Academic.

Bürkner P. C., Gabry J., & Vehtari A. (2020). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-t models. Computational Statistics, :10.1007/s00180-020-01045-4. ArXiv preprint.

Sundararajan S. & Keerthi S. S. (2001). Predictive approaches for choosing hyperparameters in Gaussian processes. Neural Computation, 13(5), 1103–1118.

Vehtari A., Mononen T., Tolvanen V., Sivula T., & Winther O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. Journal of Machine Learning Research, 17(103), 1–38. Online.

Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Online. arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

loo/inst/doc/loo2-elpd.Rmd0000644000176200001440000001723514407123455015043 0ustar liggesusers--- title: "Holdout validation and K-fold cross-validation of Stan programs with the loo package" author: "Bruno Nicenboim" date: "`r Sys.Date()`" output: html_vignette: toc: yes params: EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true") --- ```{r, child="children/SETTINGS-knitr.txt"} ``` ```{r, child="children/SEE-ONLINE.txt", eval = if (isTRUE(exists("params"))) !params$EVAL else TRUE} ``` # Introduction This vignette demonstrates how to do holdout validation and K-fold cross-validation with __loo__ for a Stan program. # Example: Eradication of Roaches using holdout validation approach This vignette uses the same example as in the vignettes [_Using the loo package (version >= 2.0.0)_](http://mc-stan.org/loo/articles/loo2-example.html) and [_Avoiding model refits in leave-one-out cross-validation with moment matching_](https://mc-stan.org/loo/articles/loo2-moment-matching.html). ## Coding the Stan model Here is the Stan code for fitting a Poisson regression model: ```{r stancode} stancode <- " data { int K; int N; matrix[N,K] x; int y[N]; vector[N] offset; real beta_prior_scale; real alpha_prior_scale; } parameters { vector[K] beta; real intercept; } model { y ~ poisson(exp(x * beta + intercept + offset)); beta ~ normal(0,beta_prior_scale); intercept ~ normal(0,alpha_prior_scale); } generated quantities { vector[N] log_lik; for (n in 1:N) log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n])); } " ``` Following the usual approach recommended in [_Writing Stan programs for use with the loo package_](http://mc-stan.org/loo/articles/loo2-with-rstan.html), we compute the log-likelihood for each observation in the `generated quantities` block of the Stan program. ## Setup In addition to __loo__, we load the __rstan__ package for fitting the model. We will also need the __rstanarm__ package for the data. ```{r setup, message=FALSE} library("rstan") library("loo") seed <- 9547 set.seed(seed) ``` # Holdout validation For this approach, the model is first fit to the "train" data and then is evaluated on the held-out "test" data. ## Splitting the data between train and test The data is divided between train (80% of the data) and test (20%): ```{r modelfit-holdout, message=FALSE} # Prepare data data(roaches, package = "rstanarm") roaches$roach1 <- sqrt(roaches$roach1) roaches$offset <- log(roaches[,"exposure2"]) # 20% of the data goes to the test set: roaches$test <- 0 roaches$test[sample(.2 * seq_len(nrow(roaches)))] <- 1 # data to "train" the model data_train <- list(y = roaches$y[roaches$test == 0], x = as.matrix(roaches[roaches$test == 0, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 0,]), K = 3, offset = roaches$offset[roaches$test == 0], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) # data to "test" the model data_test <- list(y = roaches$y[roaches$test == 1], x = as.matrix(roaches[roaches$test == 1, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$test == 1,]), K = 3, offset = roaches$offset[roaches$test == 1], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) ``` ## Fitting the model with RStan Next we fit the model to the "test" data in Stan using the __rstan__ package: ```{r fit-train} # Compile stanmodel <- stan_model(model_code = stancode) # Fit model fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) ``` We recompute the generated quantities using the posterior draws conditional on the training data, but we now pass in the held-out data to get the log predictive densities for the test data. Because we are using independent data, the log predictive density coincides with the log likelihood of the test data. ```{r gen-test} gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd <- extract_log_lik(gen_test) ``` ## Computing holdout elpd: Now we evaluate the predictive performance of the model on the test data using `elpd()`. ```{r elpd-holdout} (elpd_holdout <- elpd(log_pd)) ``` When one wants to compare different models, the function `loo_compare()` can be used to assess the difference in performance. # K-fold cross validation For this approach the data is divided into folds, and each time one fold is tested while the rest of the data is used to fit the model (see Vehtari et al., 2017). ## Splitting the data in folds We use the data that is already pre-processed and we divide it in 10 random folds using `kfold_split_random` ```{r prepare-folds, message=FALSE} # Prepare data roaches$fold <- kfold_split_random(K = 10, N = nrow(roaches)) ``` ## Fitting and extracting the log pointwise predictive densities for each fold We now loop over the 10 folds. In each fold we do the following. First, we fit the model to all the observations except the ones belonging to the left-out fold. Second, we compute the log pointwise predictive densities for the left-out fold. Last, we store the predictive density for the observations of the left-out fold in a matrix. The output of this loop is a matrix of the log pointwise predictive densities of all the observations. ```{r} # Prepare a matrix with the number of post-warmup iterations by number of observations: log_pd_kfold <- matrix(nrow = 4000, ncol = nrow(roaches)) # Loop over the folds for(k in 1:10){ data_train <- list(y = roaches$y[roaches$fold != k], x = as.matrix(roaches[roaches$fold != k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold != k,]), K = 3, offset = roaches$offset[roaches$fold != k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) data_test <- list(y = roaches$y[roaches$fold == k], x = as.matrix(roaches[roaches$fold == k, c("roach1", "treatment", "senior")]), N = nrow(roaches[roaches$fold == k,]), K = 3, offset = roaches$offset[roaches$fold == k], beta_prior_scale = 2.5, alpha_prior_scale = 5.0 ) fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0) gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test) log_pd_kfold[, roaches$fold == k] <- extract_log_lik(gen_test) } ``` ## Computing K-fold elpd: Now we evaluate the predictive performance of the model on the 10 folds using `elpd()`. ```{r elpd-kfold} (elpd_kfold <- elpd(log_pd_kfold)) ``` If one wants to compare several models (with `loo_compare`), one should use the same folds for all the different models. # References Gelman, A., and Hill, J. (2007). *Data Analysis Using Regression and Multilevel Hierarchical Models.* Cambridge University Press. Stan Development Team (2020) _RStan: the R interface to Stan, Version 2.21.1_ https://mc-stan.org Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). loo/inst/doc/loo2-large-data.html0000644000176200001440000015657414411455345016355 0ustar liggesusers Using Leave-one-out cross-validation for large data

Using Leave-one-out cross-validation for large data

Mans Magnusson, Paul Bürkner, Aki Vehtari and Jonah Gabry

2023-03-30

Introduction

This vignette demonstrates how to do leave-one-out cross-validation for large data using the loo package and Stan. There are two approaches covered: LOO with subsampling and LOO using approximations to posterior distributions. Some sections from this vignette are excerpted from the papers

  • Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. arXiv preprint arXiv:2001.00980.

  • Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 online, arXiv preprint arXiv:1904.10679.

  • Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

  • Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.04544.

which provide important background for understanding the methods implemented in the package.

Setup

In addition to the loo package, we’ll also be using rstan:

library("rstan")
library("loo")
set.seed(4711)

Example: Well water in Bangladesh

We will use the same example as in the vignette Writing Stan programs for use with the loo package. See that vignette for a description of the problem and data.

The sample size in this example is only \(N=3020\), which is not large enough to require the special methods for large data described in this vignette, but is sufficient for demonstration purposes in this tutorial.

Coding the Stan model

Here is the Stan code for fitting the logistic regression model, which we save in a file called logistic.stan:

// save in `logistic.stan`
data {
  int<lower=0> N;             // number of data points
  int<lower=0> P;             // number of predictors (including intercept)
  matrix[N,P] X;              // predictors (including 1s for intercept)
  int<lower=0,upper=1> y[N];  // binary outcome
}
parameters {
  vector[P] beta;
}
model {
  beta ~ normal(0, 1);
  y ~ bernoulli_logit(X * beta);
}

Importantly, unlike the general approach recommended in Writing Stan programs for use with the loo package, we do not compute the log-likelihood for each observation in the generated quantities block of the Stan program. Here we are assuming we have a large data set (larger than the one we’re actually using in this demonstration) and so it is preferable to instead define a function in R to compute the log-likelihood for each data point when needed rather than storing all of the log-likelihood values in memory.

The log-likelihood in R can be coded as follows:

# we'll add an argument log to toggle whether this is a log-likelihood or 
# likelihood function. this will be useful later in the vignette.
llfun_logistic <- function(data_i, draws, log = TRUE) {
  x_i <- as.matrix(data_i[, which(grepl(colnames(data_i), pattern = "X")), drop=FALSE])
  logit_pred <- draws %*% t(x_i)
  dbinom(x = data_i$y, size = 1, prob = 1/(1 + exp(-logit_pred)), log = log)
}

The function llfun_logistic() needs to have arguments data_i and draws. Below we will test that the function is working by using the loo_i() function.

Fitting the model with RStan

Next we fit the model in Stan using the rstan package:

# Prepare data
url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat"
wells <- read.table(url)
wells$dist100 <- with(wells, dist / 100)
X <- model.matrix(~ dist100 + arsenic, wells)
standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X))

# Compile
stan_mod <- stan_model("logistic.stan")

# Fit model
fit_1 <- sampling(stan_mod, data = standata, seed = 4711)
print(fit_1, pars = "beta")
         mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
beta[1]  0.00       0 0.08 -0.15 -0.05  0.00  0.06  0.16  1933    1
beta[2] -0.89       0 0.10 -1.09 -0.96 -0.89 -0.82 -0.69  2332    1
beta[3]  0.46       0 0.04  0.38  0.43  0.46  0.49  0.54  2051    1

Before we move on to computing LOO we can now test that the log-likelihood function we wrote is working as it should. The loo_i() function is a helper function that can be used to test a log-likelihood function on a single observation.

# used for draws argument to loo_i
parameter_draws_1 <- extract(fit_1)$beta

# used for data argument to loo_i
stan_df_1 <- as.data.frame(standata)

# compute relative efficiency (this is slow and optional but is recommended to allow 
# for adjusting PSIS effective sample size based on MCMC effective sample size)
r_eff <- relative_eff(llfun_logistic, 
                      log = FALSE, # relative_eff wants likelihood not log-likelihood values
                      chain_id = rep(1:4, each = 1000), 
                      data = stan_df_1, 
                      draws = parameter_draws_1, 
                      cores = 2)

loo_i(i = 1, llfun_logistic, r_eff = r_eff, data = stan_df_1, draws = parameter_draws_1)
$pointwise
    elpd_loo mcse_elpd_loo        p_loo     looic influence_pareto_k
1 -0.3314552  0.0002887608 0.0003361772 0.6629103        -0.05679886
...

Approximate LOO-CV using PSIS-LOO and subsampling

We can then use the loo_subsample() function to compute the efficient PSIS-LOO approximation to exact LOO-CV using subsampling:

set.seed(4711)
loo_ss_1 <-
  loo_subsample(
    llfun_logistic,
    observations = 100, # take a subsample of size 100
    cores = 2,
    # these next objects were computed above
    r_eff = r_eff, 
    draws = parameter_draws_1,
    data = stan_df_1
  )
print(loo_ss_1)
Computed from 4000 by 100 subsampled log-likelihood
values from 3020 total observations.

         Estimate   SE subsampling SE
elpd_loo  -1968.5 15.6            0.3
p_loo         3.1  0.1            0.4
looic      3936.9 31.2            0.6
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

The loo_subsample() function creates an object of class psis_loo_ss, that inherits from psis_loo, loo (the classes of regular loo objects).

The printed output above shows the estimates \(\widehat{\mbox{elpd}}_{\rm loo}\) (expected log predictive density), \(\widehat{p}_{\rm loo}\) (effective number of parameters), and \({\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}\) (the LOO information criterion). Unlike when using loo(), when using loo_subsample() there is an additional column giving the “subsampling SE”, which reflects the additional uncertainty due to the subsampling used.

The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the \(k\) parameter is explained in help('pareto-k-diagnostic') and in greater detail in Vehtari, Simpson, Gelman, Yao, and Gabry (2019)). In this case, the message tells us that all of the estimates for \(k\) are fine for this given subsample.

Adding additional subsamples

If we are not satisfied with the subsample size (i.e., the accuracy) we can simply add more samples until we are satisfied using the update() method.

set.seed(4711)
loo_ss_1b <-
  update(
    loo_ss_1,
    observations = 200, # subsample 200 instead of 100
    r_eff = r_eff,
    draws = parameter_draws_1,
    data = stan_df_1
  ) 
print(loo_ss_1b)
Computed from 4000 by 200 subsampled log-likelihood
values from 3020 total observations.

         Estimate   SE subsampling SE
elpd_loo  -1968.3 15.6            0.2
p_loo         3.2  0.1            0.4
looic      3936.7 31.2            0.5
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

Specifying estimator and sampling method

The performance relies on two components: the estimation method and the approximation used for the elpd. See the documentation for loo_subsample() more information on which estimators and approximations are implemented. The default implementation is using the point log predictive density evaluated at the mean of the posterior (loo_approximation="plpd") and the difference estimator (estimator="diff_srs"). This combination has a focus on fast inference. But we can easily use other estimators as well as other elpd approximations, for example:

set.seed(4711)
loo_ss_1c <-
  loo_subsample(
    x = llfun_logistic,
    r_eff = r_eff,
    draws = parameter_draws_1,
    data = stan_df_1,
    observations = 100,
    estimator = "hh_pps", # use Hansen-Hurwitz
    loo_approximation = "lpd", # use lpd instead of plpd
    loo_approximation_draws = 100,
    cores = 2
  )
print(loo_ss_1c)
Computed from 4000 by 100 subsampled log-likelihood
values from 3020 total observations.

         Estimate   SE subsampling SE
elpd_loo  -1968.9 15.4            0.5
p_loo         3.5  0.2            0.5
looic      3937.9 30.7            1.1
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

See the documentation and references for loo_subsample() for details on the implemented approximations.

Approximate LOO-CV using PSIS-LOO with posterior approximations

Using posterior approximations, such as variational inference and Laplace approximations, can further speed-up LOO-CV for large data. Here we demonstrate using a Laplace approximation in Stan.

fit_laplace <- optimizing(stan_mod, data = standata, draws = 2000, 
                          importance_resampling = TRUE)
parameter_draws_laplace <- fit_laplace$theta_tilde # draws from approximate posterior
log_p <- fit_laplace$log_p # log density of the posterior
log_g <- fit_laplace$log_g # log density of the approximation

Using the posterior approximation we can then do LOO-CV by correcting for the posterior approximation when we compute the elpd. To do this we use the loo_approximate_posterior() function.

set.seed(4711)
loo_ap_1 <-
  loo_approximate_posterior(
    x = llfun_logistic,
    draws = parameter_draws_laplace,
    data = stan_df_1,
    log_p = log_p,
    log_g = log_g,
    cores = 2
  )
print(loo_ap_1)

The function creates a class, psis_loo_ap that inherits from psis_loo, loo.

Computed from 2000 by 3020 log-likelihood matrix

         Estimate   SE
elpd_loo  -1968.4 15.6
p_loo         3.2  0.2
looic      3936.8 31.2
------
Posterior approximation correction used.
Monte Carlo SE of elpd_loo is 0.0.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     2989  99.0%   1827      
 (0.5, 0.7]   (ok)         31   1.0%   1996      
   (0.7, 1]   (bad)         0   0.0%   <NA>      
   (1, Inf)   (very bad)    0   0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

Combining the posterior approximation method with subsampling

The posterior approximation correction can also be used together with subsampling:

set.seed(4711)
loo_ap_ss_1 <-
  loo_subsample(
    x = llfun_logistic,
    draws = parameter_draws_laplace,
    data = stan_df_1,
    log_p = log_p,
    log_g = log_g,
    observations = 100,
    cores = 2
  )
print(loo_ap_ss_1)
Computed from 2000 by 100 subsampled log-likelihood
values from 3020 total observations.

         Estimate   SE subsampling SE
elpd_loo  -1968.2 15.6            0.4
p_loo         2.9  0.1            0.5
looic      3936.4 31.1            0.8
------
Posterior approximation correction used.
Monte Carlo SE of elpd_loo is 0.0.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     97    97.0%   1971      
 (0.5, 0.7]   (ok)        3     3.0%   1997      
   (0.7, 1]   (bad)       0     0.0%   <NA>      
   (1, Inf)   (very bad)  0     0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

The object created is of class psis_loo_ss, which inherits from the psis_loo_ap class previously described.

Comparing models

To compare this model to an alternative model for the same data we can use the loo_compare() function just as we would if using loo() instead of loo_subsample() or loo_approximate_posterior(). First we’ll fit a second model to the well-switching data, using log(arsenic) instead of arsenic as a predictor:

standata$X[, "arsenic"] <- log(standata$X[, "arsenic"])
fit_2 <- sampling(stan_mod, data = standata) 
parameter_draws_2 <- extract(fit_2)$beta
stan_df_2 <- as.data.frame(standata)

# recompute subsampling loo for first model for demonstration purposes

# compute relative efficiency (this is slow and optional but is recommended to allow 
# for adjusting PSIS effective sample size based on MCMC effective sample size)
r_eff_1 <- relative_eff(
  llfun_logistic,
  log = FALSE, # relative_eff wants likelihood not log-likelihood values
  chain_id = rep(1:4, each = 1000),
  data = stan_df_1,
  draws = parameter_draws_1,
  cores = 2
)

set.seed(4711)
loo_ss_1 <- loo_subsample(
  x = llfun_logistic,
  r_eff = r_eff_1,
  draws = parameter_draws_1,
  data = stan_df_1,
  observations = 200,
  cores = 2
)

# compute subsampling loo for a second model (with log-arsenic)

r_eff_2 <- relative_eff(
  llfun_logistic,
  log = FALSE, # relative_eff wants likelihood not log-likelihood values
  chain_id = rep(1:4, each = 1000),
  data = stan_df_2,
  draws = parameter_draws_2,
  cores = 2
)
loo_ss_2 <- loo_subsample(
  x = llfun_logistic,
  r_eff = r_eff_2, 
  draws = parameter_draws_2,
  data = stan_df_2,
  observations = 200,
  cores = 2
)

print(loo_ss_2)
Computed from 4000 by 100 subsampled log-likelihood
values from 3020 total observations.

         Estimate   SE subsampling SE
elpd_loo  -1952.0 16.2            0.2
p_loo         2.6  0.1            0.3
looic      3903.9 32.4            0.4
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

We can now compare the models on LOO using the loo_compare function:

# Compare
comp <- loo_compare(loo_ss_1, loo_ss_2)
print(comp)
Warning: Different subsamples in 'model2' and 'model1'. Naive diff SE is used.

       elpd_diff se_diff subsampling_se_diff
model2  0.0       0.0     0.0               
model1 16.5      22.5     0.4

This new object comp contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error. As the warning indicates, because different subsamples were used the comparison will not take the correlations between different observations into account. Here we see that the naive SE is 22.5 and we cannot see any difference in performance between the models.

To force subsampling to use the same observations for each of the models we can simply extract the observations used in loo_ss_1 and use them in loo_ss_2 by supplying the loo_ss_1 object to the observations argument.

loo_ss_2 <-
  loo_subsample(
    x = llfun_logistic,
    r_eff = r_eff_2,
    draws = parameter_draws_2,
    data = stan_df_2,
    observations = loo_ss_1,
    cores = 2
  )

We could also supply the subsampling indices using the obs_idx() helper function:

idx <- obs_idx(loo_ss_1)
loo_ss_2 <- loo_subsample(
  x = llfun_logistic,
  r_eff = r_eff_2, 
  draws = parameter_draws_2,
  data = stan_df_2,
  observations = idx,
  cores = 2
)
Simple random sampling with replacement assumed.

This results in a message indicating that we assume these observations to have been sampled with simple random sampling, which is true because we had used the default "diff_srs" estimator for loo_ss_1.

We can now compare the models and estimate the difference based on the same subsampled observations.

comp <- loo_compare(loo_ss_1, loo_ss_2)
print(comp) 
       elpd_diff se_diff subsampling_se_diff
model2  0.0       0.0     0.0               
model1 16.1       4.4     0.1

First, notice that now the se_diff is now around 4 (as opposed to 20 when using different subsamples). The first column shows the difference in ELPD relative to the model with the largest ELPD. In this case, the difference in elpd and its scale relative to the approximate standard error of the difference) indicates a preference for the second model (model2). Since the subsampling uncertainty is so small in this case it can effectively be ignored. If we need larger subsamples we can simply add samples using the update() method demonstrated earlier.

It is also possible to compare a subsampled loo computation with a full loo object.

# use loo() instead of loo_subsample() to compute full PSIS-LOO for model 2
loo_full_2 <- loo(
  x = llfun_logistic,
  r_eff = r_eff_2,
  draws = parameter_draws_2,
  data = stan_df_2,
  cores = 2
)
loo_compare(loo_ss_1, loo_full_2)
Estimated elpd_diff using observations included in loo calculations for all models.

Because we are comparing a non-subsampled loo calculation to a subsampled calculation we get the message that only the observations that are included in the loo calculations for both model1 and model2 are included in the computations for the comparison.

       elpd_diff se_diff subsampling_se_diff
model2  0.0       0.0     0.0               
model1 16.3       4.4     0.3

Here we actually see an increase in subsampling_se_diff, but this is due to a technical detail not elaborated here. In general, the difference should be better or negligible.

References

Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge University Press.

Stan Development Team (2017). The Stan C++ Library, Version 2.17.0. https://mc-stan.org/

Stan Development Team (2018) RStan: the R interface to Stan, Version 2.17.3. https://mc-stan.org/

Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2020). Leave-One-Out Cross-Validation for Model Comparison in Large Data. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), in PMLR 108. arXiv preprint arXiv:2001.00980.

Magnusson, M., Andersen, M., Jonasson, J. & Vehtari, A. (2019). Bayesian leave-one-out cross-validation for large data. Proceedings of the 36th International Conference on Machine Learning, in PMLR 97:4244-4253 online, arXiv preprint arXiv:1904.10679.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. online, arXiv preprint arXiv:1507.04544.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

loo/inst/doc/loo2-non-factorized.R0000644000176200001440000002155414411465465016524 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----more-knitr-ops, include=FALSE-------------------------------------------- knitr::opts_chunk$set( cache=TRUE, message=FALSE, warning=FALSE ) ## ----lpdf, eval=FALSE--------------------------------------------------------- # /** # * Normal log-pdf for spatially lagged responses # * # * @param y Vector of response values. # * @param mu Mean parameter vector. # * @param sigma Positive scalar residual standard deviation. # * @param rho Positive scalar autoregressive parameter. # * @param W Spatial weight matrix. # * # * @return A scalar to be added to the log posterior. # */ # real normal_lagsar_lpdf(vector y, vector mu, real sigma, # real rho, matrix W) { # int N = rows(y); # real inv_sigma2 = 1 / square(sigma); # matrix[N, N] W_tilde = -rho * W; # vector[N] half_pred; # # for (n in 1:N) W_tilde[n,n] += 1; # # half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); # # return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - # 0.5 * dot_self(half_pred) * inv_sigma2; # } ## ----setup, cache=FALSE------------------------------------------------------- library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) SEED <- 10001 set.seed(SEED) # only sets seed for R (seed for Stan set later) # loads COL.OLD data frame and COL.nb neighbor list data(oldcol, package = "spdep") ## ----data--------------------------------------------------------------------- str(COL.OLD[, c("CRIME", "HOVAL", "INC")]) ## ----fit, results="hide"------------------------------------------------------ fit <- brm( CRIME ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 4, seed = SEED ) ## ----plot-lagsar, message=FALSE----------------------------------------------- lagsar <- as.matrix(fit, pars = "lagsar") estimates <- quantile(lagsar, probs = c(0.25, 0.5, 0.75)) mcmc_hist(lagsar) + vline_at(estimates, linetype = 2, size = 1) + ggtitle("lagsar: posterior median and 50% central interval") ## ----approx------------------------------------------------------------------- posterior <- as.data.frame(fit) y <- fit$data$CRIME N <- length(y) S <- nrow(posterior) loglik <- yloo <- sdloo <- matrix(nrow = S, ncol = N) for (s in 1:S) { p <- posterior[s, ] eta <- p$b_Intercept + p$b_INC * fit$data$INC + p$b_HOVAL * fit$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y - solve(W_tilde, eta)) cbar <- diag(Cinv) yloo[s, ] <- y - g / cbar sdloo[s, ] <- sqrt(1 / cbar) loglik[s, ] <- dnorm(y, yloo[s, ], sdloo[s, ], log = TRUE) } # use loo for psis smoothing log_ratios <- -loglik psis_result <- psis(log_ratios) ## ----plot, cache = FALSE------------------------------------------------------ plot(psis_result, label_points = TRUE) ## ----checklast, cache = FALSE------------------------------------------------- yloo_sub <- yloo[S, ] sdloo_sub <- sdloo[S, ] df <- data.frame( y = y, yloo = yloo_sub, ymin = yloo_sub - sdloo_sub * 2, ymax = yloo_sub + sdloo_sub * 2 ) ggplot(data=df, aes(x = y, y = yloo, ymin = ymin, ymax = ymax)) + geom_errorbar( width = 1, color = "skyblue3", position = position_jitter(width = 0.25) ) + geom_abline(color = "gray30", size = 1.2) + geom_point() ## ----psisloo------------------------------------------------------------------ (psis_loo <- loo(loglik)) ## ----fit_dummy, cache = TRUE-------------------------------------------------- # see help("mi", "brms") for details on the mi() usage fit_dummy <- brm( CRIME | mi() ~ INC + HOVAL + sar(COL.nb, type = "lag"), data = COL.OLD, data2 = list(COL.nb = COL.nb), chains = 0 ) ## ----exact-loo-cv, results="hide", message=FALSE, warning=FALSE, cache = TRUE---- S <- 500 res <- vector("list", N) loglik <- matrix(nrow = S, ncol = N) for (i in seq_len(N)) { dat_mi <- COL.OLD dat_mi$CRIME[i] <- NA fit_i <- update(fit_dummy, newdata = dat_mi, # just for vignette chains = 1, iter = S * 2) posterior <- as.data.frame(fit_i) yloo <- sdloo <- rep(NA, S) for (s in seq_len(S)) { p <- posterior[s, ] y_miss_i <- y y_miss_i[i] <- p$Ymi eta <- p$b_Intercept + p$b_INC * fit_i$data$INC + p$b_HOVAL * fit_i$data$HOVAL W_tilde <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb) Cinv <- t(W_tilde) %*% W_tilde / p$sigma^2 g <- Cinv %*% (y_miss_i - solve(W_tilde, eta)) cbar <- diag(Cinv); yloo[s] <- y_miss_i[i] - g[i] / cbar[i] sdloo[s] <- sqrt(1 / cbar[i]) loglik[s, i] <- dnorm(y[i], yloo[s], sdloo[s], log = TRUE) } ypred <- rnorm(S, yloo, sdloo) res[[i]] <- data.frame(y = c(posterior$Ymi, ypred)) res[[i]]$type <- rep(c("pp", "loo"), each = S) res[[i]]$obs <- i } res <- do.call(rbind, res) ## ----yplots, cache = FALSE, fig.width=10, out.width="95%", fig.asp = 0.3------ res_sub <- res[res$obs %in% 1:4, ] ggplot(res_sub, aes(y, fill = type)) + geom_density(alpha = 0.6) + facet_wrap("obs", scales = "fixed", ncol = 4) ## ----loo_exact, cache=FALSE--------------------------------------------------- log_mean_exp <- function(x) { # more stable than log(mean(exp(x))) max_x <- max(x) max_x + log(sum(exp(x - max_x))) - log(length(x)) } exact_elpds <- apply(loglik, 2, log_mean_exp) exact_elpd <- sum(exact_elpds) round(exact_elpd, 1) ## ----compare, fig.height=5---------------------------------------------------- df <- data.frame( approx_elpd = psis_loo$pointwise[, "elpd_loo"], exact_elpd = exact_elpds ) ggplot(df, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + geom_point(data = df[4, ], size = 3, color = "red3") + xlab("Approximate elpds") + ylab("Exact elpds") + coord_fixed(xlim = c(-16, -3), ylim = c(-16, -3)) ## ----pt4---------------------------------------------------------------------- without_pt_4 <- c( approx = sum(psis_loo$pointwise[-4, "elpd_loo"]), exact = sum(exact_elpds[-4]) ) round(without_pt_4, 1) ## ----brms-stan-code, eval=FALSE----------------------------------------------- # // generated with brms 2.2.0 # functions { # /** # * Normal log-pdf for spatially lagged responses # * # * @param y Vector of response values. # * @param mu Mean parameter vector. # * @param sigma Positive scalar residual standard deviation. # * @param rho Positive scalar autoregressive parameter. # * @param W Spatial weight matrix. # * # * @return A scalar to be added to the log posterior. # */ # real normal_lagsar_lpdf(vector y, vector mu, real sigma, # real rho, matrix W) { # int N = rows(y); # real inv_sigma2 = 1 / square(sigma); # matrix[N, N] W_tilde = -rho * W; # vector[N] half_pred; # for (n in 1:N) W_tilde[n, n] += 1; # half_pred = W_tilde * (y - mdivide_left(W_tilde, mu)); # return 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) - # 0.5 * dot_self(half_pred) * inv_sigma2; # } # } # data { # int N; // total number of observations # vector[N] Y; // response variable # int Nmi; // number of missings # int Jmi[Nmi]; // positions of missings # int K; // number of population-level effects # matrix[N, K] X; // population-level design matrix # matrix[N, N] W; // spatial weight matrix # int prior_only; // should the likelihood be ignored? # } # transformed data { # int Kc = K - 1; # matrix[N, K - 1] Xc; // centered version of X # vector[K - 1] means_X; // column means of X before centering # for (i in 2:K) { # means_X[i - 1] = mean(X[, i]); # Xc[, i - 1] = X[, i] - means_X[i - 1]; # } # } # parameters { # vector[Nmi] Ymi; // estimated missings # vector[Kc] b; // population-level effects # real temp_Intercept; // temporary intercept # real sigma; // residual SD # real lagsar; // SAR parameter # } # transformed parameters { # } # model { # vector[N] Yl = Y; # vector[N] mu = Xc * b + temp_Intercept; # Yl[Jmi] = Ymi; # // priors including all constants # target += student_t_lpdf(temp_Intercept | 3, 34, 17); # target += student_t_lpdf(sigma | 3, 0, 17) # - 1 * student_t_lccdf(0 | 3, 0, 17); # // likelihood including all constants # if (!prior_only) { # target += normal_lagsar_lpdf(Yl | mu, sigma, lagsar, W); # } # } # generated quantities { # // actual population-level intercept # real b_Intercept = temp_Intercept - dot_product(means_X, b); # } loo/inst/doc/loo2-with-rstan.R0000644000176200001440000000430514411465510015664 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ---- eval=FALSE-------------------------------------------------------------- # library("rstan") # # # Prepare data # url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat" # wells <- read.table(url) # wells$dist100 <- with(wells, dist / 100) # X <- model.matrix(~ dist100 + arsenic, wells) # standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X)) # # # Fit model # fit_1 <- stan("logistic.stan", data = standata) # print(fit_1, pars = "beta") ## ---- eval=FALSE-------------------------------------------------------------- # library("loo") # # # Extract pointwise log-likelihood # # using merge_chains=FALSE returns an array, which is easier to # # use with relative_eff() # log_lik_1 <- extract_log_lik(fit_1, merge_chains = FALSE) # # # as of loo v2.0.0 we can optionally provide relative effective sample sizes # # when calling loo, which allows for better estimates of the PSIS effective # # sample sizes and Monte Carlo error # r_eff <- relative_eff(exp(log_lik_1), cores = 2) # # # preferably use more than 2 cores (as many cores as possible) # # will use value of 'mc.cores' option if cores is not specified # loo_1 <- loo(log_lik_1, r_eff = r_eff, cores = 2) # print(loo_1) ## ---- eval=FALSE-------------------------------------------------------------- # standata$X[, "arsenic"] <- log(standata$X[, "arsenic"]) # fit_2 <- stan(fit = fit_1, data = standata) # # log_lik_2 <- extract_log_lik(fit_2, merge_chains = FALSE) # r_eff_2 <- relative_eff(exp(log_lik_2)) # loo_2 <- loo(log_lik_2, r_eff = r_eff_2, cores = 2) # print(loo_2) ## ---- eval=FALSE-------------------------------------------------------------- # # Compare # comp <- loo_compare(loo_1, loo_2) ## ---- eval=FALSE-------------------------------------------------------------- # print(comp) # can set simplify=FALSE for more detailed print output loo/inst/doc/loo2-lfo.R0000644000176200001440000002253114411464745014356 0ustar liggesusersparams <- list(EVAL = TRUE) ## ----SETTINGS-knitr, include=FALSE-------------------------------------------- stopifnot(require(knitr)) opts_chunk$set( comment=NA, eval = if (isTRUE(exists("params"))) params$EVAL else FALSE, dev = "png", dpi = 150, fig.asp = 0.618, fig.width = 5, out.width = "60%", fig.align = "center" ) ## ----more-knitr-ops, include=FALSE-------------------------------------------- knitr::opts_chunk$set( cache = TRUE, message = FALSE, warning = FALSE ) ## ----pkgs, cache=FALSE-------------------------------------------------------- library("loo") library("brms") library("bayesplot") library("ggplot2") color_scheme_set("brightblue") theme_set(theme_default()) CHAINS <- 4 SEED <- 5838296 set.seed(SEED) ## ----hurondata---------------------------------------------------------------- N <- length(LakeHuron) df <- data.frame( y = as.numeric(LakeHuron), year = as.numeric(time(LakeHuron)), time = 1:N ) ggplot(df, aes(x = year, y = y)) + geom_point(size = 1) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)" ) ## ----fit, results = "hide"---------------------------------------------------- fit <- brm( y ~ ar(time, p = 4), data = df, prior = prior(normal(0, 0.5), class = "ar"), control = list(adapt_delta = 0.99), seed = SEED, chains = CHAINS ) ## ----plotpreds, cache = FALSE------------------------------------------------- preds <- posterior_predict(fit) preds <- cbind( Estimate = colMeans(preds), Q5 = apply(preds, 2, quantile, probs = 0.05), Q95 = apply(preds, 2, quantile, probs = 0.95) ) ggplot(cbind(df, preds), aes(x = year, y = Estimate)) + geom_smooth(aes(ymin = Q5, ymax = Q95), stat = "identity", size = 0.5) + geom_point(aes(y = y)) + labs( y = "Water Level (ft)", x = "Year", title = "Water Level in Lake Huron (1875-1972)", subtitle = "Mean (blue) and 90% predictive intervals (gray) vs. observed data (black)" ) ## ----setL--------------------------------------------------------------------- L <- 20 ## ----loo1sap, cache = FALSE--------------------------------------------------- loo_cv <- loo(log_lik(fit)[, (L + 1):N]) print(loo_cv) ## ----exact_loglik, results="hide"--------------------------------------------- loglik_exact <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_i <- update(fit, newdata = df_past, recompile = FALSE) loglik_exact[, i + 1] <- log_lik(fit_i, newdata = df_oos, oos = oos)[, oos] } ## ----helpers------------------------------------------------------------------ # some helper functions we'll use throughout # more stable than log(sum(exp(x))) log_sum_exp <- function(x) { max_x <- max(x) max_x + log(sum(exp(x - max_x))) } # more stable than log(mean(exp(x))) log_mean_exp <- function(x) { log_sum_exp(x) - log(length(x)) } # compute log of raw importance ratios # sums over observations *not* over posterior samples sum_log_ratios <- function(loglik, ids = NULL) { if (!is.null(ids)) loglik <- loglik[, ids, drop = FALSE] rowSums(loglik) } # for printing comparisons later rbind_print <- function(...) { round(rbind(...), digits = 2) } ## ----exact1sap, cache = FALSE------------------------------------------------- exact_elpds_1sap <- apply(loglik_exact, 2, log_mean_exp) exact_elpd_1sap <- c(ELPD = sum(exact_elpds_1sap[-(1:L)])) rbind_print( "LOO" = loo_cv$estimates["elpd_loo", "Estimate"], "LFO" = exact_elpd_1sap ) ## ----setkthresh--------------------------------------------------------------- k_thres <- 0.7 ## ----refit_loglik, results="hide"--------------------------------------------- approx_elpds_1sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- L + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[L + 1] <- log_mean_exp(loglik[, oos]) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - 1)) { past <- 1:i oos <- i + 1 df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) approx_elpds_1sap[i + 1] <- log_mean_exp(loglik[, oos]) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] approx_elpds_1sap[i + 1] <- log_sum_exp(lw + loglik[, oos]) } } ## ----plot_ks------------------------------------------------------------------ plot_ks <- function(ks, ids, thres = 0.6) { dat_ks <- data.frame(ks = ks, ids = ids) ggplot(dat_ks, aes(x = ids, y = ks)) + geom_point(aes(color = ks > thres), shape = 3, show.legend = FALSE) + geom_hline(yintercept = thres, linetype = 2, color = "red2") + scale_color_manual(values = c("cornflowerblue", "darkblue")) + labs(x = "Data point", y = "Pareto k") + ylim(-0.5, 1.5) } ## ----refitsummary1sap, cache=FALSE-------------------------------------------- cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - 1)) ## ----lfosummary1sap, cache = FALSE-------------------------------------------- approx_elpd_1sap <- sum(approx_elpds_1sap, na.rm = TRUE) rbind_print( "approx LFO" = approx_elpd_1sap, "exact LFO" = exact_elpd_1sap ) ## ----plot1sap, cache = FALSE-------------------------------------------------- dat_elpd <- data.frame( approx_elpd = approx_elpds_1sap, exact_elpd = exact_elpds_1sap ) ggplot(dat_elpd, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ## ----diffs1sap, cache=FALSE--------------------------------------------------- max_diff <- with(dat_elpd, max(abs(approx_elpd - exact_elpd), na.rm = TRUE)) mean_diff <- with(dat_elpd, mean(abs(approx_elpd - exact_elpd), na.rm = TRUE)) rbind_print( "Max diff" = round(max_diff, 2), "Mean diff" = round(mean_diff, 3) ) ## ----exact_loglikm, results="hide"-------------------------------------------- M <- 4 loglikm <- matrix(nrow = nsamples(fit), ncol = N) for (i in L:(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm[, i + 1] <- rowSums(loglik[, oos]) } ## ----exact4sap, cache = FALSE------------------------------------------------- exact_elpds_4sap <- apply(loglikm, 2, log_mean_exp) (exact_elpd_4sap <- c(ELPD = sum(exact_elpds_4sap, na.rm = TRUE))) ## ----refit_loglikm, results="hide"-------------------------------------------- approx_elpds_4sap <- rep(NA, N) # initialize the process for i = L past <- 1:L oos <- (L + 1):(L + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] fit_past <- update(fit, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_1sap[L + 1] <- log_mean_exp(loglikm) # iterate over i > L i_refit <- L refits <- L ks <- NULL for (i in (L + 1):(N - M)) { past <- 1:i oos <- (i + 1):(i + M) df_past <- df[past, , drop = FALSE] df_oos <- df[c(past, oos), , drop = FALSE] loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) logratio <- sum_log_ratios(loglik, (i_refit + 1):i) psis_obj <- suppressWarnings(psis(logratio)) k <- pareto_k_values(psis_obj) ks <- c(ks, k) if (k > k_thres) { # refit the model based on the first i observations i_refit <- i refits <- c(refits, i) fit_past <- update(fit_past, newdata = df_past, recompile = FALSE) loglik <- log_lik(fit_past, newdata = df_oos, oos = oos) loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_mean_exp(loglikm) } else { lw <- weights(psis_obj, normalize = TRUE)[, 1] loglikm <- rowSums(loglik[, oos]) approx_elpds_4sap[i + 1] <- log_sum_exp(lw + loglikm) } } ## ----refitsummary4sap, cache = FALSE------------------------------------------ cat("Using threshold ", k_thres, ", model was refit ", length(refits), " times, at observations", refits) plot_ks(ks, (L + 1):(N - M)) ## ----lfosummary4sap, cache = FALSE-------------------------------------------- approx_elpd_4sap <- sum(approx_elpds_4sap, na.rm = TRUE) rbind_print( "Approx LFO" = approx_elpd_4sap, "Exact LFO" = exact_elpd_4sap ) ## ----plot4sap, cache = FALSE-------------------------------------------------- dat_elpd_4sap <- data.frame( approx_elpd = approx_elpds_4sap, exact_elpd = exact_elpds_4sap ) ggplot(dat_elpd_4sap, aes(x = approx_elpd, y = exact_elpd)) + geom_abline(color = "gray30") + geom_point(size = 2) + labs(x = "Approximate ELPDs", y = "Exact ELPDs") ## ----sessioninfo-------------------------------------------------------------- sessionInfo() loo/inst/CITATION0000644000176200001440000000263714411130104013152 0ustar liggesusersyear <- sub("-.*", "", meta$Date) note <- sprintf("R package version %s", meta$Version) authors <- do.call(c, lapply(meta$Author, as.person)) authors <- grep("\\[cre|\\[aut", authors, value = TRUE) bibentry(bibtype = "Misc", title = "loo: Efficient leave-one-out cross-validation and WAIC for Bayesian models", author = authors, year = year, note = note, url = "https://mc-stan.org/loo/", header = "To cite the loo R package:" ) bibentry(bibtype = "Article", title = "Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC", author = c(person("Aki", "Vehtari"), person("Andrew", "Gelman"), person("Jonah", "Gabry")), year = "2017", journal = "Statistics and Computing", volume = 27, issue = 5, pages = "1413--1432", doi = "10.1007/s11222-016-9696-4", header = "To cite the loo paper:" ) bibentry(bibtype = "Article", title = "Using stacking to average Bayesian predictive distributions", author = c(person("Yuling", "Yao"), person("Aki", "Vehtari"), person("Daniel", "Simpson"), person("Andrew", "Gelman")), year = "2017", journal = "Bayesian Analysis", doi = "10.1214/17-BA1091", header = "To cite the stacking paper:" )