mcmc/0000755000175100001440000000000013074643067011221 5ustar hornikusersmcmc/inst/0000755000175100001440000000000013063023671012165 5ustar hornikusersmcmc/inst/designDoc/0000755000175100001440000000000013063023671014064 5ustar hornikusersmcmc/inst/designDoc/Makefile0000644000175100001440000000031413074514645015532 0ustar hornikusers all : metrop.pdf temper.pdf clean metrop.pdf : metrop.tex pdflatex metrop.tex pdflatex metrop.tex temper.pdf : temper.tex pdflatex temper.tex pdflatex temper.tex clean : rm -f *.dvi *.aux *.log mcmc/inst/designDoc/metrop.tex0000644000175100001440000003622113063023671016120 0ustar hornikusers \documentclass{article} \usepackage{indentfirst} \begin{document} \title{An MCMC Package for R} \author{Charles J. Geyer} \maketitle \section{Introduction} This package is a simple first attempt at a sensible \emph{general} MCMC package. It doesn't do much yet. It only does ``normal random-walk'' Metropolis for continuous distributions. No non-normal proposals. No Metropolis-Hastings or Metropolis-Hastings-Green. No discrete state. No dimension jumping. No simulated tempering. No perfect sampling. There's a lot left to do. Still, limited as it is, it does equilibrium distributions that no other R package does. Its basic idea is the following. Given an R function \verb@fred@ that calculates the unnormalized density of the desired equilibrium distribution of the Markov chain, or, better yet, \emph{log} unnormalized density, so we avoid overflow and underflow, the \verb@metrop@ function should generate a Markov chain having this stationary distribution. The package does not do any of the following. \begin{itemize} \item \textbf{Theory.} (What R package does?) It doesn't prove the Markov chain is irreducible or ergodic or positive recurrent or Harris recurrent or geometrically ergodic or uniformly ergodic or satisfies conditions for the central limit theorem. \item \textbf{Diagnostics.} There are no non-bogus Markov chain diagnostics (except for perfect sampling). This package doesn't do any bogus diagnostics (other R packages do them). \item \textbf{Calculus.} If the putative unnormalized density specified by \verb@fred@ is not integrable, then it does not specify an equilibrium distribution. But this package doesn't check that either. \end{itemize} Thus the only requirement the package has to satisfy is that given a function \verb@fred@ it correctly simulates a Markov chain that actually has \verb@fred@ as its equilibrium distribution (when \verb@fred@ actually does specify some equilibrium distribution) \section{Design Issues} \subsection{First Try} For a start we have a function with signature \begin{verbatim} metrop(lud, initial, niter, ...) \end{verbatim} such that when \begin{itemize} \item \verb@initial@ is a real vector, the initial state of the Markov chain, \item \verb@lud@ is a function, the log unnormalized density of the equilibrium distribution of the Markov chain, such that \begin{itemize} \item \verb@lud(initial, ...)@ works and produces a finite scalar value and \item \verb@lud(x, ...)@ works for any real vector \verb@x@ having the same length as \verb@initial@ and all elements finite and and produces a scalar value that is finite or \verb@-Inf@, \end{itemize} \end{itemize} then the function produces an \verb@niter@ by \verb@length(initial)@ matrix whose rows are the iterations of the Markov chain. \subsubsection{Checks} If \begin{verbatim} logh <- lud(initial, ...) \end{verbatim} then \verb@is.finite(logh)@ is \verb@TRUE@. Moreover, if \verb@x@ is any vector such that \verb@length(x) == length(initial)@ and \verb@all(is.finite(x))@ are \verb@TRUE@ and \begin{verbatim} logh <- lud(x, ...) \end{verbatim} then \begin{verbatim} is.finite(logh) | (logh == -Inf) \end{verbatim} is \verb@TRUE@. Points \verb@x@ having log unnormalized density \verb@-Inf@ have density zero (normalized or unnormalized, since a constant times zero is zero) hence cannot occur. Thus if \begin{verbatim} path <- metrop(fred, x, n, some, extra, arguments) \end{verbatim} then \begin{verbatim} all(is.finite(apply(path, 1, fred, some, extra, arguments))) \end{verbatim} is \verb@TRUE@. This is how we specify log unnormalized densities for distribution having support that is not all of Euclidean space. The value of the log unnormalized density off the support is \verb@-Inf@. Thus when coding a log unnormalized density, we should normally do something like \begin{verbatim} fred <- function(x, ...) { if (! is.numeric(x)) stop("argument x not numeric") if (length(x) != d) stop("argument x wrong length") if (! all(is.finite(x))) stop("elements of argument x not all finite") if (! is.in.the.support(x)) return(-Inf) return(log.unnormalized.density(x)) } \end{verbatim} where \verb@d@ is the dimension of the state space of the Markov chain (defined in the global environment or in the \verb@...@ arguments), \verb@is.in.the.support(x)@ returns \verb@TRUE@ if \verb@x@ is in the support of the desired equilibrium distribution and \verb@FALSE@ otherwise and \verb@log.unnormalized.density(x)@ calculates the log unnormalized density of the desired equilibrium distribution at the point \verb@x@, which is guaranteed to be finite because \verb@x@ is in the support if the this code is executed. Of course, you needn't actually have functions named \verb@is.in.the.support@ and \verb@log.unnormalized.density@. The point is that you use this logic. First you check whether \verb@x@ is in the support. If not return \verb@-Inf@. If it is, return a finite value. Do not crash. Do not return \verb@NA@, \verb@NaN@, or \verb@Inf@. If you do, then \verb@metrop@ crashes, and it's your fault. Of course, a crash is no big deal. Lots of first efforts in R crash. You just fix the problem and retry. Error messages are your friends. \subsection{Proposal} We also need to specify the proposal distribution (the preceding stuff assumed some default proposal). This can be any multivariate normal distribution on the Euclidean space of dimension \verb@length(initial)@ having mean zero. Thus it is specified by specifying its covariance matrix. But to avoid having to check whether the specified covariance matrix actually is a covariance matrix, we make the specification an arbitrary \verb@d@ by \verb@d@ matrix, call it \verb@scale@, where \verb@d@ is the dimension of the state space, specified by \verb@length(initial)@, and use the proposal \verb@x + scale %*% z@, where \verb@x@ is the current state and \verb@z@ is a standard normal random vector (``standard'' meaning its covariance matrix is the identity matrix). Thus we need to add this to the argument list of our function. It is now \begin{verbatim} metrop(lud, initial, niter, scale, ...) \end{verbatim} The covariance matrix specified by this is, of course, \verb@scale %*% t(scale)@. If you want the proposal to have covariance matrix \verb@melvin@, then specifying \verb@scale = t(chol(melvin))@ will do the job. (Of course, many other specifications will also do the job.) For convenience, we also allow \verb@scale@ to be a vector of length \verb@d@ and in this case take \verb@scale = sally@ to mean the same thing as \verb@scale = diag(sally)@. For convenience, we also allow \verb@scale@ to be a vector of length 1 and in this case take \verb@scale = sally@ to mean the same thing as \verb@scale = sally * diag(d)@ where \verb@d@ is still the dimension of the state space \verb@length(initial)@. We can use this last convenience option to give \verb@scale@ a default \begin{verbatim} metrop(lud, initial, niter, scale = 1, ...) \end{verbatim} In order to tell what is sensible scaling, we need to return the acceptance rate (the proportion of proposals that are accepted). The only criterion known for choosing sensible scaling is to adjust so that the acceptance rate is about 20\%. Of course, that recommendation was derived for a specific toy model that is not very much like what people do in real MCMC applications, so 20\% is only a very rough guideline. But acceptance rate is all we know to use, so that's what we will output. Thus the result of \verb@metrop@, assuming we write it in R will be something like \begin{verbatim} return(structure(list(path = path, rate = rate), class = "mcmc")) \end{verbatim} and if we write it in C will be whatever does the same job. \subsection{Output I} Generally we don't want \verb@path@ to be as described above. It may be way too big. We might have \verb@d@, the dimension of the state space $10^3$ or even larger and we might have \verb@niter@ $10^7$ or even larger, the resulting \verb@path@ matrix would be $10^{10}$ doubles or $8 \times 10^{10}$ bytes. Too big to fit in my computer! Thus we facilitate subsampling and batching of the output. \subsubsection{Subsampling} If the Markov chain exhibits high autocorrelation, subsampling the chain may lose little information. (Most users way overdo the subsampling, but it's not the job of a computer program to keep users from overdoing things). Thus we add an argument \verb@nspac@ that specifies subsampling. Only every \verb@nspac@ iterate is output. \subsubsection{Batching} The method of batch means uses ``batches'' which are sums over consecutive blocks of output. For most purposes batching is better than subsampling. It loses no information while reducing the amount of output even more than subsampling. So we introduce arguments \verb@nbatch@ specifying the number of batches and \verb@blen@ specifying the length of the batches. Our function now has signature \begin{verbatim} metrop(lud, initial, nbatch, blen = 1, nspac = 1, scale = 1, ...) \end{verbatim} Note that the argument \verb@niter@ formerly present has vanished. The number of iterations that will now be done is \verb@nbatch * blen * nspac@. If we accept the defaults \verb@blen = 1@ and \verb@nspac = 1@, then \verb@nbatch@ is the same as the former argument \verb@niter@. Otherwise, it is quite different. \subsection{Output II} The preceding section takes care of of the problem of \verb@niter@ being too big. This section deals with the dimension of the state space being too big. When the dimension of the state space is large, we generally do not want to output the whole state, but only some function of the state. Thus we need another function (besides \verb@lud@) to produce the output vector. Call it \verb@outfun@. The requirements on \verb@outfun@ are \begin{itemize} \item If \verb@is.finite(lud(x, ...))@, then \verb@outfun(x, ...)@ works (it does not crash) and produces a vector having all elements finite and always of the same length (say \verb@k@). \end{itemize} \verb@outfun@ will never be called in any other situation (that is, never when \verb@x@ is not in the support of the equilibrium distribution). Now we can describe the \verb@path@ component of the output. We'll use a little math here. Write $L$ for \verb@blen@ and $M$ for \verb@nspac@. Write $x_i$ for the $i$-th iterate of the Markov chain, and write $g$ for \verb@outfun@. Then \verb@path[@$j$\verb@, ]@ is the vector $$ \frac{1}{L} \sum_{i = 1}^L g(x_{M [L (j - 1) + i]}) $$ For convenience, we also allow \verb@outfun@ to be a logical vector of length \verb@d@ or an integer vector having elements in \verb@1:d@ or in \verb@-(1:d)@ and in this case take \verb@outfun = fred@ to mean the same thing as \verb@outfun = function(x) x[fred]@. For convenience, we also allow \verb@outfun@ to be missing take this to mean the same thing as \verb@outfun = function(x) x@, that is, the ``outfun'' is the identity function and we are back to outputting the entire state. Our function now has signature \begin{verbatim} metrop(lud, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, ...) \end{verbatim} \subsection{Restarting} It should be possible to restart the Markov chain and get exactly the same results. It should be possible to continue the Markov chain and get exactly the same results. Thus we need to save the initial and final state of the Markov chain and the initial and final state of the random number generator (the R object \verb@.Random.seed@). Thus the result of \verb@metrop@ now looks like \begin{verbatim} return(structure(list(path = path, rate = rate, initial = initial, final = final, initial.seed = iseed, final.seed = .Random.seed), class = "mcmc")) \end{verbatim} We also need to add arguments to \verb@metrop@. It now has signature \begin{verbatim} metrop(lud, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, object, restart = FALSE, ...) \end{verbatim} Here \verb@object@ is an R object of class \verb@"mcmc"@, the output a previous call to \verb@metrop@, from which we take either initial or final state and seed depending the value of \verb@restart@. While we are at it, it is convenient to allow any or all of the other arguments to be missing if \verb@object@ is supplied. We just take the argument from \verb@object@. Thus we can make calls like \begin{verbatim} out <- metrop(fred, x, 1e3, scale = 4, blen = 3) out.too <- metrop(object = out, nbatch = 1e4) \end{verbatim} Woof! I now see (how embarrasing) after four earlier versions how to use the R class system to make this convenient. We have three functions. \begin{verbatim} metrop.default <- function(o, ...) UseMethod("metrop") metrop.mcmc <- function(o, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, restart = FALSE, ...) { if (missing(nbatch)) nbatch <- o$nbatch if (missing(blen)) blen <- o$blen if (missing(nspac)) nspac <- o$nspac if (missing(scale)) scale <- o$scale if (missing(outfun)) outfun <- o$outfun if (restart) { .Random.seed <- o$final.seed return(metrop.function(o$lud, o$final, nbatch, blen, nspac, scale, outfun)) } else { .Random.seed <- o$initial.seed return(metrop.function(o$lud, o$initial, nbatch, blen, nspac, scale, outfun)) } } metrop.function <- function(o, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, restart = FALSE, ...) { if (! exists(".Random.seed")) runif(1) initial.seed <- .Random.seed func1 <- function(state) o(state, ...) func2 <- function(state) outfun(state, ...) .Call("metrop", func1, initial, nbatch, blen, nspac, scale, func2, environment(fun = func1), environment(fun = func2)) } \end{verbatim} Note that \verb@restart@ is ignored in \verb@metrop.function@. We can't ``restart'' when we have no saved state in an \verb@"mcmc"@ object. Note also that our \verb@"mcmc"@ objects must now store a lot more stuff (and more to come in the next section). \subsection{Testing and Debugging} It is nearly impossible to test or debug a random algorithm (any Monte Carlo) from looking at its designed (useful to the user) output. In order to do any serious testing or debugging, it is necessary to look under the hood. For the Metropolis algorithm, we need to look at the current state, the proposal, the log odds ratio, the uniform random variate (if any) used in the Metropolis rejection, and the result (accept or reject) of the Metropolis rejection. Hence we need to add one final argument \verb@debug = FALSE@ to our functions and a lot of debugging output to the result. In debugging a Metropolis (etc.)\ algorithm there is a very important principle. Debugging should use Markov chain theory! Just enlarge the state space of the Markov chain to include \begin{itemize} \item the proposal (vector), \item details of the calculation of the Metropolis-Hastings-Green ratio (for Metropolis this is just the log unnormalized density at the current state and proposal, for others it includes proposal densities) and the calculated ratio, \item the uniform random number (if any) used in the decision, and \item the decision (\verb@TRUE@ or \verb@FALSE@) in the Metropolis rejection. \end{itemize} With all that it is easy to tell whether the algorithm is doing the Right Thing. Without all that, it's nearly impossible. \end{document} mcmc/inst/designDoc/temper.tex0000644000175100001440000002477313063023671016117 0ustar hornikusers \documentclass[11pt]{article} \usepackage{amsmath} \usepackage{indentfirst} \usepackage{natbib} \usepackage{url} \RequirePackage{amsfonts} \newcommand{\real}{\mathbb{R}} \newcommand{\fatdot}{\,\cdot\,} \begin{document} \title{Simulated Tempering for the MCMC Package} \author{Charles J. Geyer} \maketitle \section{Parallel and Serial Tempering} Serial tempering \citep{marinari-parisi,geyer-thompson} runs a Markov chain whose state is $(i, x)$, where $i$ is a positive integer between 1 and $k$ and $x$ is an element of $\real^p$. The unnormalized density of the equilibrium distribution is $h(i, x)$. The integer $i$ is called the \emph{index of the component of the mixture}, and the integer $k$ is called the \emph{number of components of the mixture}. The reason for this terminology is that \begin{equation} \label{eq:mix} h(x) = \sum_{i = 1}^k h(i, x), \end{equation} which is the unnormalized marginal density of $x$ derived from the unnormalized joint density $h(i, x)$ of the equilibrium distribution of the Markov chain, is a mixture of $k$ component distributions having unnormalized density $h(i, \fatdot)$ for different $i$. Parallel tempering \citep{geyer} runs a Markov chain whose state is $(x_1, \ldots, x_k)$ where each $x_i$ is an element of $\real^p$. Thus the state is a vector whose elements are vectors, which may be thought of as a $k \times p$ matrix. The unnormalized density of the equilibrium distribution is \begin{equation} \label{eq:parallel-joint} h(x_1, \ldots, x_k) = \prod_{i \in I} h(i, x_i). \end{equation} This joint equilibrium distribution is the product of the marginals $h(i, \fatdot)$ for different $i$. This the $x_i$ are asymptotically independent in parallel tempering. \section{Sensitivity to Normalization} So long as one is only interested in one of the component distributions $h(i, \fatdot)$, both parallel and serial tempering do the job. And this job is what gives them the name ``tempering'' by analogy with simulated annealing \citep{marinari-parisi}. The other component distributions only help in sampling the component of interest. In this job, parallel tempering is easier to set up because it is insensitive to normalizing constants in the following sense. Suppose we change the normalization for each component distribution using $$ h^*(i, x) = a_i h(i, x). $$ This greatly changes the mixture distribution \eqref{eq:mix} sampled by simulated tempering. We now get $$ h^*(x) = \sum_{i = 1}^k h^*(i, x) = \sum_{i = 1}^k a_i h(i, x), $$ which may be very different from \eqref{eq:mix}, even considered as an unnormalized density (which it is). But \eqref{eq:parallel-joint}, considered as an unnormalized density (which it is), does not change at all \begin{align*} h^*(x_1, \ldots, x_k) & = \prod_{i = 1}^k h^*(i, x) \\ & = \prod_{i = 1}^k a_i h(i, x) \\ & = \left( \prod_{i = 1}^k a_i \right) \left( \prod_{i = 1}^k h(i, x) \right) \\ & = \left( \prod_{i = 1}^k a_i \right) h(x_1, \ldots, x_k) \end{align*} (the normalizing constant changes, but that does not matter for an unnormalized density; it still specifies the same probability distribution). All this is to say that serial tempering is very sensitive to the choices of normalizing constants of the individual component distributions (the $a_i$ in the preceding discussion) and parallel tempering is totally insensitive to them. Thus parallel tempering is easier to set up and get working. \citet{geyer-thompson}, however, independently invented serial tempering because it worked for a problem where parallel tempering failed. So for this ``tempering'' job, where one is only interested in sampling one component distribution (and the others are just helpers) parallel tempering is easier to use but serial tempering works better. \section{Umbrella Sampling} Sometimes one is actually interested in sampling a particular mixture distribution having unnormalized density \eqref{eq:mix}. This arises in Bayesian and frequentist model averaging and for other reasons. An umbrella term for this application is ``umbrella sampling'' \citep{torrie-valleau}. In this application only serial tempering does more than parallel tempering. Parallel tempering can simulate any directly specified mixture. If $$ f(i, x) = \frac{h(i, x)}{\int h(i, x) \, d x} $$ are the normalized component distributions and $b_1$, $\dots$, $b_k$ are nonnegative and sum to one, then $$ f(x) = \sum_{i = 1}^k b_i f(i, x) $$ is a normalized mixture distribution, and parallel tempering can be used to sample it. However, this ``directly specified'' mixture is often not of interest because the individual component normalizing constants \begin{equation} \label{eq:norm-con} c_i = \int h(i, x) \, d x \end{equation} are unknown. Suppose we are doing Bayesian model averaging and $h(i, x)$ is the unnormalized posterior density (likelihood times prior). This means $i$ and $x$ are parameters to the Bayesian; $i$ denotes the model and $x$ denotes the within-model parameters. Then the normalizing constants \eqref{eq:norm-con} are unnormalized Bayes factors, which Bayesians use for model comparison. The function $i \mapsto c_i$ is the unnormalized density of the marginal distribution of the random variable $i$ derived from the joint distribution $h(i, x)$, which is the equilibrium distribution of the Markov chain. It is therefore estimated, up to a constant of proportionality, by the marginal distribution of $i$. Thus serial tempering, unlike parallel tempering, provides simple and direct estimates of Bayes factors and other normalizing constants. \section{Update Mechanisms} Traditionally, tempering makes two kinds of elementary updates, one changes only $x$ in serial tempering or one $x_i$ in parallel tempering. We call them within-component updates, and will use normal random walk Metropolis updates analogous to those used by the \texttt{metrop} function. The other kind changes $i$ in serial tempering or swaps $x_i$ and $x_j$ in parallel tempering. We call them jump/swap updates (jump in serial tempering, swap in parallel tempering). \subsection{Serial Tempering} The combined update is a 50-50 mixture of within-component elementary updates and jump updates. Suppose the current state is $(i, x)$. A within-component update proposes $x^*$ which is normally distributed centered at $x$. Then Metropolis rejection of the proposal is done with Metropolis ratio $$ \frac{h(i, x^*)}{h(i, x)} $$ This is valid because the proposal is symmetric by symmetry of the normal distribution. A jump update proposes $i^*$, which is chosen uniformly at random from the ``neighbors'' of $i$ (the neighbor relation is specified by a user-supplied logical matrix). This proposal need not be symmetric, because $i$ and $i^*$ need not have the same number of neighbors. Write $n(i)$ and $n(i^*)$ for these neighbor counts. Then the probability of proposing $i^*$ when the current state is $i$ is $1 / n(i)$, and the probability of proposing $i$ when the current state is $i^*$ is $1 / n(i^*)$. Hence the appropriate Hastings ratio for Metropolis-Hastings rejection is $$ \frac{h(i^*, x) / n(i^*)}{h(i, x) / n(i)} = \frac{h(i^*, x)}{h(i, x)} \cdot \frac{n(i)}{n(i^*)} $$ \subsection{Parallel Tempering} The combined update is a 50-50 mixture of within-component elementary updates and swap updates. Suppose the current state is $(x_1, \ldots, x_k)$. A within-component chooses $i$ uniformly at random in $\{ 1, \ldots, k \}$, and then proposes $x_i^*$ which is normally distributed centered at $x_i$. Then Metropolis rejection of the proposal is done with Metropolis ratio $$ \frac{h(i, x_i^*)}{h(i, x_i)} $$ This is valid because the proposal is symmetric by symmetry of the normal distribution. A swap update chooses $i$ uniformly at random in $\{ 1, \ldots, k \}$ and then $j$, which is chosen uniformly at random from the neighbors of $i$. This proposal is automatically symmetric, because a swap move is its own inverse. Hence the appropriate Hastings ratio for Metropolis-Hastings rejection is $$ \frac{h(i, x_j) h(j, x_i)}{h(i, x_i) h(j, x_j)} $$ \section{Acceptance Rates} Metropolis-Hastings acceptance rates are not comparable to Metropolis acceptance rates. For serial tempering $$ E \left\{ 1 \wedge \frac{h(i^*, x)}{h(i, x)} \cdot \frac{n(i)}{n(i^*)} \right\} \neq E \left\{ 1 \wedge \frac{h(i^*, x)}{h(i, x)} \right\} $$ where the expectations are taken with respect to $(i, x)$ having the equilibrium distribution of the Markov chain and the conditional distribution of $i^*$ given $i$ being uniform over neighbors of $i$. For parallel tempering, $$ E \left\{ 1 \wedge \frac{h(i, x_j) h(j, x_i)}{h(i, x_i) h(j, x_j)} \cdot \frac{n(i)}{n(j)} \right\} \neq E \left\{ 1 \wedge \frac{h(i, x_j) h(j, x_i)}{h(i, x_i) h(j, x_j)} \right\} $$ where the expectations are taken with respect to $(x_1, \ldots, x_k)$ having the equilibrium distribution of the Markov chain, $(i, j)$ being independent of $(x_1, \ldots, x_k)$, the marginal distribution of $i$ being uniform on $\{ 1, \ldots, k \}$, and the conditional distribution of $j$ given $i$ being uniform over neighbors of $i$. Thus we need to report rates going both ways, for example, for serial tempering, when $i = 1$ and $i^* = 2$ as well as when $i = 2$ and $i^* = 1$. And similarly for parallel tempering. \begin{thebibliography}{} \bibitem[Geyer(1991)]{geyer} Geyer, C.~J. (1991). \newblock Markov chain Monte Carlo maximum likelihood. \newblock \emph{Computing Science and Statistics: Proceedings of the Symposium on Interface Critical Applications of Scientific Computing (23rd): Biology, Engineering, Medicine, Speech Held in Seattle, Washington on 21-24 April 1991}, J.~R. Kettenring and E.~M. Keramidas, eds., 156--163. \newblock \url{http://www.stat.umn.edu/geyer/f05/8931/c.ps} \bibitem[Geyer and Thompson(1995)]{geyer-thompson} Geyer, C.~J., and Thompson, E.~A. (1995). \newblock Annealing Markov chain Monte Carlo with applications to ancestral inference. \newblock \emph{Journal of the American Statistical Association}, \textbf{90}, 909--920. \bibitem[Marinari and Parisi(1992)]{marinari-parisi} Marinari, E., and Parisi G. (1992). \newblock Simulated tempering: A new Monte Carlo Scheme. \newblock \emph{Europhysics Letters}, \textbf{19}, 451--458. \bibitem[Torrie and Valleau(1977)]{torrie-valleau} Torrie, G.~M., and Valleau, J.~P. (1977). \newblock Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. \newblock \emph{Journal of Computational Physics}, \textbf{23}, 187--199. \end{thebibliography} \end{document} mcmc/inst/doc/0000755000175100001440000000000013074514644012741 5ustar hornikusersmcmc/inst/doc/metrop.pdf0000644000175100001440000046473213062557432014761 0ustar hornikusers%PDF-1.5 % 3 0 obj << /Length 1805 /Filter /FlateDecode >> stream xڭXK6WDbU$E=NEy)t6m-%e΃%YiH ɷ? y,N7w qYM*=T"/޾_2~jc7 =ZKfqQNqg+{G/,"QFii\o|"{29,4ViF/v*2)X:z]o:߀}皚2.3B@^ Կ;;=@". {u*\OD.w<9[wllW>H*$2 խ*s1PQ;&d.b |;3,'=MD-z%$'[,RQSW&" J_>4#J1Շo=cSD{ toz$Itpm}oA MX:>ؚjI?ꓓ>WNLK]0^zkkx S =ĠLgo(.]V4LK*vEbYH* zIXgvIkaa}ڲD$Uy&ڹtYUP fRdQ)dڳ遚B _C*w]i.PvgD M;>M鋜DCNѽi?:X6^cSU Ȯut)(̄]I: Uj(G cY ;M'C(GSTWPVRpȤ#kl]t5p]2F>PiZRyK'Xx!BJqFC=: "IDai6lZ炀Ud4x(@w2rR[2jŅKmQaN3"\EPe[fxcx"B0ޞ a' P5?*NQT;IJsYIT%*@2yTJ'>DN^355f2ۀeŷ1IU\1rU.T 0KZԽgwpA 1yBU ΊP=A 7PQOfhԮ%^{AX_ibn ,uNp3Pl)[+1AFez! rY淄-Ł0,?ozRgQx{YXS$oWXduNG΍p0pb;R~Mo[ aAGȅiQ.&(tC5S:q)$<8Pd;REgdT>PJõ8W@tMytյ?GN?~_1Msb5V^f22\D2а;˛0. endstream endobj 19 0 obj << /Length 2349 /Filter /FlateDecode >> stream xڥYY~ׯX 0M4mHlD8JldNGrC =u4ٞ?,Guu_~7U.D&W7H)&US!~l_}#7;CFU[n~k޴#CK9ji |#kwq7 ̂|z'׻((xⅺcӴݱ'SLeָ;mG*6t(𪏶s"-xz}0!,? ;]oP 1JjGQ#c8堻{83aޝ-diy6=NXE /B9 !bq.9q.Ls©,;Is8ґ~c~pI$1h%) <GL.d(&$KI˱L{^"tF.^d_V|uW ]4YgvUv E뭑̥J]%_xPvU!LKU;_fq \!%_+yhyL>#Hf{2J.OU\G|O ]{q^HJN`PЕ)'cߡ끣K@,UXEi2r@n{kxPƾ@jQ44VSS=.G',Iqq 'koوH1vTq F5@+,)}qgL'q*iޡV"]@0EA,zAAL }ˣ/7߻,kOnӥh x)M;J:\0^FnRHx[cxz 5svn/ϔ!a3,qDMIˡT )w{crHKZ iw)=U[u[Z> uK<[ X<[~'D3]{yd֡ qSgS ftSMLhZ3t2oJx>|?bn3V=imK7 K0kҔ$, ?{XyfqS^9VBz{҆yq?MV}`\'/W'TmhnݓTD&C6߲q2j,J=V%2 cU)`RPI`&YC&8wQZH4KUs 7SO_'8pV;Dt::q )^H3G MJ8`gU0ʉV"5 <,Tto18Ww6S*gѹ~i8LقI*6@i\GH~^*RnO_ faYzc-k9dj 0!`6L#zEmbǺ/j]S'/նw٩uU2XpcI"k(>0Pעe̿¥o^7>A endstream endobj 22 0 obj << /Length 2375 /Filter /FlateDecode >> stream xڝYɎF+j f4svh3 aC6]EEظU-ɗܘ"Eo~yk]ĻǝV%qDo0 .2QPe_Jz)Ziw i+3߿]TNÐcOZ$}-}Ӹ ξo˽pLy1Ԯ8{{ w)yEy3HJ­`Viv! 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QAp~ɿԶrOI!Ciz~un7Q˓A>C2Q>.*o0.A}6N6xʼn!&hr8Ȃ_.__V ] 5)HiB_tQ&`WWy'He+c_F0o7 󯞾ytJڤ(Cx^`$V8Dn@p=@id=X]" rڗ:] liQ=Ը' U_9<ͽy:i`>79~ߛ#{DX7"/•F*t\de>^Ά_ )Zw>.(Ŧ3RͪBE闿d39Dhq_-}8F/PN@WWF11hw_MU^U ?MC)t_I)f-6t%LQ8<|bkތXj\-(R)y>p5::@Eh%%D7 x(~Ufu,6eq樥FO>O A@v{8sז};(a(ߎs~pcGˎl&%Qn»m+\£hd; <43444718EAD7017BCE21423F9E63935E>] /Length 228 /Filter /FlateDecode >> stream xM3qs~*J\+z6cgξW`ojkh 1:|{0@RP8Ǣ,"đSQg~\L@į?zl&r%D(~y<I1CT͊91/R"-ĢX,VĪȊQ5kxyrmk޻ dεEFmSrD_vQϵP{u17b!> endstream endobj startxref 162057 %%EOF mcmc/inst/doc/debug.R0000644000175100001440000000037613074514644014160 0ustar hornikusers### R code from vignette source 'debug.Rnw' ################################################### ### code chunk number 1: foo ################################################### options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") mcmc/inst/doc/debug.Rnw0000644000175100001440000003135013074514644014521 0ustar hornikusers \documentclass{article} \usepackage{amstext} % \VignetteIndexEntry{Debugging MCMC Code} \begin{document} <>= options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") @ \title{Debugging MCMC Code} \author{Charles J. Geyer} \maketitle \section{Introduction} This document discusses debugging Markov chain Monte Carlo code using the R contributed package \texttt{mcmc} (Version \Sexpr{foo$Version}) for examples. It also documents the debugging output of the functions \texttt{mcmc} and \texttt{temper}. Debugging MCMC code if the code is taken as a black box is basically impossible. In interesting examples, the only thing one knows about the equilibrium distribution of an MCMC sampler is what one learns from the samples. This obviously doesn't help with testing. If the sampler is buggy, then the only thing you know about the equilibrium distribution is wrong, but if you don't know it is buggy, then you don't know it is wrong. So you don't know anything. There is no way to tell whether random output has the correct distribution when you don't know anything about the distribution it is supposed to have. The secret to debugging MCMC code lies in two principles: \begin{itemize} \item take the randomness out, and \item expose the innards. \end{itemize} The first slogan means consider the algorithm a deterministic function of the elementary pseudo-random numbers that are trusted (for example, the outputs of the R random number generators, which you aren't responsible for debugging and are also well tested). The second slogan means output, at least for debugging purposes, enough intermediate state so that testing is straightforward. For a Gibbs sampler, this means outputting all of the trusted elementary pseudo-random numbers used, the state before and after each elementary Gibbs update, and which update is being done if a random scan is used. Also one needs to output the initial seeds of the pseudo-random number generator (this is true for all situations and will not be mentioned again). For a Metropolis-Hastings sampler, this means outputting all of the trusted elementary pseudo-random numbers used, the state before and after each elementary Metropolis-Hastings update, the proposal for that update, the Hastings ratio for that update, decision (accept or reject) in that update. For more complicated MCMC samplers, there is more ``innards'' to ``expose'' (see the discussion of the \texttt{temper} function below), but you get the idea. You can't output too much debugging information. \section{The Metrop Function} The R function \texttt{metrop} in the \texttt{mcmc} package has an argument \verb@debug = FALSE@ that when \verb@TRUE@ causes extra debugging information to be output. Let \texttt{niter} be the number of iterations \verb@nbatch * blen * nspac@, and let \texttt{d} be the dimension of the state vector. The result of invoking \texttt{metrop} is a list. When \verb@debug = TRUE@ it has the following additional components \begin{itemize} \item \texttt{current}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@current[i, ]@ \item \texttt{proposal}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ \item \texttt{z}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ \item \texttt{log.green}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{u}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{debug.accept}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \end{itemize} (The components \texttt{z} and \texttt{debug.accept} were added in version 0.7-3 of the \texttt{mcmc} package. Before that only the others were output.) Two components of the list returned by the \texttt{metrop} function always (whether \verb@debug = TRUE@ or \verb@debug = FALSE@) are also necessary for debugging. They are \begin{itemize} \item \texttt{initial.seed} the value of the variable \texttt{.Random.seed} that contains the seeds of the R random number generator system before invocation of the \texttt{metrop} function \item \texttt{final}, a vector of length \texttt{d} and mode \verb@"numeric"@, the state after the last iteration \end{itemize} All of the files in the \texttt{tests} directory of the source for the package (not installed but found in the source tarball on CRAN) test the \texttt{metrop} function except those beginning \texttt{temp}, which test the \texttt{temper} function. Since these tests were written many years ago, are spread out over many files, and are not commented, we will not describe them in detail. Suffice it to say that they check every aspect of the functioning of the \texttt{metrop} function. \section{The Temper Function} The R function \texttt{temper} in the \texttt{mcmc} package has an argument \verb@debug = FALSE@ that when \verb@TRUE@ causes extra debugging information to be output. Let \texttt{niter} be the number of iterations \verb@nbatch * blen * nspac@, let \texttt{d} be the dimension of the state vector, and let \texttt{ncomp} be the number of components of the tempering mixture. The result of invoking \texttt{temper} is a list. When \verb@debug = TRUE@ and \verb@parallel = TRUE@ it has the following additional components % which % unif.which % state % log.hastings % unif.hastings % proposal % acceptd % norm % unif.choose % coproposal \begin{itemize} \item \texttt{which}, a vector of length \texttt{niter} and mode \verb@"logical"@ the type of update for each iteration, within component (\texttt{TRUE}) or swap components (\texttt{FALSE}). \item \texttt{unif.which}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate used to decide which type of update is done. \item \texttt{state}, an \texttt{niter} by \texttt{ncomp} by \texttt{d} array of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@state[i, , ]@ \item \texttt{proposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ (explanation below) \item \texttt{coproposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@coproposal[i, ]@ (explanation below) \item \texttt{log.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{unif.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{acceptd}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \item \texttt{norm}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ unless none are needed (for swap updates) when it is \texttt{NA} \item \texttt{unif.choose}, an \texttt{niter} by 2 matrix of mode \verb@"numeric"@, the vector of $\text{Uniform}(0, 1)$ random variates used to choose the components to update in iteration \texttt{i} is \verb@unif.choose[i, ]@; in a swap update two are used; in a within-component update only one is used and the second is \texttt{NA} \end{itemize} In a within-component update, one component say \texttt{j} is chosen for update. The \emph{coproposal} is the current value of the state for this component, which is a vector of length \verb@d + 1@, the first component of which is \texttt{j} and the rest of which is \verb@state[i, j, ]@ if we are in iteration \texttt{i}. The \emph{proposal} is a similar vector, the first component of which is again \texttt{j} and the rest of which is a multivariate normal random vector centered at \verb@state[i, j, ]@. The coproposal is the current state; the proposal is the possible value (if accepted) of the state at the next time. In a swap update, two components say \texttt{j1} and \texttt{j2} are chosen for update. Strictly, speaking the coproposal is the pair of vectors \verb@c(j1, state[i, j1, ])@ and \verb@c(j2, state[i, j2, ])@ and the proposal is these swapped, that is, the pair of vectors \verb@c(j2, state[i, j1, ])@ and \verb@c(j1, state[i, j2, ])@ if we are in iteration \texttt{i}. Since, however, there is a lot of redundant information here, the vector \verb@c(j1, state[i, j1, ])@ is output as \verb@coproposal[i, ]@ and the vector \verb@c(j2, state[i, j2, ])@ is output as \verb@proposal[i, ]@. When \verb@debug = TRUE@ and \verb@parallel = FALSE@ the result of invoking \texttt{temper} is a list having the following additional components % which % unif.which % state % log.hastings % unif.hastings % proposal % acceptd % norm % unif.choose \begin{itemize} \item \texttt{which}, a vector of length \texttt{niter} and mode \verb@"logical"@ the type of update for each iteration, within component (\texttt{TRUE}) or jump from one component to another (\texttt{FALSE}). \item \texttt{unif.which}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate used to decide which type of update is done. \item \texttt{state}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@state[i, ]@ \item \texttt{proposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ \item \texttt{log.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{unif.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{acceptd}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \item \texttt{norm}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ unless none are needed (for jump updates) when it is \texttt{NA} \item \texttt{unif.choose}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variates used to choose the component to update in iteration \texttt{i} is \verb@unif.choose[i, ]@; in a jump update one is used; in a within-component update none is used and \texttt{NA} is output \end{itemize} All of the files in the \texttt{tests} directory of the source for the package (not installed but found in the source tarball on CRAN) beginning \texttt{temp} test the \texttt{temper} function. They check every aspect of the functioning of the \texttt{temper} function. In the file \texttt{temp-par.R} in the \texttt{tests} directory, the following checks are made according to the comments in that file \begin{enumerate} \item check decision about within-component or jump/swap \item check proposal and coproposal are actually current state or part thereof \item check hastings ratio calculated correctly \item check hastings rejection decided correctly \item check acceptance carried out or not (according to decision) correctly \item check within-component proposal \item check swap proposal \item check standard normal and uniform random numbers are as purported \item check batch means \item check acceptance rates \item check scale vector \item check scale matrix \item check scale list \item check outfun \end{enumerate} In the file \texttt{temp-ser.R} in the \texttt{tests} directory, the all of the same checks are made according to the comments in that file except for check number 2 above, which would make no sense because there is no \texttt{coproposal} component in the serial (\verb@parallel = FALSE@) case. \end{document} mcmc/inst/doc/bfst.pdf0000644000175100001440000070031113074514644014374 0ustar hornikusers%PDF-1.5 % 3 0 obj << /Length 1899 /Filter /FlateDecode >> stream xYKo7WP17EM SujӃlIZKjFC[Bi֜O?a!,~! 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Ex,E\'U3)qwj'c_\`\Yu='SOWȎAWd $zɰåb^,0|I\ ; endstream endobj startxref 228955 %%EOF mcmc/inst/doc/bfst.R0000644000175100001440000001536613074514644014035 0ustar hornikusers### R code from vignette source 'bfst.Rnw' ### Encoding: UTF-8 ################################################### ### code chunk number 1: foo ################################################### options(keep.source = TRUE, width = 65) ################################################### ### code chunk number 2: library ################################################### library(mcmc) ################################################### ### code chunk number 3: baz ################################################### baz <- library(help = "mcmc") baz <- baz$info[[1]] baz <- baz[grep("Version", baz)] baz <- sub("^Version: *", "", baz) bazzer <- paste(R.version$major, R.version$minor, sep = ".") ################################################### ### code chunk number 4: set-seed ################################################### set.seed(42) ################################################### ### code chunk number 5: frequentist ################################################### data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial, x = TRUE) summary(out) ################################################### ### code chunk number 6: models ################################################### varnam <- names(coefficients(out)) varnam <- varnam[varnam != "(Intercept)"] nvar <- length(varnam) models <- NULL foo <- seq(0, 2^nvar - 1) for (i in 1:nvar) { bar <- foo %/% 2^(i - 1) bar <- bar %% 2 models <- cbind(bar, models, deparse.level = 0) } colnames(models) <- varnam models ################################################### ### code chunk number 7: neighbor ################################################### neighbors <- matrix(FALSE, nrow(models), nrow(models)) for (i in 1:nrow(neighbors)) { for (j in 1:ncol(neighbors)) { foo <- models[i, ] bar <- models[j, ] if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE } } ################################################### ### code chunk number 8: ludfun ################################################### modmat <- out$x y <- logit$y ludfun <- function(state, log.pseudo.prior) { stopifnot(is.numeric(state)) stopifnot(length(state) == ncol(models) + 2) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= nrow(models)) stopifnot(is.numeric(log.pseudo.prior)) stopifnot(length(log.pseudo.prior) == nrow(models)) beta <- state[-1] inies <- c(TRUE, as.logical(models[icomp, ])) beta.logl <- beta beta.logl[! inies] <- 0 eta <- as.numeric(modmat %*% beta.logl) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) logl + sum(dnorm(beta, 0, 2, log = TRUE)) + log.pseudo.prior[icomp] } ################################################### ### code chunk number 9: try1 ################################################### state.initial <- c(nrow(models), out$coefficients) qux <- rep(0, nrow(models)) out <- temper(ludfun, initial = state.initial, neighbors = neighbors, nbatch = 1000, blen = 100, log.pseudo.prior = qux) names(out) out$time ################################################### ### code chunk number 10: what ################################################### ibar <- colMeans(out$ibatch) ibar ################################################### ### code chunk number 11: adjust ################################################### qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux ################################################### ### code chunk number 12: iterate ################################################### lout <- suppressWarnings(try(load("bfst1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { qux.save <- qux time.save <- out$time repeat{ out <- temper(out, log.pseudo.prior = qux) ibar <- colMeans(out$ibatch) qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux.save <- rbind(qux.save, qux, deparse.level = 0) time.save <- rbind(time.save, out$time, deparse.level = 0) if (max(ibar) / min(ibar) < 2) break } save(out, qux, qux.save, time.save, file = "bfst1.rda") } else { .Random.seed <- out$final.seed } print(qux.save, digits = 3) print(qux, digits = 3) apply(time.save, 2, sum) ################################################### ### code chunk number 13: accept-i-x ################################################### print(out$accepti, digits = 3) print(out$acceptx, digits = 3) ################################################### ### code chunk number 14: accept-i-min ################################################### min(as.vector(out$accepti), na.rm = TRUE) ################################################### ### code chunk number 15: scale ################################################### out <- temper(out, scale = 0.5, log.pseudo.prior = qux) time.save <- rbind(time.save, out$time, deparse.level = 0) print(out$acceptx, digits = 3) ################################################### ### code chunk number 16: try6 ################################################### lout <- suppressWarnings(try(load("bfst2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out <- temper(out, blen = 10 * out$blen, log.pseudo.prior = qux) save(out, file = "bfst2.rda") } else { .Random.seed <- out$final.seed } time.save <- rbind(time.save, out$time, deparse.level = 0) foo <- apply(time.save, 2, sum) foo.min <- floor(foo[1] / 60) foo.sec <- foo[1] - 60 * foo.min c(foo.min, foo.sec) ################################################### ### code chunk number 17: doit ################################################### log.10.unnorm.bayes <- (qux - log(colMeans(out$ibatch))) / log(10) k <- seq(along = log.10.unnorm.bayes)[log.10.unnorm.bayes == min(log.10.unnorm.bayes)] models[k, ] log.10.bayes <- log.10.unnorm.bayes - log.10.unnorm.bayes[k] log.10.bayes ################################################### ### code chunk number 18: doit-se-one ################################################### fred <- var(out$ibatch) / out$nbatch sally <- colMeans(out$ibatch) mcse.log.10.bayes <- (1 / log(10)) * sqrt(diag(fred) / sally^2 - 2 * fred[ , k] / (sally * sally[k]) + fred[k, k] / sally[k]^2) mcse.log.10.bayes foompter <- cbind(models, log.10.bayes, mcse.log.10.bayes) round(foompter, 5) ################################################### ### code chunk number 19: doit-too ################################################### ibar <- colMeans(out$ibatch) herman <- sweep(out$ibatch, 2, ibar, "/") herman <- sweep(herman, 1, herman[ , k], "-") mcse.log.10.bayes.too <- (1 / log(10)) * apply(herman, 2, sd) /sqrt(out$nbatch) all.equal(mcse.log.10.bayes, mcse.log.10.bayes.too) mcmc/inst/doc/morph.R0000644000175100001440000002702213074514644014214 0ustar hornikusers### R code from vignette source 'morph.Rnw' ### Encoding: UTF-8 ################################################### ### code chunk number 1: foo ################################################### options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") ################################################### ### code chunk number 2: morph.Rnw:98-100 ################################################### library(mcmc) h2 <- morph(b=1) ################################################### ### code chunk number 3: morph.Rnw:105-107 ################################################### lud <- function(x) dt(x, df=3, log=TRUE) lud.induced <- h2$lud(lud) ################################################### ### code chunk number 4: morph.Rnw:110-114 ################################################### curve(exp(Vectorize(lud.induced)(x)), from = -3, to = 3, lty = 2, xlab = "t", ylab = "density") curve(exp(lud(x)), add = TRUE) legend("topright", c("t density", "induced density"), lty=1:2) ################################################### ### code chunk number 5: morph.Rnw:123-129 ################################################### lud(1:4) lud(1) foo <- try(lud.induced(1:4)) class(foo) cat(foo, "\n") lud.induced(1) ################################################### ### code chunk number 6: set-seed ################################################### set.seed(42) ################################################### ### code chunk number 7: morph.Rnw:146-147 ################################################### out <- morph.metrop(lud, 0, blen=100, nbatch=100, morph=morph(b=1)) ################################################### ### code chunk number 8: morph.Rnw:153-155 ################################################### # adjust scale to find a roughly 20% acceptance rate out$accept ################################################### ### code chunk number 9: morph.Rnw:161-163 ################################################### out <- morph.metrop(out, scale=4) out$accept ################################################### ### code chunk number 10: fig0too ################################################### acf(out$batch) ################################################### ### code chunk number 11: fig0 ################################################### acf(out$batch) ################################################### ### code chunk number 12: morph.Rnw:187-188 ################################################### t.test(out$batch) ################################################### ### code chunk number 13: morph.Rnw:191-193 ################################################### colMeans(out$batch) apply(out$batch, 2, sd) / sqrt(out$nbatch) ################################################### ### code chunk number 14: unmorph-metrop-adjust ################################################### out.unmorph <- metrop(lud, 0, blen=1000, nbatch=1) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=4) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=6) out.unmorph$accept ################################################### ### code chunk number 15: unmorph-metrop-t-long-run ################################################### lout <- suppressWarnings(try(load("morph1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.unmorph <- metrop(out.unmorph, blen = 1e5, nbatch = 1e3) save(out.unmorph, file = "morph1.rda") } else { .Random.seed <- out.unmorph$final.seed } out.unmorph$accept ################################################### ### code chunk number 16: fig4too ################################################### foo <- as.vector(out.unmorph$batch) qqnorm(foo) qqline(foo) ################################################### ### code chunk number 17: fig4 ################################################### foo <- as.vector(out.unmorph$batch) qqnorm(foo) qqline(foo) ################################################### ### code chunk number 18: shapiro-wilk ################################################### shapiro.test(foo) ################################################### ### code chunk number 19: morph-metrop-t-long-run ################################################### lout <- suppressWarnings(try(load("morph2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.morph <- morph.metrop(out, blen = 1e5, nbatch = 1e3) save(out.morph, file = "morph2.rda") } else { .Random.seed <- out.morph$final.seed } out.morph$accept ################################################### ### code chunk number 20: fig5too ################################################### foo <- as.vector(out.morph$batch) qqnorm(foo) qqline(foo) ################################################### ### code chunk number 21: fig5 ################################################### foo <- as.vector(out.morph$batch) qqnorm(foo) qqline(foo) ################################################### ### code chunk number 22: shapiro-wilk ################################################### shapiro.test(foo) ################################################### ### code chunk number 23: def-posterior-binom ################################################### lud.binom <- function(beta, M, x, n) { MB <- M %*% beta sum(x * MB) - sum(n * log(1 + exp(MB))) } ################################################### ### code chunk number 24: convert ################################################### dat <- as.data.frame(UCBAdmissions) dat.split <- split(dat, dat$Admit) dat.split <- lapply(dat.split, function(d) { val <- as.character(d$Admit[1]) d["Admit"] <- NULL names(d)[names(d) == "Freq"] <- val d }) dat <- merge(dat.split[[1]], dat.split[[2]]) ################################################### ### code chunk number 25: build-model-matrix ################################################### formula <- cbind(Admitted, Rejected) ~ (Gender + Dept)^2 mf <- model.frame(formula, dat) M <- model.matrix(formula, mf) ################################################### ### code chunk number 26: morph.Rnw:396-398 ################################################### xi <- 0.30 nu <- 5 ################################################### ### code chunk number 27: lud-binom ################################################### lud.berkeley <- function(B) lud.binom(B, M, dat$Admitted + xi * nu, dat$Admitted + dat$Rejected + nu) ################################################### ### code chunk number 28: morph.Rnw:410-419 ################################################### berkeley.out <- morph.metrop(lud.berkeley, rep(0, ncol(M)), blen=1000, nbatch=1, scale=0.1, morph=morph(p=3)) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.05) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.02) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, blen=10000) berkeley.out$accept ################################################### ### code chunk number 29: morph.Rnw:422-423 ################################################### berkeley.out <- morph.metrop(berkeley.out, blen=1, nbatch=100000) ################################################### ### code chunk number 30: morph.Rnw:428-433 ################################################### beta <- setNames(colMeans(berkeley.out$batch), colnames(M)) MB <- M %*% beta dat$p <- dat$Admitted / (dat$Admitted + dat$Rejected) dat$p.post <- exp(MB) / (1 + exp(MB)) dat ################################################### ### code chunk number 31: calculate-posterior-probabilities ################################################### posterior.probabilities <- t(apply(berkeley.out$batch, 1, function(r) { eMB <- exp(M %*% r) eMB / (1 + eMB) })) quants <- apply(posterior.probabilities, 2, quantile, prob=c(0.05, 0.95)) quants.str <- matrix(apply(quants, 2, function(r) sprintf("[%0.2f, %0.2f]", r[1], r[2])), nrow=2, byrow=TRUE) ################################################### ### code chunk number 32: fig1 ################################################### x <- (0:5) * 2 + 1 plot(x[c(1, 6)] + 0.5 * c(-1, 1), 0:1, xlab="Department", ylab="Probability", xaxt="n", type="n") axis(1, x, LETTERS[1:6]) for(i in 1:6) { lines((x[i]-0.25)*c(1, 1), quants[1:2, i], lwd=2, col="gray") lines((x[i] + 0.25) * c(1, 1), quants[1:2, i + 6], lwd=2, col="gray") points(x[i] + 0.25 * c(-1, 1), dat$p.post[i + c(0, 6)], pch=c("F", "M")) } ################################################### ### code chunk number 33: cauchy-data ################################################### n <- 15 mu0 <- 50 sigma0 <- 10 x <- rcauchy(n, mu0, sigma0) round(sort(x), 1) ################################################### ### code chunk number 34: cauchy-log-unnormalized-posterior ################################################### lup <- function(theta) { if (any(is.na(theta))) stop("NA or NaN in input to log unnormalized density function") mu <- theta[1] sigma <- theta[2] if (sigma <= 0) return(-Inf) if (any(! is.finite(theta))) return(-Inf) result <- sum(dcauchy(x, mu, sigma, log = TRUE)) - log(sigma) if (! is.finite(result)) { warning(paste("Oops! mu = ", mu, "and sigma =", sigma)) } return(result) } ################################################### ### code chunk number 35: cauchy-robust ################################################### mu.twiddle <- median(x) sigma.twiddle <- IQR(x) c(mu.twiddle, sigma.twiddle) ################################################### ### code chunk number 36: cauchy-posterior-mode ################################################### oout <- optim(c(mu.twiddle, sigma.twiddle), lup, control = list(fnscale = -1), hessian = TRUE) stopifnot(oout$convergence == 0) mu.hat <- oout$par[1] sigma.hat <- oout$par[2] c(mu.hat, sigma.hat) ################################################### ### code chunk number 37: cauchy-hessian ################################################### oout$hessian ################################################### ### code chunk number 38: cauchy-se ################################################### sqrt(- 1 / diag(oout$hessian)) ################################################### ### code chunk number 39: cauchy-doit ################################################### moo <- morph(b = 0.5, r = 7, center = c(mu.hat, sigma.hat)) mout <- morph.metrop(lup, c(mu.hat, sigma.hat), 1e4, scale = 3, morph = moo) mout$accept mout <- morph.metrop(mout) ################################################### ### code chunk number 40: cfig1too ################################################### acf(mout$batch) ################################################### ### code chunk number 41: cfig1 ################################################### acf(mout$batch) ################################################### ### code chunk number 42: cfig2too ################################################### mu <- mout$batch[ , 1] i <- seq(1, mout$nbatch, by = 15) out.sub <- density(mu[i]) out <- density(mu, bw = out.sub$bw) plot(out) ################################################### ### code chunk number 43: cfig2 ################################################### mu <- mout$batch[ , 1] i <- seq(1, mout$nbatch, by = 15) out.sub <- density(mu[i]) out <- density(mu, bw = out.sub$bw) plot(out) ################################################### ### code chunk number 44: cfig3 ################################################### sigma <- mout$batch[ , 2] out.sub <- density(sigma[i]) out <- density(sigma, bw = out.sub$bw) plot(out) mcmc/inst/doc/demo.R0000644000175100001440000002152713074514644014017 0ustar hornikusers### R code from vignette source 'demo.Rnw' ### Encoding: UTF-8 ################################################### ### code chunk number 1: foo ################################################### options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") ################################################### ### code chunk number 2: frequentist ################################################### library(mcmc) data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial(), x = TRUE) summary(out) ################################################### ### code chunk number 3: log.unnormalized.posterior ################################################### x <- out$x y <- out$y lupost <- function(beta, x, y) { eta <- as.numeric(x %*% beta) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) return(logl - sum(beta^2) / 8) } ################################################### ### code chunk number 4: metropolis-try-1 ################################################### set.seed(42) # to get reproducible results beta.init <- as.numeric(coefficients(out)) out <- metrop(lupost, beta.init, 1e3, x = x, y = y) names(out) out$accept ################################################### ### code chunk number 5: metropolis-try-2 ################################################### out <- metrop(out, scale = 0.1, x = x, y = y) out$accept out <- metrop(out, scale = 0.3, x = x, y = y) out$accept out <- metrop(out, scale = 0.5, x = x, y = y) out$accept out <- metrop(out, scale = 0.4, x = x, y = y) out$accept ################################################### ### code chunk number 6: metropolis-try-3 ################################################### out <- metrop(out, nbatch = 1e4, x = x, y = y) out$accept out$time ################################################### ### code chunk number 7: fig1too ################################################### plot(ts(out$batch)) ################################################### ### code chunk number 8: fig1 ################################################### plot(ts(out$batch)) ################################################### ### code chunk number 9: fig2too ################################################### acf(out$batch) ################################################### ### code chunk number 10: fig2 ################################################### acf(out$batch) ################################################### ### code chunk number 11: metropolis-try-4 ################################################### out <- metrop(out, nbatch = 1e2, blen = 100, outfun = function(z, ...) c(z, z^2), x = x, y = y) out$accept out$time ################################################### ### code chunk number 12: metropolis-batch ################################################### apply(out$batch, 2, mean) ################################################### ### code chunk number 13: metropolis-batch-too ################################################### foo <- apply(out$batch, 2, mean) mu <- foo[1:5] sigmasq <- foo[6:10] - mu^2 mu sigmasq ################################################### ### code chunk number 14: metropolis-mcse-mu ################################################### mu.mcse <- apply(out$batch[ , 1:5], 2, sd) / sqrt(out$nbatch) mu.mcse ################################################### ### code chunk number 15: metropolis-mcse-sigmasq ################################################### u <- out$batch[ , 1:5] v <- out$batch[ , 6:10] ubar <- apply(u, 2, mean) vbar <- apply(v, 2, mean) deltau <- sweep(u, 2, ubar) deltav <- sweep(v, 2, vbar) foo <- sweep(deltau, 2, ubar, "*") sigmasq.mcse <- sqrt(apply((deltav - 2 * foo)^2, 2, mean) / out$nbatch) sigmasq.mcse ################################################### ### code chunk number 16: metropolis-mcse-sigmasq-too ################################################### sqrt(mean(((v[ , 2] - vbar[2]) - 2 * ubar[2] * (u[ , 2] - ubar[2]))^2) / out$nbatch) ################################################### ### code chunk number 17: metropolis-mcse-sigma ################################################### sigma <- sqrt(sigmasq) sigma.mcse <- sigmasq.mcse / (2 * sigma) sigma sigma.mcse ################################################### ### code chunk number 18: metropolis-try-5 ################################################### out <- metrop(out, nbatch = 5e2, blen = 400, x = x, y = y) out$accept out$time foo <- apply(out$batch, 2, mean) mu <- foo[1:5] sigmasq <- foo[6:10] - mu^2 mu sigmasq mu.mcse <- apply(out$batch[ , 1:5], 2, sd) / sqrt(out$nbatch) mu.mcse u <- out$batch[ , 1:5] v <- out$batch[ , 6:10] ubar <- apply(u, 2, mean) vbar <- apply(v, 2, mean) deltau <- sweep(u, 2, ubar) deltav <- sweep(v, 2, vbar) foo <- sweep(deltau, 2, ubar, "*") sigmasq.mcse <- sqrt(apply((deltav - 2 * foo)^2, 2, mean) / out$nbatch) sigmasq.mcse sigma <- sqrt(sigmasq) sigma.mcse <- sigmasq.mcse / (2 * sigma) sigma sigma.mcse ################################################### ### code chunk number 19: tab1 ################################################### foo <- rbind(mu, mu.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) ################################################### ### code chunk number 20: tab1 ################################################### foo <- rbind(sigmasq, sigmasq.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) ################################################### ### code chunk number 21: tab1 ################################################### foo <- rbind(sigma, sigma.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) ################################################### ### code chunk number 22: time ################################################### cat(out$time[1], "\n") ################################################### ### code chunk number 23: x ################################################### n <- 2e4 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) ################################################### ### code chunk number 24: figgamtoo ################################################### out <- initseq(x) plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l") lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, lty = "dotted") lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, lty = "dashed") ################################################### ### code chunk number 25: figgam ################################################### out <- initseq(x) plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l") lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, lty = "dotted") lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, lty = "dashed") ################################################### ### code chunk number 26: assvar ################################################### out$var.con (1 + rho) / (1 - rho) * 1 / (1 - rho^2) ################################################### ### code chunk number 27: batx ################################################### blen <- 5 x.batch <- apply(matrix(x, nrow = blen), 2, mean) bout <- initseq(x.batch) ################################################### ### code chunk number 28: figgambattoo ################################################### plot(seq(along = bout$Gamma.con) - 1, bout$Gamma.con, xlab = "k", ylab = expression(Gamma[k]), type = "l") ################################################### ### code chunk number 29: figgambat ################################################### plot(seq(along = bout$Gamma.con) - 1, bout$Gamma.con, xlab = "k", ylab = expression(Gamma[k]), type = "l") ################################################### ### code chunk number 30: compvar ################################################### out$var.con bout$var.con * blen ################################################### ### code chunk number 31: ci-con ################################################### mean(x) + c(-1, 1) * qnorm(0.975) * sqrt(out$var.con / length(x)) mean(x.batch) + c(-1, 1) * qnorm(0.975) * sqrt(bout$var.con / length(x.batch)) mcmc/inst/doc/morph.pdf0000644000175100001440000102114613074514644014566 0ustar hornikusers%PDF-1.5 % 3 0 obj << /Length 2775 /Filter /FlateDecode >> stream xڵY[o~_!4p$ Ei.bMnhDWw?6CRN>"ggΜwywJf5֪*Գ,g,+]rw.fq57u]\GCS|,}] тze]ӕ7E sku<.pJ.~LkxoJ+:U\~n಺[xH4k ٶG~1`\ EG}8pn*x,C]wKR<-ͭDzG!!7U̬䗐mR?aM 12,8ΦA1RMSM&Z:nZ"|"!ԓHD:HlY΂j*[*m&tk Wg:3Z#d,[LF ޲&i r3-m 7G4'~gz$3ᗇգ(/ k/T\ !4YuTʃϏh D{ۛA뜨:"$R[D?r:n30M2d L+o\$}v ep^2 "/yJGkh,nDz<zΒL̤9)!F 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Geyer} \maketitle \section{The Problem} This is an example of using the \verb@mcmc@ package in R. The problem comes from a take-home question on a (take-home) PhD qualifying exam (School of Statistics, University of Minnesota). Simulated data for the problem are in the dataset \verb@logit@. There are five variables in the data set, the response \verb@y@ and four predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. A frequentist analysis for the problem is done by the following R statements <>= library(mcmc) data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial(), x = TRUE) summary(out) @ But this problem isn't about that frequentist analysis, we want a Bayesian analysis. For our Bayesian analysis we assume the same data model as the frequentist, and we assume the prior distribution of the five parameters (the regression coefficients) makes them independent and identically normally distributed with mean 0 and standard deviation 2. The log unnormalized posterior (log likelihood plus log prior) density for this model is calculated by the following R function (given the preceding data definitions) <>= x <- out$x y <- out$y lupost <- function(beta, x, y) { eta <- as.numeric(x %*% beta) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) return(logl - sum(beta^2) / 8) } @ The tricky calculation of the log likelihood avoids overflow and catastrophic cancellation in calculation of $\log(p)$ and $\log(q)$ where \begin{align*} p & = \frac{\exp(\eta)}{1 + \exp(\eta)} = \frac{1}{1 + \exp(- \eta)} \\ q & = \frac{1}{1 + \exp(\eta)} = \frac{\exp(- \eta)}{1 + \exp(- \eta)} \end{align*} so taking logs gives \begin{align*} \log(p) & = \eta - \log(1 + \exp(\eta)) = - \log(1 + \exp(- \eta)) \\ \log(q) & = - \log(1 + \exp(\eta)) = - \eta - \log(1 + \exp(- \eta)) \end{align*} To avoid overflow, we always chose the case where the argument of $\exp$ is negative. We have also avoided catastrophic cancellation when $\lvert\eta\rvert$ is large. If $\eta$ is large and positive, then \begin{align*} p & \approx 1 \\ q & \approx 0 \\ \log(p) & \approx - \exp(- \eta) \\ \log(q) & \approx - \eta - \exp(- \eta) \end{align*} and our use of the R function \texttt{log1p}, which calculates the function $x \mapsto \log(1 + x)$ correctly for small $x$ avoids all problems. The case where $\eta$ is large and negative is similar. \section{Beginning MCMC} With those definitions in place, the following code runs the Metropolis algorithm to simulate the posterior. <>= set.seed(42) # to get reproducible results beta.init <- as.numeric(coefficients(out)) out <- metrop(lupost, beta.init, 1e3, x = x, y = y) names(out) out$accept @ The arguments to the \verb@metrop@ function used here (there are others we don't use) are \begin{itemize} \item an R function (here \verb@lupost@) that evaluates the log unnormalized density of the desired stationary distribution of the Markov chain (here a posterior distribution). Note that (although this example does not exhibit the phenomenon) that the unnormalized density may be zero, in which case the log unnormalized density is \verb@-Inf@. \item an initial state (here \verb@beta.init@) of the Markov chain. \item a number of batches (here \verb@1e3@) for the Markov chain. This combines with batch length and spacing (both 1 by default) to determine the number of iterations done. \item additional arguments (here \verb@x@ and \verb@y@) supplied to provided functions (here \verb@lupost@). \item there is no ``burn-in'' argument, although burn-in is easily accomplished, if desired (more on this below). \end{itemize} The output is in the component \verb@out$batch@ returned by the \verb@metrop@ function. We'll look at it presently, but first we need to adjust the proposal to get a higher acceptance rate (\verb@out$accept@). It is generally accepted \citep*{grg} that an acceptance rate of about 20\% is right, although this recommendation is based on the asymptotic analysis of a toy problem (simulating a multivariate normal distribution) for which one would never use MCMC and is very unrepresentative of difficult MCMC applications. \citet{geyer-temp} came to a similar conclusion, that a 20\% acceptance rate is about right, in a very different situation. But they also warned that a 20\% acceptance rate could be very wrong and produced an example where a 20\% acceptance rate was impossible and attempting to reduce the acceptance rate below 70\% would keep the sampler from ever visiting part of the state space. So the 20\% magic number must be considered like other rules of thumb we teach in intro courses (like $n > 30$ means means normal approximation is valid). We know these rules of thumb can fail. There are examples in the literature where they do fail. We keep repeating them because we want something simple to tell beginners, and they are all right for some problems. Be that as it may, we try for 20\%. <>= out <- metrop(out, scale = 0.1, x = x, y = y) out$accept out <- metrop(out, scale = 0.3, x = x, y = y) out$accept out <- metrop(out, scale = 0.5, x = x, y = y) out$accept out <- metrop(out, scale = 0.4, x = x, y = y) out$accept @ Here the first argument to each instance of the \verb@metrop@ function is the output of a previous invocation. The Markov chain continues where the previous run stopped, doing just what it would have done if it had kept going, the initial state and random seed being the final state and random seed of the previous invocation. Everything stays the same except for the arguments supplied (here \verb@scale@). \begin{itemize} \item The argument \verb@scale@ controls the size of the Metropolis ``normal random walk'' proposal. The default is \verb@scale = 1@. Big steps give lower acceptance rates. Small steps give higher. We want something about 20\%. It is also possible to make \verb@scale@ a vector or a matrix. See \verb@help(metrop)@. \end{itemize} Because each run starts where the last one stopped (when the first argument to \verb@metrop@ is the output of the previous invocation), each run serves as ``burn-in'' for its successor (assuming that any part of that run was worth anything at all). \section{Diagnostics} O.~K. That does it for the acceptance rate. So let's do a longer run and look at the results. <>= out <- metrop(out, nbatch = 1e4, x = x, y = y) out$accept out$time @ Figure~\ref{fig:fig1} (page~\pageref{fig:fig1}) shows the time series plot made by the R statement <>= plot(ts(out$batch)) @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Time series plot of MCMC output.} \label{fig:fig1} \end{figure} Another way to look at the output is an autocorrelation plot. Figure~\ref{fig:fig2} (page~\pageref{fig:fig2}) shows the time series plot made by the R statement <>= acf(out$batch) @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot of MCMC output.} \label{fig:fig2} \end{figure} As with any multiplot plot, these are a bit hard to read. Readers are invited to make the separate plots to get a better picture. As with all ``diagnostic'' plots in MCMC, these don't ``diagnose'' subtle problems. As \begin{verbatim} http://www.stat.umn.edu/~charlie/mcmc/diag.html \end{verbatim} says \begin{quotation} The purpose of regression diagnostics is to find obvious, gross, embarrassing problems that jump out of simple plots. \end{quotation} The time series plots will show \emph{obvious} nonstationarity. They will not show \emph{nonobvious} nonstationarity. They provide no guarantee whatsoever that your Markov chain is sampling anything remotely resembling the correct stationary distribution (with log unnormalized density \verb@lupost@). In this very easy problem, we do not expect any convergence difficulties and so believe what the diagnostics seem to show, but one is a fool to trust such diagnostics in difficult problems. The autocorrelation plots seem to show that the the autocorrelations are negligible after about lag 25. This diagnostic inference is reliable if the sampler is actually working (has nearly reached equilibrium) and worthless otherwise. Thus batches of length 25 should be sufficient, but let's use length 100 to be safe. \section{Monte Carlo Estimates and Standard Errors} <>= out <- metrop(out, nbatch = 1e2, blen = 100, outfun = function(z, ...) c(z, z^2), x = x, y = y) out$accept out$time @ We have added an argument \verb@outfun@ that gives the ``functional'' of the state we want to average. For this problem we are interested in both posterior mean and variance. Mean is easy, just average the variables in question. But variance is a little tricky. We need to use the identity $$ \var(X) = E(X^2) - E(X)^2 $$ to write variance as a function of two things that can be estimated by simple averages. Hence we want to average the state itself and the squares of each component. Hence our \verb@outfun@ returns \verb@c(z, z^2)@ for an argument (the state vector) \verb@z@. The \verb@...@ argument to \verb@outfun@ is required, since the function is also passed the other arguments (here \verb@x@ and \verb@y@) to \verb@metrop@. \subsection{Simple Means} The grand means (means of batch means) are <>= apply(out$batch, 2, mean) @ The first 5 numbers are the Monte Carlo estimates of the posterior means. The second 5 numbers are the Monte Carlo estimates of the posterior ordinary second moments. We get the posterior variances by <>= foo <- apply(out$batch, 2, mean) mu <- foo[1:5] sigmasq <- foo[6:10] - mu^2 mu sigmasq @ Monte Carlo standard errors (MCSE) are calculated from the batch means. This is simplest for the means. <>= mu.mcse <- apply(out$batch[ , 1:5], 2, sd) / sqrt(out$nbatch) mu.mcse @ The extra factor \verb@sqrt(out$nbatch)@ arises because the batch means have variance $\sigma^2 / b$ where $b$ is the batch length, which is \verb@out$blen@, whereas the overall means \verb@mu@ have variance $\sigma^2 / n$ where $n$ is the total number of iterations, which is \verb@out$blen * out$nbatch@. \subsection{Functions of Means} To get the MCSE for the posterior variances we apply the delta method. Let $u_i$ denote the sequence of batch means of the first kind for one parameter and $\bar{u}$ the grand mean (the estimate of the posterior mean of that parameter), let $v_i$ denote the sequence of batch means of the second kind for the same parameter and $\bar{v}$ the grand mean (the estimate of the posterior second absolute moment of that parameter), and let $\mu = E(\bar{u})$ and $\nu = E(\bar{v})$. Then the delta method linearizes the nonlinear function $$ g(\mu, \nu) = \nu - \mu^2 $$ as $$ \Delta g(\mu, \nu) = \Delta \nu - 2 \mu \Delta \mu $$ saying that $$ g(\bar{u}, \bar{v}) - g(\mu, \nu) $$ has the same asymptotic normal distribution as $$ (\bar{v} - \nu) - 2 \mu (\bar{u} - \mu) $$ which, of course, has variance \verb@1 / nbatch@ times that of $$ (v_i - \nu) - 2 \mu (u_i - \mu) $$ and this variance is estimated by $$ \frac{1}{n_{\text{batch}}} \sum_{i = 1}^{n_{\text{batch}}} \bigl[ (v_i - \bar{v}) - 2 \bar{u} (u_i - \bar{u}) \bigr]^2 $$ So <>= u <- out$batch[ , 1:5] v <- out$batch[ , 6:10] ubar <- apply(u, 2, mean) vbar <- apply(v, 2, mean) deltau <- sweep(u, 2, ubar) deltav <- sweep(v, 2, vbar) foo <- sweep(deltau, 2, ubar, "*") sigmasq.mcse <- sqrt(apply((deltav - 2 * foo)^2, 2, mean) / out$nbatch) sigmasq.mcse @ does the MCSE for the posterior variance. Let's just check that this complicated \verb@sweep@ and \verb@apply@ stuff does do the right thing. <>= sqrt(mean(((v[ , 2] - vbar[2]) - 2 * ubar[2] * (u[ , 2] - ubar[2]))^2) / out$nbatch) @ \paragraph{Comment} Through version 0.5 of this vignette it contained an incorrect procedure for calculating this MCSE, justified by a handwave (which was incorrect). Essentially, it said to use the standard deviation of the batch means called \verb@v@ here, which appears to be very conservative. \subsection{Functions of Functions of Means} If we are also interested in the posterior standard deviation (a natural question, although not asked on the exam problem), the delta method gives its standard error in terms of that for the variance <>= sigma <- sqrt(sigmasq) sigma.mcse <- sigmasq.mcse / (2 * sigma) sigma sigma.mcse @ \section{A Final Run} So that's it. The only thing left to do is a little more precision (the exam problem directed ``use a long enough run of your Markov chain sampler so that the MCSE are less than 0.01'') <>= out <- metrop(out, nbatch = 5e2, blen = 400, x = x, y = y) out$accept out$time <> <> <> <> @ and some nicer output, which is presented in three tables constructed from the R variables defined above using the R \verb@xtable@ command in the \verb@xtable@ library. First the posterior means, \begin{table}[ht] \caption{Posterior Means} \label{tab:mu} \begin{center} <>= foo <- rbind(mu, mu.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} then the posterior variances (table on page~\pageref{tab:sigmasq}), \begin{table}[ht] \caption{Posterior Variances} \label{tab:sigmasq} \begin{center} <>= foo <- rbind(sigmasq, sigmasq.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} and finally the posterior standard deviations (table on page~\pageref{tab:sigma}). \begin{table}[ht] \caption{Posterior Standard Deviations} \label{tab:sigma} \begin{center} <>= foo <- rbind(sigma, sigma.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} Note for the record that the all the results presented in the tables are from ``one long run'' where long here took only <>= cat(out$time[1], "\n") @ seconds (on whatever computer it was run on). \section{New Variance Estimation Functions} A new function \texttt{initseq} estimates variances in the Markov chain central limit theorem (CLT) following the methodology introduced by \citet[Section~3.3]{practical}. These methods only apply to scalar-valued functionals of reversible Markov chains, but the Markov chains produced by the \texttt{metrop} function satisfy this condition, even, as we shall see below, when batching is used. Rather than redo the Markov chains in the preceding material, we just look at a toy problem, an AR(1) time series, which can be simulated in one line of R. This is the example on the help page for \texttt{initseq}. <>= n <- 2e4 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) @ The time series \texttt{x} is a reversible Markov chain and trivially a scalar-valued functional of a Markov chain. Define \begin{equation} \label{eq:little} \gamma_k = \cov(X_i, X_{i + k}) \end{equation} where the covariances refer to the stationary Markov chain having the same transition probabilities as \texttt{x}. Then the variance in the CLT is $$ \sigma^2 = \gamma_0 + 2 \sum_{k = 1}^\infty \gamma_k $$ \citep[Theorem~2.1]{practical}, that is, $$ \bar{x}_n \approx \text{Normal}\left(\mu, \frac{\sigma^2}{n}\right), $$ where $\mu = E(X_i)$ is the quantity being estimated by MCMC (in this toy problem we know $\mu = 0$). Naive estimates of $\sigma^2$ obtained by plugging in empirical estimates of the gammas do not provide consistent estimation \citep[Section~3.1]{practical}. Thus the scheme implemented by the R function \texttt{initseq}. Define \begin{equation} \label{eq:big} \Gamma_k = \gamma_{2 k} + \gamma_{2 k + 1} \end{equation} \citet[Theorem~3.1]{practical} says that $\Gamma_k$ considered as a function of $k$ is strictly positive, strictly decreasing, and strictly convex (provided we are, as stated above, working with a reversible Markov chain). Thus it makes sense to use estimators that use these properties. The estimators implemented by the R function \texttt{initseq} and described by \citet[Section~3.3]{practical} are conservative-consistent in the sense of Theorem~3.2 of that section. Figure~\ref{fig:gamma} (page~\pageref{fig:gamma}) shows the time series plot made by the R statement <>= out <- initseq(x) plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l") lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, lty = "dotted") lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, lty = "dashed") @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Plot ``Big Gamma'' defined by \eqref{eq:little} and \eqref{eq:big}. Solid line, initial positive sequence estimator. Dotted line, initial monotone sequence estimator. Dashed line, initial convex sequence estimator.} \label{fig:gamma} \end{figure} One can use whichever curve one chooses, but now that the \texttt{initseq} function makes the computation trivial, it makes sense to use the initial convex sequence. Of course, one is not interested in Figure~\ref{fig:gamma}, except perhaps when explaining the methodology. What is actually important is the estimate of $\sigma^2$, which is given by <>= out$var.con (1 + rho) / (1 - rho) * 1 / (1 - rho^2) @ where for comparison we have given the exact theoretical value of $\sigma^2$, which, of course, is never available in a non-toy problem. These initial sequence estimators seem, at first sight to be a competitor for the method of batch means. However, appearances can be deceiving. The two methods are complementary. The sequence of batch means is itself a scalar-valued functional of a reversible Markov chain. Hence the initial sequence estimators can be applied to it. <>= blen <- 5 x.batch <- apply(matrix(x, nrow = blen), 2, mean) bout <- initseq(x.batch) @ Because the batch length is too short, the variance of the batch means does not estimate $\sigma^2$. We must account for the autocorrelation of the batches, shown in Figure~\ref{fig:gambat}. <>= plot(seq(along = bout$Gamma.con) - 1, bout$Gamma.con, xlab = "k", ylab = expression(Gamma[k]), type = "l") @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Plot ``Big Gamma'' defined by \eqref{eq:little} and \eqref{eq:big} for the sequence of batch means (batch length \Sexpr{blen}). Only initial convex sequence estimator is shown.} \label{fig:gambat} \end{figure} Because the the variance is proportional to one over the batch length, we need to multiply by the batch length to estimate the $\sigma^2$ for the original series. <>= out$var.con bout$var.con * blen @ Another way to look at this is that the MCMC estimator of $\mu$ is either \texttt{mean(x)} or \texttt{mean(x.batch)}. And the variance must be divided by the sample size to give standard errors. So either <>= mean(x) + c(-1, 1) * qnorm(0.975) * sqrt(out$var.con / length(x)) mean(x.batch) + c(-1, 1) * qnorm(0.975) * sqrt(bout$var.con / length(x.batch)) @ is an asymptotic 95\% confidence interval for $\mu$. Just divide by the relevant sample size. \begin{thebibliography}{} \bibitem[Gelman et al.(1996)Gelman, Roberts, and Gilks]{grg} Gelman, A., G.~O. Roberts, and W.~R. Gilks (1996). \newblock Efficient Metropolis jumping rules. \newblock In \emph{Bayesian Statistics, 5 (Alicante, 1994)}, pp.~599--607. Oxford University Press. \bibitem[Geyer(1992)]{practical} Geyer, C.~J. (1992). \newblock Practical Markov chain Monte Carlo (with discussion). \newblock \emph{Statistical Science}, 7, 473--511. \bibitem[Geyer and Thompson(1995)]{geyer-temp} Geyer, C.~J. and E.~A. Thompson (1995). \newblock Annealing Markov chain Monte Carlo with applications to ancestral inference. \newblock \emph{Journal of the American Statistical Association}, 90, 909--920. \end{thebibliography} \end{document} mcmc/inst/doc/demo.pdf0000644000175100001440000135474413074514644014402 0ustar hornikusers%PDF-1.5 % 3 0 obj << /Length 1564 /Filter /FlateDecode >> stream xڵX[F~WmW- ./Rg!n ;28v+$&s;w.sf0>SlȔ7: Sl:~_?()T, \B]50-ϡ3[mu( -z/mRy݅e~ִ@ЮC)/T)%j4k cEe=z e7@ꕀ/0x2oP*f^8iBaJK8O7D\#jД*̚LYaGf2Nx*3.coic 8fCٶXRfY{V~$tB2F5L _/qA2$3sQé *{hN{,L؉2y''c[UV6Bl']q(ubGA8 WI}ptj S_bO̠uc.ipIGîpu HdC/ vxϡ*DysQ'&*S .{ƓIA.^JIJ:>&#e5&{Zў?˅W 4Ut}?OFɈ

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Geyer} \maketitle <>= options(keep.source = TRUE, width = 65) @ \section{Introduction} \subsection{Bayes Factors} \label{sec:bayes-factors} Let $\mathcal{M}$ be a finite or countable set of models (here we only deal with finite $\mathcal{M}$ but Bayes factors make sense for countable $\mathcal{M}$). For each model $m \in \mathcal{M}$ we have the prior probability of the model $\prior(m)$. It does not matter if this prior on models is unnormalized. Each model $m$ has a parameter space $\Theta_m$ and a prior $$ g(\theta \mid m), \qquad \theta \in \Theta_m $$ The spaces $\Theta_m$ can and usually do have different dimensions. That's the point. These within model priors must be normalized proper priors. The calculations to follow make no sense if these priors are unnormalized or improper. Each model $m$ has a data distribution $$ f(y \mid \theta, m) $$ and the observed data $y$ may be either discrete or continuous (it makes no difference to the Bayesian who treats $y$ as fixed after it is observed and treats only $\theta$ and $m$ as random). The unnormalized posterior for everything (for models and parameters within models) is $$ f(y \mid \theta, m) g(\theta \mid m) \prior(m) $$ To obtain the conditional distribution of $y$ given $m$, we must integrate out the nuisance parameter $\theta$ \begin{align*} q(y \mid m) & = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \prior(m) \, d \theta \\ & = \prior(m) \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta \end{align*} These are the unnormalized posterior probabilities of the models. The normalized posterior probabilities are $$ \posterior(m \mid y) = \frac{ q(y \mid m) }{ \sum_{m \in \mathcal{M}} q(y \mid m) } $$ It is considered useful to define $$ b(y \mid m) = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta $$ so $$ q(y \mid m) = b(y \mid m) \prior(m) $$ Then the ratio of posterior probabilities of models $m_1$ and $m_2$ is $$ \frac{\posterior(m_1 \mid y)}{\posterior(m_2 \mid y)} = \frac{q(y \mid m_1)}{q(y \mid m_2)} = \frac{b(y \mid m_1)}{b(y \mid m_2)} \cdot \frac{\prior(m_1)}{\prior(m_2)} $$ This ratio is called the \emph{posterior odds} of the models (a ratio of probabilities is called an \emph{odds}) of these models. The \emph{prior odds} is $$ \frac{\prior(m_1)}{\prior(m_2)} $$ The term we have not yet named in $$ \frac{\posterior(m_1 \mid y)}{\posterior(m_2 \mid y)} = \frac{b(y \mid m_1)}{b(y \mid m_2)} \cdot \frac{\prior(m_1)}{\prior(m_2)} $$ is called the \emph{Bayes factor} \begin{equation} \label{eq:factor} \frac{b(y \mid m_1)}{b(y \mid m_2)} \end{equation} the ratio of posterior odds to prior odds. The prior odds tells how the prior compares the probability of the models. The Bayes factor tells us how the data shifts that comparison going from prior to posterior via Bayes rule. Bayes factors are the primary tool Bayesians use for model comparison, the competitor for frequentist $P$-values in frequentist hypothesis tests of model comparison. Note that our clumsy multiple letter notation for priors and posteriors $\prior(m)$ and $\posterior(m \mid y)$ does not matter because neither is involved in the actual calculation of Bayes factors \eqref{eq:factor}. Priors and posteriors are involved in motivating Bayes factors but not in calculating them. \subsection{Tempering} \label{sec:temper} Simulated tempering \citep{marinari-parisi,geyer-thompson} is a method of Markov chain Monte Carlo (MCMC) simulation of many distributions at once. It was originally invented with the primary aim of speeding up MCMC convergence, but was also recognized to be useful for sampling multiple distributions \citep{geyer-thompson}. In the latter role it is sometimes referred to as ``umbrella sampling'' which is a term coined by \citet{torrie-valleau} for sampling multiple distributions via MCMC. We have a finite set of unnormalized distributions we want to sample, all related in some way. The R function \texttt{temper} in the CRAN package \texttt{mcmc} requires all to have continuous distributions for random vectors of the same dimension (all distributions have the same domain $\real^p$). Let $h_i$, $i \in \mathcal{I}$ denote the unnormalized densities of these distributions. Simulated tempering (called ``serial tempering'' by the \texttt{temper} function to distinguish from a related scheme not used in this document called ``parallel tempering'' and in either case abbreviated ST) runs a Markov chain whose state is a pair $(i, x)$ where $i \in \mathcal{I}$ and $x \in \real^p$. The unnormalized density of stationary distribution of the ST chain is \begin{equation} \label{eq:st-joint} h(i, x) = h_i(x) c_i \end{equation} where the $c_i$ are arbitrary constants chosen by the user (more on this later). The equilibrium distribution of the ST state $(I, X)$ --- both bits random --- is such that conditional distribution of $X$ given $I = i$ is the distribution with unnormalized density $h_i$. This is obvious from $h(i, x)$ being the unnormalized conditional density --- the same function thought of as a function of both variables is the unnormalized joint density and thought of as a function of just one of the variables is an unnormalized conditional density --- and $h(i, x)$ thought of as a function of $x$ for fixed $i$ being proportional to $h_i$. The equilibrium unnormalized marginal distribution of $I$ is \begin{equation} \label{eq:margin} \int h(i, x) \, d x = c_i \int h_i(x) \, d x = c_i d_i \end{equation} where $$ d_i = \int h_i(x) \, d x $$ is the normalizing constant for $h_i$, that is, $h_i / d_i$ is a normalized distribution. It is clear from \eqref{eq:margin} being the unnormalized marginal distribution that in order for the marginal distribution to be uniform we must choose the tuning constants $c_i$ to be proportional to $1 / d_i$. It is not important that the marginal distribution be exactly uniform, but unless it is approximately uniform, the sampler will not visit each distribution frequently. Thus we do need to have the $c_i$ to be approximately proportional to $1 / d_i$. This is accomplished by trial and error (one example is done in this document) and is easy for easy problems and hard for hard problems \citep[have much to say about adjusting the $c_i$]{geyer-thompson}. For the rest of this section we will assume the tuning constants $c_i$ have been so adjusted: we do not have the $c_i$ exactly proportional to $1 / d_i$ but do have them approximately proportional to $1 / d_i$. \subsection{Tempering and Bayes Factors} Bayes factors are very important in Bayesian inference and many methods have been invented to calculate them. No method except the one described here using ST is anywhere near as accurate and straightforward. Thus no competitors will be discussed. In using ST for Bayes factors we identify the index set $\mathcal{I}$ with the model set $\mathcal{M}$ and use the integers 1, $\ldots$, $k$ for both. We would like to identify the within model parameter vector $\theta$ with the vector $x$ that is the continuous part of the state of the ST Markov chain, but cannot because the dimension of $\theta$ depends on $m$ and this is not allowed. Thus we have to do something a bit more complicated. We ``pad'' $\theta$ so that it always has the same dimension, doing so in a way that does not interfere with the Bayes factor calculation. Write $\theta = (\theta_{\text{actual}}, \theta_{\text{pad}})$, the dimension of both parts depending on the model $m$. Then we insist on the following conditions: $$ f(y \mid \theta, m) = f(y \mid \theta_{\text{actual}}, m) $$ so the data distribution does not depend on the ``padding'' and $$ g(\theta \mid m) = g_{\text{actual}}(\theta_{\text{actual}} \mid m) \cdot g_{\text{pad}}(\theta_{\text{pad}} \mid m) $$ so the two parts are \emph{a priori} independent and both parts of the prior are normalized proper priors. This assures that \begin{equation} \label{eq:unnormalized-bayes-factors} \begin{split} b(y \mid m) & = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta \\ & = \iint f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) g_{\text{pad}}(\theta_{\text{pad}} \mid m) \, d \theta_{\text{actual}} \, d \theta_{\text{pad}} \\ & = \int_{\Theta_m} f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) \, d \theta_{\text{actual}} \end{split} \end{equation} so the calculation of the unnormalized Bayes factors is the same whether or not we ``pad'' $\theta$, and we may then take \begin{align*} h_m(\theta) & = f(y \mid \theta, m) g(\theta \mid m) \\ & = f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) g_{\text{pad}}(\theta_{\text{pad}} \mid m) \end{align*} to be the unnormalized densities for the component distributions of the ST chain, in which case the unnormalized Bayes factors are proportional to the normalizing constants $d_i$ in Section~\ref{sec:temper}. \subsection{Tempering and Normalizing Constants} Let $d$ be the normalizing constant for the joint equilibrium distribution of the ST chain \eqref{eq:st-joint}. When we are running the ST chain we know neither $d$ nor the $d_i$ but we do know the $c_i$, which are constants we have chosen based on the results of previous runs but are fixed known numbers for the current run. Let $(I_t, X_t)$, $t = 1$, 2, $\ldots$ be the sample path of the ST chain. Recall that (somewhat annoyingly) we are using the notation $(i, x)$ for the state vector of a general ST chain and the notation $(m, \theta)$ for ST chains used to calculate Bayes factors, identifying $i = m$ and $x = \theta$. Let $\indicator(\fatdot)$ denote the function that maps logical values to numerical values, false to zero and true to one. Normalizing constants are estimated by averaging the time spent in each model \begin{equation} \label{eq:st-estimates} \hat{\delta}_n(m) = \frac{1}{n} \sum_{t = 1}^n \indicator(I_t = m) \end{equation} For the purposes of approximating Bayes factors the $X_t$ are ignored. The $X_t$ may be useful for other purposes, such as Bayesian model averaging \citep*{bma}, but this is not discussed here. The Monte Carlo approximations \eqref{eq:st-estimates} converge to their expected values under the equilibrium distribution \begin{equation} \label{eq:st-expectations} E\{ \indicator(I_t = m) \} = \int \frac{h(m, x)}{d} \, d x = \frac{c_m d_m}{d} = \delta(m) \end{equation} We want to estimate the unnormalized Bayes factors \eqref{eq:unnormalized-bayes-factors}, which are in this context proportional to the $d_m$. The $c_m$ are known, $d$ is unknown but does not matter since we only need to estimate the $d_m = b(m \mid y)$ up to an overall unknown constant of proportionality, which cancels out of Bayes factors \eqref{eq:factor}. Note that our discussion here applies unchanged to the general problem of estimating normalizing constants up to an unknown constant of proportionality, which has applications other than Bayes factors, for example, missing data maximum likelihood \citep{thompson-guo,geyer,sung-geyer}. The ST method approximates normalizing constants up to an overall constant of proportionality with high accuracy regardless of how large or small they are (whether they are $10^{100}$ or $10^{-100}$), and no other method that does not use essentially the same idea can do this. The key is what seems at first sight to be a weakness of ST, the need to adjust the tuning constants $c_i$ by trial and error. In this context the weakness is actually a strength: the adjusted $c_i$ contain most of the information about the size of the normalizing constants $d_i$ and the Monte Carlo averages \eqref{eq:st-estimates} add only the finishing touch. Thus multiple runs of the ST chain with different choices of the $c_i$ used in each run are needed (the ``trial and error''), but the information from all are incorporated in the final run used for final approximation of the normalizing constants (Bayes factors). It is perhaps surprising that the Monte Carlo error approximation is trivial. In the context of the last run of the ST chain the $c_i$ are known constants and contribute no error. The Monte Carlo error of the averages \eqref{eq:st-estimates} is straightforwardly estimated by batch means or competing methods. \citet{geyer-thompson} note that the $c_i$ enter formally like a prior: one can think of $h_i(x) c_i$ as likelihood times prior. But one should not think of the $c_i$ as representing prior information, informative, non-informative, or in between. The $c_i$ are adjusted to make the ST distribution sample all the models $h_i$, and that is the only criterion for the adjustment. For this reason \citet{geyer-thompson} call the $c_i$ the \emph{pseudoprior}. This is a special case of a general principle of MCMC. When doing MCMC one should forget the statistical motivation (in this case Bayes factors). One should set up a Markov chain that does a good job of simulating the required equilibrium distribution, whatever it is. Thinking about the statistical motivation of the equilibrium does not help and can hurt (if one thinks of the pseudoprior as an actual prior, one may be tempted to adjust it to represent prior information). \section{R Package MCMC} We use the R statistical computing environment \citep{rcore} in our analysis. It is free software and can be obtained from \url{http://cran.r-project.org}. Precompiled binaries are available for Windows, Macintosh, and popular Linux distributions. We use the contributed package \verb@mcmc@ \citep{mcmc-R-package} If R has been installed, but this package has not yet been installed, do \begin{verbatim} install.packages("mcmc") \end{verbatim} from the R command line (or do the equivalent using the GUI menus if on Apple Macintosh or Microsoft Windows). This may require root or administrator privileges. Assuming the \verb@mcmc@ package has been installed, we load it <>= library(mcmc) @ <>= baz <- library(help = "mcmc") baz <- baz$info[[1]] baz <- baz[grep("Version", baz)] baz <- sub("^Version: *", "", baz) bazzer <- paste(R.version$major, R.version$minor, sep = ".") @ The version of the package used to make this document is \Sexpr{baz} (which is available on CRAN). The version of R used to make this document is \Sexpr{bazzer}. We also set the random number generator seed so that the results are reproducible. <>= set.seed(42) @ To get different results, change the setting or don't set the seed at all. \section{Logistic Regression Example} We use the same logistic regression example used in the \texttt{mcmc} package vignette for the \texttt{metrop} function (file \texttt{demo.pdf}. Simulated data for the problem are in the data set \verb@logit@. There are five variables in the data set, the response \verb@y@ and four predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. A frequentist analysis for the problem is done by the following R statements <>= data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial, x = TRUE) summary(out) @ But this example isn't about frequentist analysis, we want a Bayesian analysis. For our Bayesian analysis we assume the same data model as the frequentist, and we assume the prior distribution of the five parameters (the regression coefficients) makes them independent and identically normally distributed with mean 0 and standard deviation 2. Moreover, we wish to calculate Bayes factors for the $16 = 2^4$ possible submodels that include or exclude each of the predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. \subsection{Setup} We set up a matrix that indicates these models. <>= varnam <- names(coefficients(out)) varnam <- varnam[varnam != "(Intercept)"] nvar <- length(varnam) models <- NULL foo <- seq(0, 2^nvar - 1) for (i in 1:nvar) { bar <- foo %/% 2^(i - 1) bar <- bar %% 2 models <- cbind(bar, models, deparse.level = 0) } colnames(models) <- varnam models @ In each row, 1 indicates the predictor is in the model and 0 indicates it is out. The function \texttt{temper} in the \text{mcmc} package that does tempering requires a notion of neighbors among models. It attempts jumps only between neighboring models. Here we choose models to be neighbors if they differ only by one predictor. <>= neighbors <- matrix(FALSE, nrow(models), nrow(models)) for (i in 1:nrow(neighbors)) { for (j in 1:ncol(neighbors)) { foo <- models[i, ] bar <- models[j, ] if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE } } @ Now we specify the equilibrium distribution of the ST chain. Its state vector is $(i, x)$ or $(m, \theta)$ in our alternative notations, where $i$ is an integer between $1$ and \verb@nrow(models)@ = \Sexpr{nrow(models)} and $\theta$ is the parameter vector ``padded'' to always be the same length, so we take it to be the length of the parameter vector of the full model which is \verb@length(out$coefficients)@ or \verb@ncol(models) + 1@ which makes the length of the state of the ST chain \verb@ncol(models) + 2@. We take the within model priors for the ``padded'' components of the parameter vector to be the same as those for the ``actual'' components, normal with mean 0 and standard deviation 2 for all cases. As is seen in \eqref{eq:unnormalized-bayes-factors} the priors for the ``padded'' components (parameters not in the model for the current state) do not matter because they drop out of the Bayes factor calculation. The choice does not matter much for this toy example. See the discussion section for more on this issue. It is important that we use normalized log priors, the term \verb@dnorm(beta, 0, 2, log = TRUE)@ in the function, unlike when we are simulating only one model as in the \texttt{mcmc} package vignette where it would be o.~k.\ to use unnormalized log priors \verb@- beta^2 / 8@. The \texttt{temper} function wants the log unnormalized density of the equilibrium distribution. We include an additional argument \texttt{log.pseudo.prior}, which is $\log(c_i)$ in our mathematical development, because this changes from run to run as we adjust it by trial and error. Other ``arguments'' are the model matrix of the full model \texttt{modmat}, the matrix \texttt{models} relating integer indices (the first component of the state vector of the ST chain) to which predictors are in or out of the model, and the data vector \texttt{y}, but these are not passed as arguments to our function and instead are found in the R global environment. <>= modmat <- out$x y <- logit$y ludfun <- function(state, log.pseudo.prior) { stopifnot(is.numeric(state)) stopifnot(length(state) == ncol(models) + 2) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= nrow(models)) stopifnot(is.numeric(log.pseudo.prior)) stopifnot(length(log.pseudo.prior) == nrow(models)) beta <- state[-1] inies <- c(TRUE, as.logical(models[icomp, ])) beta.logl <- beta beta.logl[! inies] <- 0 eta <- as.numeric(modmat %*% beta.logl) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) logl + sum(dnorm(beta, 0, 2, log = TRUE)) + log.pseudo.prior[icomp] } @ \subsection{Trial and Error} Now we are ready to try it out. We start in the full model at its MLE, and we initialize \texttt{log.pseudo.prior} at all zeros, having no idea \emph{a priori} what it should be. <>= state.initial <- c(nrow(models), out$coefficients) qux <- rep(0, nrow(models)) out <- temper(ludfun, initial = state.initial, neighbors = neighbors, nbatch = 1000, blen = 100, log.pseudo.prior = qux) names(out) out$time @ So what happened? <>= ibar <- colMeans(out$ibatch) ibar @ The ST chain did not mix well, several models not being visited even once. So we adjust the pseudo priors to get uniform distribution. <>= qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux @ The new pseudoprior should be proportional to \verb@1 / ibar@ if \texttt{ibar} is an accurate estimate of \eqref{eq:st-expectations}, but this makes no sense when the estimates are bad, in particular, when the are exactly zero. Thus we put an upper bound, chosen arbitrarily (here 10) on the maximum increase of the log pseudoprior. The statement \begin{verbatim} qux <- qux - min(qux) \end{verbatim} is unnecessary. An overall arbitrary constant can be added to the log pseudoprior without changing the equilibrium distribution of the ST chain. We do this only to make \texttt{qux} more comparable from run to run. Now we repeat this until the log pseudoprior ``converges'' roughly. Because this loop takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("bfst1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { qux.save <- qux time.save <- out$time repeat{ out <- temper(out, log.pseudo.prior = qux) ibar <- colMeans(out$ibatch) qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux.save <- rbind(qux.save, qux, deparse.level = 0) time.save <- rbind(time.save, out$time, deparse.level = 0) if (max(ibar) / min(ibar) < 2) break } save(out, qux, qux.save, time.save, file = "bfst1.rda") } else { .Random.seed <- out$final.seed } print(qux.save, digits = 3) print(qux, digits = 3) apply(time.save, 2, sum) @ Now that the pseudoprior is adjusted well enough, we need to perhaps make other adjustments to get acceptance rates near 20\%. <>= print(out$accepti, digits = 3) print(out$acceptx, digits = 3) @ The acceptance rates for swaps seem o. k. <>= min(as.vector(out$accepti), na.rm = TRUE) @ and there is nothing simple we can do to adjust them (adjustment is possible, see the discussion section for more on this issue). We adjust the acceptance rates for within model moves by adjusting the scaling. <>= out <- temper(out, scale = 0.5, log.pseudo.prior = qux) time.save <- rbind(time.save, out$time, deparse.level = 0) print(out$acceptx, digits = 3) @ Looks o.~k.\ now. Inspection of autocorrelation functions for components of \verb@out$ibatch@ (not shown) says batch length needs to be at least 4 times longer. We make it 10 times longer for safety. Because this run takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("bfst2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out <- temper(out, blen = 10 * out$blen, log.pseudo.prior = qux) save(out, file = "bfst2.rda") } else { .Random.seed <- out$final.seed } time.save <- rbind(time.save, out$time, deparse.level = 0) foo <- apply(time.save, 2, sum) foo.min <- floor(foo[1] / 60) foo.sec <- foo[1] - 60 * foo.min c(foo.min, foo.sec) @ The total time for all runs of the temper function was \Sexpr{foo.min} minutes and \Sexpr{round(foo.sec, 1)} seconds. \subsection{Bayes Factor Calculations} Now we calculate log 10 Bayes factors relative to the model with the highest unnormalized Bayes factor. <>= log.10.unnorm.bayes <- (qux - log(colMeans(out$ibatch))) / log(10) k <- seq(along = log.10.unnorm.bayes)[log.10.unnorm.bayes == min(log.10.unnorm.bayes)] models[k, ] log.10.bayes <- log.10.unnorm.bayes - log.10.unnorm.bayes[k] log.10.bayes @ These are base 10 logarithms of the Bayes factors against the $k$-th model where $k = \Sexpr{k}$. For example, the Bayes factor for the $k$-th model divided by the Bayes factor for the first model is $10^{\Sexpr{round(log.10.bayes[1], 3)}}$. Now we calculate Monte Carlo standard errors two different ways. One is the way the delta method is usually taught. To simplify notation, denote the Bayes factors $$ b_m = b(y \mid m) $$ and their Monte Carlo approximations $\hat{b}_m$. Then the log Bayes factors are $$ g_i(b) = \log_{10} b_i - \log_{10} b_k $$ hence we need to apply the delta method with the function $g_i$, which has derivatives \begin{align*} \frac{\partial g_i(b)}{\partial b_i} & = \frac{1}{b_i \log_e(10)} \\ \frac{\partial g_i(b)}{\partial b_k} & = - \frac{1}{b_k \log_e(10)} \\ \frac{\partial g_i(b)}{\partial b_j} & = 0, \qquad \text{$j \neq i$ and $j \neq k$} \end{align*} <>= fred <- var(out$ibatch) / out$nbatch sally <- colMeans(out$ibatch) mcse.log.10.bayes <- (1 / log(10)) * sqrt(diag(fred) / sally^2 - 2 * fred[ , k] / (sally * sally[k]) + fred[k, k] / sally[k]^2) mcse.log.10.bayes foompter <- cbind(models, log.10.bayes, mcse.log.10.bayes) round(foompter, 5) @ An alternative calculation of the MCSE replaces the actual function of the raw Bayes factors with its best linear approximation $$ \frac{1}{\log_e(10)} \left(\frac{\hat{b}_i - b_i}{b_i} - \frac{\hat{b}_k - b_k}{b_k} \right) $$ and calculates the standard deviation of this quantity by batch means <>= ibar <- colMeans(out$ibatch) herman <- sweep(out$ibatch, 2, ibar, "/") herman <- sweep(herman, 1, herman[ , k], "-") mcse.log.10.bayes.too <- (1 / log(10)) * apply(herman, 2, sd) /sqrt(out$nbatch) all.equal(mcse.log.10.bayes, mcse.log.10.bayes.too) @ \section{Discussion} We hope readers are impressed with the power of this method. The key to the method is pseudopriors adjusted by trial and error. The method could have been invented by any Bayesian who realized that the priors on models, $\prior(m)$ in our notation in Section~\ref{sec:bayes-factors}, do not affect the Bayes factors and hence are irrelevant to calculating Bayes factors. Thus the priors (or pseudopriors in our terminology) should be chosen for reasons of computational convenience, as we have done, rather than to incorporate prior information. The rest of the details of the method are unimportant. The \texttt{temper} function in R is convenient to use for this purpose, but there is no reason to believe that it provides optimal sampling. Samplers carefully designed for each particular application would undoubtedly do better. Our notion of ``padding'' so that the within model parameters have the same dimension for all models follows \citet{carlin-chib} but ``reversible jump'' samplers \citep{green} would undoubtedly do better. Unfortunately, there seems to be no way to code up a function like \texttt{temper} that uses ``reversible jump'' and requires no theoretical work from users that if messed up destroys the algorithm. The \texttt{temper} function is foolproof in the sense that if the log unnormalized density function written by the user (like our \texttt{ludfun}) is correct, then the ST Markov chain has the equilibrium distribution is supposed to have. There is nothing the user can mess up except this user written function. No analog of this for ``reversible jump'' chains is apparent (to your humble author). Two issues remain where the text above said ``see the discussion section for more on this issue.'' The first was about within model priors for the ``padding'' components of within model parameter vectors $g_{\text{pad}}(\theta_{\text{pad}} \mid m)$ in the notation in \eqref{eq:unnormalized-bayes-factors}. Rather than choose these so that they do not depend on the data (as we did), it would be better (if more trouble) to choose them differently for each ``padding'' component, centering $g_{\text{pad}}(\theta_{\text{pad}} \mid m)$ so the distribution of a component of $\theta_{\text{pad}}$ is near to the marginal distribution of the same component in neighboring models (according to the \texttt{neighbors} argument of the \texttt{temper} function). The other remaining issue is adjusting acceptance rates for jumps. There is no way to adjust this other than by changing the number of models and their definitions. But the models we have cannot be changed; if we are to calculate Bayes factors for them, then we must sample them as they are. But we can insert new models between old models. For example, if the acceptance for swaps between model $i$ and model $j$ is too low, then we can insert distribution $k$ between them that has unnormalized density $$ h_k(x) = \sqrt{h_i(x) h_j(x)}. $$ This idea is inherited from simulated tempering; \citep{geyer-thompson} have much discussion of how to insert additional distributions into a tempering network. It is another key issue in using tempering to speed up sampling. It is less obvious in the Bayes factor context, but still an available technique if needed. \begin{thebibliography}{} \bibitem[Carlin and Chib(1995)]{carlin-chib} Carlin, B.~P. and Chib, S. (1995). \newblock Bayesian model choice via Markov chain Monte Carlo methods. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{57}, 473--484. \bibitem[Geyer(1994)]{geyer} Geyer, C.~J. (1994). \newblock On the convergence of Monte Carlo maximum likelihood calculations. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{56} 261--274. \bibitem[Geyer., 2009]{mcmc-R-package} Geyer., C.~J. (2009). \newblock \emph{mcmc: Markov Chain Monte Carlo}. \newblock R package version 0.7-2, available from CRAN. \bibitem[Geyer and Thompson(1995)]{geyer-thompson} Geyer, C.~J., and Thompson, E.~A. (1995). \newblock Annealing Markov chain Monte Carlo with applications to ancestral inference. \newblock \emph{Journal of the American Statistical Association}, \textbf{90}, 909--920. \bibitem[Green(1995)]{green} Green, P.~J. (1995). \newblock Reversible jump {M}arkov chain {M}onte {C}arlo computation and {B}ayesian model determination. \newblock \emph{Biometrika}, \textbf{82}, 711--732. \bibitem[Hoeting et al.(1999)Hoeting, Madigan, Raftery, and Volinsky]{bma} Hoeting, J.~A., Madigan, D., Raftery, A.~E. and Volinsky, C.~T. (1999). \newblock Bayesian model averaging: A tutorial (with discussion). \newblock \emph{Statical Science}, \textbf{19}, 382--417. \newblock The version printed in the journal had the equations messed up in the production process; a corrected version is available at \url{http://www.stat.washington.edu/www/research/online/1999/hoeting.pdf}. \bibitem[Marinari and Parisi(1992)]{marinari-parisi} Marinari, E., and Parisi G. (1992). \newblock Simulated tempering: A new Monte Carlo Scheme. \newblock \emph{Europhysics Letters}, \textbf{19}, 451--458. \bibitem[R Development Core Team(2010)]{rcore} R Development Core Team (2010). \newblock R: A language and environment for statistical computing. \newblock R Foundation for Statistical Computing, Vienna, Austria. \newblock \url{http://www.R-project.org}. \bibitem[Sung and Geyer(2007)]{sung-geyer} Sung, Y.~J. and Geyer, C.~J. (2007). \newblock Monte Carlo likelihood inference for missing data models. \newblock \emph{Annals of Statistics}, \textbf{35}, 990--1011. \bibitem[Thompson and Guo(1991)]{thompson-guo} Thompson, E. A. and Guo, S. W. (1991). \newblock Evaluation of likelihood ratios for complex genetic models. \newblock \emph{IMA J. Math. Appl. Med. Biol.}, \textbf{8}, 149--169. \bibitem[Torrie and Valleau(1977)]{torrie-valleau} Torrie, G.~M., and Valleau, J.~P. (1977). \newblock Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. \newblock \emph{Journal of Computational Physics}, \textbf{23}, 187--199. \end{thebibliography} \end{document} mcmc/inst/doc/debug.pdf0000644000175100001440000027720713074514644014541 0ustar hornikusers%PDF-1.5 % 3 0 obj << /Length 2495 /Filter /FlateDecode >> stream xڝɒ`bJ@c͑-[%JU*$AI}`F${yKoo~xe SƩaHMWi(Lgux.4Q|{sr%&nI_4q\ymy^y6\*j7;>|x"=<ŧeZF-AE<꼗jЉm_Ņ^ 64RB⥽,m?ǑWEA14 8q&:egXԷڰg *}QqIZ GKWA"( gIO<Ҍ1w/Pl %so.Htǧ`n(0ggIƦ 8ՍƥTa!JI1BY xF}b{)(4v 5u^IMC9fI+'_tss*VI@dW^,.8ȶCLæ?/L`# b2~J(BR}x{W#o2^$g:; vhF17=;Si+6DBPI|*UX]epw%הL䨳4!| oTkx*:1QrXpi73ir/ի Ye--K) J%3aa'I LqRJ )5G&p|@)4gͳfQk>Q[tPzMH[HogC?S7)Pڤ؁%~O-, mB2n1)a' OpvHd*(34i>HL&[-eSe&jɔp}dv+WA/jQ゜ߩ2^ ki8{%fx՗ι-ZQQj,;s*eyya蹏r=)e3^~ЃҗMtV 0Oj͑(V-/D5Y"7pʥL^uIo_m29&Fy$EáG)(AXj4 L+pH}1'lXNNMQKo8 3#AkbҌ}='N鷺Ƚ*\ղ}9BZWULN^M˲-FdZ4KGhWbhh5gRzn K*T|v )89Ij}Ke7W_J :$U^Rݤuupa{-I戹7ə L6sE~wIW;cXbcYw,1uVڢ^DﻗEҜЛ&/8uo$,h%h%̟ &sk83o$p{CЮɾ+Iԑ]6k_h'=+'qSn- > stream xZK6WL$mY$Y=8[qmmes'8F#if-S&o?$aI~߾)n8usZU7ΖBi{s{wLR*;jХ{~^~y_oS/nJ{x:f~79;1eK)1:Nv?Oq:Ux=c:3. %deqGxYU9C{\-lwj]A]5Q١y8듒EFxj80r`M{`u#+(FRmb0 F% !vMv`SÖ.u!S0Ť_7SQԕ*`$6nYtKpS3ޘ2Ldu54늢P1#gxa޲=y=VX}p;OȔdA xUBC(x` {%-tb4`dy:kָ4p$C0|!ES\#Tː ^ANg`R8, e%iUB/%< Hȗ8 ] ӫ2(*+$q C!Dp hξ]YjC|ݱ ysQ>ފQ_xO7D47L+ к CR:gIE{4_f-U_[o*lO0(QKb\5"R!`ÐVT.1Z"*{(gLXQW5TCDmia7հLrDϦ;8uP5by$|bX\ |$xXu1#݆4'?CISPOC_?` {K=/ڂpDм ;ɮf^=\ǣoFGYi =s 1x~**FzKUd~$6xzkp6 *Ȅ.$_ev<d].8݇ _ c810*<^~+246Snke"<:?;R@%^%qTTX7|ԍl8Ѓ-?vE:IL(2Ŭ{w4Sz=҇Mq\|0Ш/ޛE 8W|ENҗua,q1q{ݨK\'5lƤ!*}IMzR}[ctj*i2^yyQ3=\w0VsEO4/)byn*[kޅ#z<;,˪+8ZY@O&S} xo%,ԐhqA5/STs|&fV w0R\ք5ֿ~bo]gygoƗ£k{[0M h w&`.y(yHe#Yt)Vk~Yeܳݳ'XJ*g?;gmT>~]*uön0u*BtG1fc=v31u}?auܷZڧ܎?0$nLD40{4zd \{.aa*./bpN[ￅ{L8&eL`M| '%@-}!u!t>1JW/nC?tUbϤ#pY}RX{F_ CPɫӨ jGҗ^eGpoI endstream endobj 18 0 obj << /Length 3173 /Filter /FlateDecode >> stream x[KW $%+n"X# 9laׇyxgIקWz MboUsʈ+.%]mzuL =z[5[?>rY}6᭻jYkN5cBgr ЈE+3$4IG=ͣHfM+G^>#~Or{-3ďei9A(ftEsagϵuawݎN:߿u[ڋ+p4UǷr:;7~˺V3CeYw)~J_9f[eAlw@+NCqw(;ճ0[T34T>ɮux 4mcN q}i.lPTƹcj@hiK/="-e0ArBypq#Ã~Cv*vD|DOQP$W YQL mr J[okTn/ >$)?Zu%\"`-I$;q^|!K9V~1ti$/l xRbME9x(%L}Q$Bߒ 4O3@Y+m{|o&"CP ߥ@GL|7Y7L8SQC_IE6JJ ?%kM8s7Ab1qrjʑ8b:7`(I0`9xjf5n-?7:is !$8v/ctB O3ƔPM0^0$H!"Fo|҂<B%ѨOS1mW 3Į\vH jWߙ]m?޲*v:y}-]~":´rS"{4x;Ĺ7@;FAz38nzyW~K.~R !^ƺ m+|\G%v DOr!q2D ^Av%D7kڭږ<mG^&KaSn&s *rlFɚf KSަ eXIL<'kΚ j8OA}R`0>g֔W\itai9qg:SR3|GӶ o&~u2tXaGqQ!WcȏgWko<I#(0 gԕn6A#[C}V d-ܾNuḭ`ޤ_m</،1c4RUTYJ$ma`roOp(S-Of X%1Y2pT`Ue0[[ɜ~wA]Vll!6$\Es~9 K?LBKPYMBOw^% =S ~ _J|cgfl>>u)5ÚA!kbV qMYVxYD!҇u -m|pCu<'帑?"[(n9/Rp5b)wf/L?i+I4L2?<5;?ǼMԜYmǞC3<hj$ՐFqkp;@?b@ajAX?i#>Twq`+Jxt_[c{1!iʱEuʋM_>vZoez^,d9F.;YXUQY%ZA@!?x e'ﻼKZ+ pMrxz@LJwƥWZGbo}䛝ӚnG?~'-{x< z= !qV$ !'|U{!bH5Y;V].&NC9QYS͋I5/"鬔o(e}0y\VS,,fbL*0Ӌe:CUju0H$?d [=|dZ 1ipA9 iH 9ljE!ЍI[)lc=IU`1v;Œ"VfgTfYɴ0y:AE)F0KCk`AԷogԺ}:V!8_b3m#/yܚN  +=g# 엖1P970GFU{Wt-VnR|+AпGwiҙڊMc;:n_\#8ԎK8fI|%awZ,C:8v&~݇,9ڨA6܂hRB0fKZ/֘ v*a+|f??aw۴=,c DgN=b6U@G,zӗ6b'wXA(䊲MӑÆxs}~0' 8 endstream endobj 22 0 obj << /Length 2935 /Filter /FlateDecode >> stream xZKsWDU5T*'$Z>P$%9*`^n p4_w_^~/*Q 'EiP^\.~cw97-rzl/ nnqճ#5V8[x=innvUԤ{~95/6{Q W ÐQ7Ca H*MKze+0ФGk`mqԖZc)DTg W榨XBslLiCC4`Ŋ3rЌb(Z /;(LHp#zOb]ªhbK/<ܽNwȆ]~E>.(+^6Dړ?'n`zP*  d|/FE\pPrn(-Hا .WhX}G(qFylv(6zMim4JJ\}UB}z1t7sm@Cfw5M}M]}?eRJuʖ9.s }!`|UmH^XʸбV#:6›Ɖ?_ @|rRuohiYE]#/|'{i@"+*N{Jh'sE,2“7/{7,/M1 ߰~ATQU)$lyV< GHZ> ~X $ Fʅn6vWktL em6Xلe-2aWd1&kkX tӐ Y+&m dbd&LoۺK#B25Hs0a Br(1e1Ze|JYSp ]k#&P,qW}09H5fBޒ3 ٬ !䪡4OLM[! :22-툁@&427Whz{ދRF¦6 `0U2!L LmOg<՟cX'UB 4N -܅(mFKj0N*A 2s2Dle%0Ŧ? & e%F5m|%L\ڒ ()[ +Z 2=`A0=܊ӌX"z6dtx1"|(6u=@m5ih$EO|zݒ>4Y ;%6EU{pi 5Ä!F#ޤӠNF $?zao3Dx=c h_E1r:yi-$r^{ՕgQUk*ȕ- #U3F+VK=k]Ȳg2aNqlEP" lePDZ";Elb1?%Ȕ9<)(]yw? zS@)KФvab4d]Cٶcr%}qh-L ޭ$\!z.Z߀VU=VO/V٧YߝAUPU /ʧj٣61S d! /o-r1y%[Fxv`aRUPϔKZ8{)- Y R LdID}[}*@bM?Wg)69:KX"x!&'Cp0r(n]S&mN,Ua䄸 ~FYL6y4udV8#qmgu7ySGC,;Ճl+tO=:*gSJE BX̪̀](gҙ.bO pM*5rgVA6%7HD(R?Bz"$:yѐNT*3G\5#ıB*DfK[.%k4CpRF6u|~K75?gdbvx./DR;sRnwf6"UN# X^F R};H EXG7 Ŕ$ (3en5n: JokZhpj8AfG'tw[?M2gZG'6w\,k@/M|ic ! 2nC!1pdʫS_M>Ʌf =z퓆G12k I{Jg@RJKB((>rя#V1IFo|$MrH9hs uj7(rE< . erggBF$E"']FS%D*`i SZ]qO J.Y,ZSpx*hv50d'~uׄ%b·HH~$R_5ŴtN $b=Od. ,\f1/0{ϓVUoBi3lBILyN?2BHd>D!\{W?],\ endstream endobj 25 0 obj << /Length 912 /Filter /FlateDecode >> stream xڥVKs0Wpgb$C;̴S[18qc8I}wW+;@"}.R?T6 E&6*~/aBq!ED<-ZkB__,_pvS"r{XŴȐQy pc)OtX]9%fYE QoV_[JboEo%#xh-dQ[ښ| w]!}Yw3aPېnΉEQfZfKB> )rW5~&@5"M}2QkyD=>F}|ATMΖ%"%VwJA؅LD6߽ '۟v2yf)wr͘ ސLi e:J %Ӧv9P [:gMʂ|m ni%'z8)L<)bSt1| ܹff <񡐨lV`3̭P:ksA 2 ?#00\rJ*/)(K:fQD3veN|ħj]S@=֌6E5ֽj2$aMdD||> stream x3135R0P0Bc3csCB.c46K$r9yr+p{E=}JJS ]  b<]00 @0?`d=0s@f d'n.WO@.sud endstream endobj 37 0 obj << /Length1 1640 /Length2 8887 /Length3 0 /Length 9955 /Filter /FlateDecode >> stream xڍP. 0w d h !NݽޫbOwO, Z>?p8@T0+6<s{8??VY'jnv]_)%ܜŁ@OOO^+/a+ t e Wk,=;]'<VC;<UVh:C`$p ;_ѿAaN`7f :Bjn^n07 {`Cj:BaOap''w} Pa.@a6۰vwà.e8&l7xY m~ yh<|a= 7;ߎF jBad0ClzLAap??W;^_! 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Johnson \and Charles J. Geyer} \maketitle \section{Overview} This is an example how to use morphometric Markov chains as implemented in the \verb@mcmc@ package in R. Let $X$ be an $\mathbb{R}^k$ valued random variable with probability density function, $f_X$. Let $g$ be a diffeomorphism, and $Y=g(X)$. Then the probability density function of $Y$, $f_Y$ is given by \begin{equation}\label{eq:def-fy} f_Y(y) = f_X\bigl(g^{-1}(y)\bigr) \det\bigl( \nabla g^{-1}(y) \bigr). \end{equation} Since $g$ is a diffeomorphism, we can draw inference about $X$ from information about $Y$ (and vice versa). It is not unusual for $f_X$ to either be known only up to a normalizing constant, or to be analytically intractable in other ways --- such as being high dimensional. A common solution to this problem is to use Markov chain Monte Carlo (MCMC) methods to learn about $f_X$. When using MCMC, a primary concern of the practitioner should be the question ``Does the Markov chain converge fast enough to be useful?'' One very useful convergence rate is called \emph{geometrically ergodic} \citep[Chapter~1]{johnson-thesis}. The \texttt{mcmc} package implements the Metropolis random-walk algorithm for arbitrary log unnormalized probability densities. But the Metropolis random-walk algorithm does not always perform well. As is demonstrated in \citet{johnson-geyer}, for $f_X$ and $f_Y$ related by diffeomorphism as in \eqref{eq:def-fy}, a Metropolis random-walk for $f_Y$ can be geometrically ergodic even though a Metropolis random-walk for $f_X$ is not. Since the transformation is one-to-one, inference about $f_X$ can be drawn from the Markov chain for $f_Y$. The \texttt{morph.metrop} and \texttt{morph} functions in the \texttt{mcmc} package provide this functionality, and this vignette gives a demonstration on how to use them. \section{T Distribution} \label{sec:toy} We start with a univariate example, which is a Student $t$ distribution with three degrees of freedom. Of course, one doesn't need MCMC to simulate this distribution (the R function \texttt{rt} does that), so this is just a toy problem. But it does illustrate some aspects of using variable transformation. A necessary condition for geometric ergodicity of a random-walk Metropolis algorithm is that the target density $\pi$ have a moment generating function \citep{jarner-tweedie}. For a univariate target density, which we have in this section, a sufficient condition for geometric ergodicity of a random-walk Metropolis algorithm is that the target density $\pi$ be exponentially light \citet{mengersen-tweedie}. Thus if we do not use variable transformation, the Markov chain simulated by the \texttt{metrop} function will not be geometrically ergodic. \citet[Example 4.2]{johnson-geyer} show that a $t$ distribution is sub-exponentially light. Hence using the transformations described in their Corollaries~1 and~2 will induce a target density $\pi_\gamma$ for which a Metropolis random-walk will be geometrically ergodic. using the transformation described as $h_2$ in \citet[Corollary~2]{johnson-geyer} will induce a target density for which a Metropolis random-walk will be geometrically ergodic. Passing a positive value for \texttt{b} to \texttt{morph} function will create the aforementioned transformation, $h_2$. It's as simple as <<>>= library(mcmc) h2 <- morph(b=1) @ We can now see the induced density. Note that \texttt{morph} works for log unnormalized densities, so we need exponentiate the induced density to plot it on the usual scale. <<>>= lud <- function(x) dt(x, df=3, log=TRUE) lud.induced <- h2$lud(lud) @ We can plot the two densities, <>= curve(exp(Vectorize(lud.induced)(x)), from = -3, to = 3, lty = 2, xlab = "t", ylab = "density") curve(exp(lud(x)), add = TRUE) legend("topright", c("t density", "induced density"), lty=1:2) @ The \texttt{Vectorize} in this example is necessary because the function \texttt{lud.induced} is not vectorized. Instead, it treats any vector passed as a single input, which is rescaled (using the specified diffeomorphism) and passed to \texttt{lud}. Compare the behavior of \texttt{lud} and \texttt{lud.induced} in the following example. <<>>= lud(1:4) lud(1) foo <- try(lud.induced(1:4)) class(foo) cat(foo, "\n") lud.induced(1) @ Because the function \texttt{dt} is vectorized, the function \texttt{lud} is also vectorized, mapping vectors to vectors, whereas the function \texttt{lud.induced} is not vectorized, mapping vectors to scalars. Before we start using random numbers, we set the seed of the random number generator so this document always produces the same results. <>= set.seed(42) @ To change the results, change the seed or delete the \texttt{set.seed} statement. Running a Markov chain for the induced density is done with \texttt{morph.metrop}. <<>>= out <- morph.metrop(lud, 0, blen=100, nbatch=100, morph=morph(b=1)) @ The content of \texttt{out\$batch} is on the scale of used by \texttt{lud}. Once the transformation has been set, no adjustment is needed (unless you want to change transformations). We start by adjusting the scale. <<>>= # adjust scale to find a roughly 20% acceptance rate out$accept @ An acceptance rate of \Sexpr{round(100 * out$accept, 1)}\% %$ to fix emacs highlighting is probably too high. By increasing the scale of the proposal distribution we can bring it down towards 20\%. <<>>= out <- morph.metrop(out, scale=4) out$accept @ We now use this Markov chain to estimate the expectation of the target distribution. But first we need to check whether our batch length is good. The following code <>= acf(out$batch) @ makes the autocorrelation plot (Figure~\ref{fig:fig0}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot for the sequence of batch means.} \label{fig:fig0} \end{figure} It looks like there is no significant autocorrelation among the batches so the following produces a valid confidence interval for the true unknown mean of the target distribution (since this is a toy problem we actually know the true ``unknown'' mean is zero, but we pretend we don't know that for the purposes of the toy problem) <<>>= t.test(out$batch) @ If we want a point estimate and a Monte Carlo standard error, those are <<>>= colMeans(out$batch) apply(out$batch, 2, sd) / sqrt(out$nbatch) @ If a shorter confidence interval is desired, the Markov chain can be run longer (increase either the number of batches or the batch length, or both). Note that when calculating our estimate and the Monte Carlo standard error we are not concerned with what was happening on the transformed scale. The \texttt{morph.metrop} function seamlessly does this for us. \subsection{Comparison of Morphed and Unmorphed} To show the utility of the transformation, we will study the behavior of the Markov chain with and without the transformation for the same problem as in the preceding section. We will consider two different estimation methods. \begin{enumerate} \item \label{enum:rw} Estimate the mean of the target distribution using a random-walk Metropolis algorithm implemented by the \texttt{metrop} function. \citet{jarner-roberts} demonstrate that a central limit theorem does not hold for these estimates. \item \label{enum:rw-induced} Estimate the mean of the target distribution using a random-walk Metropolis algorithm implemented by the \texttt{morph.metrop} function with argument \texttt{morph = morph(b=1)}. \citet{johnson-geyer} demonstrate that a central limit theorem does hold for these estimates. \end{enumerate} For the former, we need to adjust the scale. <>= out.unmorph <- metrop(lud, 0, blen=1000, nbatch=1) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=4) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=6) out.unmorph$accept @ A scale of 6 appears to be about right. Now we do a long run for this sampler. Because this run takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("morph1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.unmorph <- metrop(out.unmorph, blen = 1e5, nbatch = 1e3) save(out.unmorph, file = "morph1.rda") } else { .Random.seed <- out.unmorph$final.seed } out.unmorph$accept @ Let's look at the distribution of batch means. The following code <>= foo <- as.vector(out.unmorph$batch) qqnorm(foo) qqline(foo) @ makes a Q-Q plot of the batch means (Figure~\ref{fig:fig4}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Q-Q plot of batch means (batch length \Sexpr{out.unmorph$blen}) for the unmorphed chain.} \label{fig:fig4} \end{figure} We see bad behavior of the unmorphed chain. These batch means (or at least some batch means for sufficiently long batch length) should look normally distributed, and these don't. Not even close. We do a formal test just to check our interpretation of the plot <>= shapiro.test(foo) @ Now for comparison, we check the morphed chain. <>= lout <- suppressWarnings(try(load("morph2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.morph <- morph.metrop(out, blen = 1e5, nbatch = 1e3) save(out.morph, file = "morph2.rda") } else { .Random.seed <- out.morph$final.seed } out.morph$accept @ The following code <>= foo <- as.vector(out.morph$batch) qqnorm(foo) qqline(foo) @ makes a Q-Q plot of the batch means (Figure~\ref{fig:fig5}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Q-Q plot of batch means (batch length \Sexpr{out.unmorph$blen}) for the morphed chain.} \label{fig:fig5} \end{figure} We see good behavior of the morphed chain. These batch means do look normally distributed. We do a formal test just to check our interpretation of the plot <>= shapiro.test(foo) @ \section{Binomial Distribution with a Conjugate Prior} We demonstrate a morphometric Markov chain using the \texttt{UCBAdmisions} data set included in \texttt{R}, (use \texttt{help(UCBAdmissions)} to see details of this data set). We will model the probability of a student being admitted or rejected, using the sex of the student and the department that the student applied to as predictor variables. For our prior, we naively assume that 30\% of all students are admitted, independent of sex or department. As this is a naive prior, we will only add 5 students to each gender-department combination. This will not give the prior much weight, most of the information in the posterior distribution will be from the data. If we have $L$ observations from a multinomial distribution, then using a multinomial logit-link, with model matrices $M^1,\dots,M^L$, regression parameter $\beta$, observed counts $Y^1,\dots,Y^N$ with observed sample sizes $N^1,\dots,N^L$ and prior probabilities $\xi^1, \dots, \xi^L$ and prior ``sample sizes'' $\nu^1,\dots,\nu^L$ then the posterior distribution of $\beta$ is given by \citep[Sec. 5.1.2]{johnson-thesis} \begin{equation}\label{eq:mult-post-conj-complicated} \pi(\beta|y,n,\xi,\nu) \propto \exp\biggl\{ \sum_{l=1}^L \inner{y^l + \xi^l \nu^l, M^l \beta} - (n^l + \nu^l) \log\bigl( \sum_j e^{M_{j\cdot} \beta} \bigr) \biggr\} \end{equation} where $\inner{a, b}$ denotes the usual inner product between vectors $a$ and $b$. For our application, we can simplify this in two ways. First, we use the posterior counts instead of the sum of the prior and data counts, i.e. use $y^{*l} = y^l + \xi^l \nu^l$ and $n^{*l} = n^l + \nu^l$. Second, to avoid having a direction of recession in $\pi(\beta|\cdot)$, we need to fix the elements of $\beta$ that correspond with one of the response categories. Since we are going to fitting a binomial response, if we set these elements of $\beta$ to be $0$, we may then replace the sequence of model matrices with a single model matrix; $M$ instead of $M^1,\dots,M^L$. The $l$-th row of $M$ will correspond to $M^l$. Label the two response categories $A$ and $B$. Without loss of generality, we will fix the elements of $\beta$ corresponding to category $B$ to 0. Let $x_1,\dots,x_L$ represent the posterior counts of category $A$, and $\beta^*$ represent the corresponding elements of $\beta$ --- these are the elements of $\beta$ we did not fix as 0. The meaning of $n^{*1},\dots,n^{*L}$ is unchanged. Then our simplified unnormalized posterior density is \begin{equation}\label{eq:simplified-posterior} \pi(\beta|x,n^*) \propto \exp\biggl\{ \inner{x, M \beta^*} - \sum_{l=1}^L n^{*l} \log\bigl(1 + e^{(M \beta^*)_l}\bigr) \biggr\}. \end{equation} This can be computed with a very simple \texttt{R} function, we implement it in log form. <>= lud.binom <- function(beta, M, x, n) { MB <- M %*% beta sum(x * MB) - sum(n * log(1 + exp(MB))) } @ Now that we have a function to calculate a log-unnormalized posterior density, we can run the Markov chain. To that, we need the model matrix. First we convert the \texttt{UCAdmissions} data to a \texttt{data.frame}. <>= dat <- as.data.frame(UCBAdmissions) dat.split <- split(dat, dat$Admit) dat.split <- lapply(dat.split, function(d) { val <- as.character(d$Admit[1]) d["Admit"] <- NULL names(d)[names(d) == "Freq"] <- val d }) dat <- merge(dat.split[[1]], dat.split[[2]]) @ Next we build the model matrix. Our model specification allows for an interaction between gender and department, even though our prior assumes that they are independent. <>= formula <- cbind(Admitted, Rejected) ~ (Gender + Dept)^2 mf <- model.frame(formula, dat) M <- model.matrix(formula, mf) @ As stated above, we will take $\nu = 5$ and $\xi=0.30$. That is, we will add 5 students to each gender-department combination, where each combination has a 30\% acceptance rate. <<>>= xi <- 0.30 nu <- 5 @ <>= lud.berkeley <- function(B) lud.binom(B, M, dat$Admitted + xi * nu, dat$Admitted + dat$Rejected + nu) @ This function is suitable for passing to \texttt{metrop} or \texttt{morph.metrop}. We know that using \texttt{morph.metrop} with \texttt{morph=morph(p=3)} will run a geometrically ergodic Markov chain \citep{johnson-geyer}. <<>>= berkeley.out <- morph.metrop(lud.berkeley, rep(0, ncol(M)), blen=1000, nbatch=1, scale=0.1, morph=morph(p=3)) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.05) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.02) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, blen=10000) berkeley.out$accept @ <<>>= berkeley.out <- morph.metrop(berkeley.out, blen=1, nbatch=100000) @ Estimate the posterior mean acceptance probabilities for each gender-department combination. <<>>= beta <- setNames(colMeans(berkeley.out$batch), colnames(M)) MB <- M %*% beta dat$p <- dat$Admitted / (dat$Admitted + dat$Rejected) dat$p.post <- exp(MB) / (1 + exp(MB)) dat @ The small difference between the data and posterior probabilities is expected, our prior was given very little weight. Using \texttt{morph.metrop} with the setting \texttt{morph=morph(p=3)} in this setting is an efficient way of sampling from the posterior distribution. We can also compare the posterior distribution of admittance probability for each gender-department combination. Table~\ref{tab:post-quant} gives the 5\% and 95\% quantiles for the posterior distribution of the admittance probabilities for each gender-department combination. Figure~\ref{fig:posterior-probs} gives the same quantiles, plus the mean posterior-probability for each gender-department combination. From these we can see that for each department, there is considerable overlap of the distributions of probabilities for males and females. <>= posterior.probabilities <- t(apply(berkeley.out$batch, 1, function(r) { eMB <- exp(M %*% r) eMB / (1 + eMB) })) quants <- apply(posterior.probabilities, 2, quantile, prob=c(0.05, 0.95)) quants.str <- matrix(apply(quants, 2, function(r) sprintf("[%0.2f, %0.2f]", r[1], r[2])), nrow=2, byrow=TRUE) @ \begin{table}[ht] \caption{5\% and 95\% posterior quantiles for admittance probability for each gender-department combination} \begin{center} \begin{tabular}{|l|c|c|c|c|c|c|} \hline Gender & Dept. A & Dept. B & Dept. C & Dept. D & Dept. E. & Dept. F \\ \hline Female & \Sexpr{paste(quants.str[1, 1:6], collapse=" & ")} \\ Male & \Sexpr{paste(quants.str[2, 1:6], collapse=" & ")} \\ \hline \end{tabular} \label{tab:post-quant} \end{center} \end{table} \begin{figure} \begin{center} <>= x <- (0:5) * 2 + 1 plot(x[c(1, 6)] + 0.5 * c(-1, 1), 0:1, xlab="Department", ylab="Probability", xaxt="n", type="n") axis(1, x, LETTERS[1:6]) for(i in 1:6) { lines((x[i]-0.25)*c(1, 1), quants[1:2, i], lwd=2, col="gray") lines((x[i] + 0.25) * c(1, 1), quants[1:2, i + 6], lwd=2, col="gray") points(x[i] + 0.25 * c(-1, 1), dat$p.post[i + c(0, 6)], pch=c("F", "M")) } @ \end{center} \caption{Posterior 5\% and 95\% quantiles and mean, by department and gender.} \label{fig:posterior-probs} \end{figure} \section{Cauchy Location-Scale Model} We are going to do a Cauchy location-scale family objective Bayesianly. \subsection{Data} First we generate some data. <>= n <- 15 mu0 <- 50 sigma0 <- 10 x <- rcauchy(n, mu0, sigma0) round(sort(x), 1) @ \texttt{mu0} and \texttt{sigma0} are the true unknown parameter values (since the data are simulated we actually know these ``unknown'' parameter values, but we must pretend we don't know them and estimate them). \subsection{Prior} The standard objective prior distribution for this situation (insofar as any prior distribution can be said to be an objective standard) is the improper prior $$ g(\mu, \sigma) = \frac{1}{\sigma} $$ which is right Haar measure for the location-scale group, and is the standard prior that comes from the group invariance argument \citep[Section~3.2]{kass-wasserman}. \subsection{Log Unnormalized Posterior} We need a function whose argument is a two-vector <>= lup <- function(theta) { if (any(is.na(theta))) stop("NA or NaN in input to log unnormalized density function") mu <- theta[1] sigma <- theta[2] if (sigma <= 0) return(-Inf) if (any(! is.finite(theta))) return(-Inf) result <- sum(dcauchy(x, mu, sigma, log = TRUE)) - log(sigma) if (! is.finite(result)) { warning(paste("Oops! mu = ", mu, "and sigma =", sigma)) } return(result) } @ \subsection{Laplace Approximation} To have some idea what we are doing, we first maximize the log unnormalized posterior. To do it helps to have good starting points for the optimization. Robust estimators of location and scale are <>= mu.twiddle <- median(x) sigma.twiddle <- IQR(x) c(mu.twiddle, sigma.twiddle) @ The posterior mode is <>= oout <- optim(c(mu.twiddle, sigma.twiddle), lup, control = list(fnscale = -1), hessian = TRUE) stopifnot(oout$convergence == 0) mu.hat <- oout$par[1] sigma.hat <- oout$par[2] c(mu.hat, sigma.hat) @ and the hessian evaluated at the posterior mode (calculated by \texttt{optim} using finite differences) is <>= oout$hessian @ The hessian is nearly diagonal and one can check that theoretically is exactly diagonal. Thus approximate (asymptotic) posterior standard deviations are <>= sqrt(- 1 / diag(oout$hessian)) @ \subsection{Theory} To use the theory in \citet{johnson-geyer} we must verify that the target distribution (the unnormalized posterior) is everywhere positive, and it isn't (it is zero for $\sigma \le 0$). We tried making $\log(\sigma)$ the parameter but this didn't work either because $\log(\sigma)$ goes to infinity so slowly that this stretches out the tails so much that the transformations introduced by \citet{johnson-geyer} can't pull them back in again. We do know \citep[Example~3.4]{johnson-geyer} that if we fix $\sigma$ this is a sub-exponentially light target distribution. Letting $\sigma$ vary can only make this worse. Thus, if we don't do anything and just use the \texttt{metrop} function, then performance will be very bad. So we are going to use the transformations and the \texttt{morph.metrop} function, even though the theory that motivates them does not hold. \subsection{Morph} We want to center the transformation at the posterior mode, and use a radius $r$ that doesn't transform until several approximate standard deviations <>= moo <- morph(b = 0.5, r = 7, center = c(mu.hat, sigma.hat)) mout <- morph.metrop(lup, c(mu.hat, sigma.hat), 1e4, scale = 3, morph = moo) mout$accept mout <- morph.metrop(mout) @ Good enough. An attempt to increase the scale led to error when the transformation functions overflowed. Can't take steps too big with this stuff. The following code <>= acf(mout$batch) @ makes an autocorrelation plot (Figure~\ref{fig:cfig1}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot. First component is $\mu$, second is $\sigma$.} \label{fig:cfig1} \end{figure} It looks like lag 10 to 15 is enough to get near independence. Now we want to make marginal density plots. If we just feed our MCMC output to the R function \texttt{density} it undersmooths because it expects independent and identically distributed data rather than autocorrelated data. Thus we feed it subsampled, nearly uncorrelated data to select the bandwidth and then use that bandwidth on the full data. Here's how that works. The following code <>= mu <- mout$batch[ , 1] i <- seq(1, mout$nbatch, by = 15) out.sub <- density(mu[i]) out <- density(mu, bw = out.sub$bw) plot(out) @ makes the density plot (Figure~\ref{fig:cfig2}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Density plot for the marginal posterior for $\mu$.} \label{fig:cfig2} \end{figure} And a similar plot for $\sigma$ (Figure~\ref{fig:cfig3}) \begin{figure} \begin{center} <>= sigma <- mout$batch[ , 2] out.sub <- density(sigma[i]) out <- density(sigma, bw = out.sub$bw) plot(out) @ \end{center} \caption{Density plot for the marginal posterior for $\sigma$.} \label{fig:cfig3} \end{figure} \begin{thebibliography}{} \bibitem[Jarner and Roberts(2007)]{jarner-roberts} Jarner, S.F., and G.O. Roberts (2007). \newblock Convergence of heavy-tailed Monte Carlo Markov chain algorithms. \newblock \emph{Scandinavian Journal of Statistics}, 34, 781--815. \bibitem[Jarner and Tweedie(2003)]{jarner-tweedie} Jarner, S.~F., and Tweedie, R.~L. (2003). \newblock Necessary conditions for geometric and polynomial ergodicity of random-walk-type Markov chains. \newblock \emph{Bernoulli}, 9, 559--578. \bibitem[Johnson(2011)]{johnson-thesis} Johnson, L.~T. (2011). \newblock Geometric Ergodicity of a Random-Walk Metropolis Algorithm via Variable Transformation and Computer Aided Reasoning in Statistics. \newblock Ph.~D. thesis. University of Minesota. \url{http://purl.umn.edu/113140} \bibitem[Johnson and Geyer(submitted)]{johnson-geyer} Johnson, L.~T., and C.~J. Geyer (submitted). \newblock Variable Transformation to Obtain Geometric Ergodicity in the Random-walk Metropolis Algorithm. \newblock Revised and resubmitted to \emph{Annals of Statistics}. \bibitem[Kass and Wasserman(1996)]{kass-wasserman} Kass, R.~E., and Wasserman, L. (1996). \newblock Formal rules for selecting prior distributions: A review and annotated bibliography. \newblock \emph{Journal of the American Statistical Association}, 435, 1343--1370. \bibitem[Mengersen and Tweedie(1996)]{mengersen-tweedie} Mengersen, K.L., ad R. L. Tweedie (1996). \newblock Rates of convergence of the Hastings and Metropolis algorithms. \newblock \emph{Annals of Statistics}, 24, 101--121. \end{thebibliography} \end{document} mcmc/tests/0000755000175100001440000000000013074514645012362 5ustar hornikusersmcmc/tests/logitnegidx.Rout.save0000644000175100001440000001001013045741600016466 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test outfun (negative index vector) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, outfun = - 2) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- my.path[ , out.metro$outfun] > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/logitfunarg.Rout.save0000644000175100001440000000600713045741600016505 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test outfun (function) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, outfun = function(x) c(x, x^2)) > > out.metro <- metrop(out.metro) > out.metro$outfun function (x) c(x, x^2) > dim(out.metro$batch) [1] 100 6 > > logl <- function(beta, x1, x2, y) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, x1 = x1, x2 = x2, y = y) > out.metro$lud function (beta, x1, x2, y) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta)/(1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } > out.metro <- metrop(out.metro, x1 = x1, x2 = x2, y = y) > out.metro$lud function (beta, x1, x2, y) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta)/(1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } > > mcmc/tests/logitfun.R0000644000175100001440000000617513045741240014334 0ustar hornikusers # test outfun (function) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, outfun = function(x) c(x, x^2)) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- t(apply(my.path, 1, out.metro$outfun)) k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon goom <- cbind(my.path, my.path^2) all(dim(goom) == dim(out.metro$batch)) max(abs(goom - out.metro$batch)) < epsilon mcmc/tests/logitsub.Rout.save0000644000175100001440000000746413045741600016024 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test spacing (nspac) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, nspac = 3) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > my.path <- my.path[seq(nspac, niter, by = nspac), ] > all(dim(my.path) == dim(out.metro$batch)) [1] TRUE > > all(my.path == out.metro$batch) [1] TRUE > > mcmc/tests/morphtoo.Rout.save0000644000175100001440000000412713045741600016034 0ustar hornikusers R version 2.15.0 (2012-03-30) Copyright (C) 2012 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: i686-pc-linux-gnu (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > x <- seq(0, 10, length = 10001) > > ### sub-exponentially light transformation > > b <- 0.5 > fsub <- morph(b = b) > > y <- unlist(Map(fsub$inverse, x)) > > myfsub <- function(x) ifelse(x > 1 / b, exp(b * x) - exp(1) / 3, + (x * b)^3 * exp(1) / 6 + x * b * exp(1) / 2) > y2 <- myfsub(x) > all.equal(y, y2, tolerance = 1e-14) [1] TRUE > > z <- unlist(Map(fsub$transform, y)) > all.equal(z, x, tolerance = 1e-14) [1] TRUE > > ### exponentially light transformation > > r <- 5 > p <- 3 > fp3 <- morph(r = r) > > y <- unlist(Map(fp3$inverse, x)) > > myfp3 <- function(x) ifelse(x < r, x, x + (x - r)^p) > y2 <- myfp3(x) > all.equal(y, y2, tolerance = 1e-14) [1] TRUE > > z <- unlist(Map(fp3$transform, y)) > all.equal(z, x, tolerance = 1e-12) [1] TRUE > > ### both together > > fboth <- morph(b = b, r = r) > > y <- unlist(Map(fboth$inverse, x)) > y2 <- myfsub(myfp3(x)) > all.equal(y, y2, tolerance = 1e-14) [1] TRUE > > z <- unlist(Map(fboth$transform, y)) > all.equal(z, x, tolerance = 1e-12) [1] TRUE > > ### exponentially light transformation with p != 3 > > r <- 5 > p <- 2.2 > fpo <- morph(r = r, p = p) > > y <- unlist(Map(fpo$inverse, x)) > > myfpo <- function(x) ifelse(x < r, x, x + (x - r)^p) > y2 <- myfpo(x) > all.equal(y, y2, tolerance = 1e-14) [1] TRUE > > z <- unlist(Map(fpo$transform, y)) > all.equal(z, x, tolerance = 1e-14) [1] TRUE > > > proc.time() user system elapsed 4.024 0.036 4.038 mcmc/tests/isotropic.R0000644000175100001440000000243513045741600014513 0ustar hornikuserslibrary(mcmc) isotropic <- mcmc:::isotropic isotropic.logjacobian <- mcmc:::isotropic.logjacobian # create identity test function identity <- function(x) x d.identity <- function(x) 1 # check that isotropic is length preserving for vectors of lengths 1--1000 all(sapply(1:1000, function(x) length(isotropic(identity)(rep(1, x))) == x)) # test that isotropic(identity) is an identity function all.equal(isotropic(identity)(1:10), 1:10) x <- seq(0, 1, length.out=200) all.equal(isotropic(identity)(x), x) # make sure that isotropic.logjacobian(identity, d.identity) is a 0 function all.equal(isotropic.logjacobian(identity, d.identity)(1:10), 0) # make sure that 0 as an input does not cause divide-by-zero errors all.equal(isotropic(identity)(0), 0) all.equal(isotropic(identity)(0 * 1:4), rep(0, 4)) all.equal(isotropic.logjacobian(identity, d.identity)(0), 0) all.equal(isotropic.logjacobian(identity, d.identity)(0 * 1:4), 0) # try isotropic with f(x) = x^2, then we should get # istropic(f)(x) := |x| * x f <- function(x) x^2 all.equal(isotropic(f)(1), 1) all.equal(isotropic(f)(c(1, 1)), sqrt(2) * c(1, 1)) all.equal(isotropic(f)(c(1, 0, 1)), sqrt(2) * c(1, 0, 1)) # make sure lazy-loading works properly. g <- function(x) x^2 g.iso <- isotropic(g) g <- function(x) x all.equal(g.iso(2), 2*2) mcmc/tests/circle.Rout.save0000644000175100001440000000555713045741600015436 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > d <- 5 > > logh <- function(x) { + if (! is.numeric(x)) stop("x not numeric") + if (length(x) != d) stop("length(x) != d") + fred <- 1 - sum(x^2) + if (fred > 0) return(log(fred)) else return(-Inf) + } > > out.metro <- metrop(logh, rep(0, d), 1e3, scale = 0.01) > out.metro$accept [1] 0.979 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.72 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.16 > > out.metro <- metrop(out.metro, scale = 0.4) > out.metro$accept [1] 0.228 > > out.metro <- metrop(out.metro, nbatch = 1e2, debug = TRUE) > > all(out.metro$batch[- out.metro$nbatch, ] == out.metro$current[- 1, ]) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$batch[out.metro$nbatch, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logh) > my.prop.log.green <- apply(out.metro$proposal, 1, logh) > all(is.na(out.metro$u) == ((my.prop.log.green == -Inf) | + (my.prop.log.green > my.curr.log.green))) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > blurfle <- foo - out.metro$log.green > blurfle[foo == -Inf & out.metro$log.green == -Inf] <- 0 > max(blurfle) < epsilon [1] TRUE > > my.accept <- (my.prop.log.green > -Inf) & (is.na(my.u) | my.u < exp(foo)) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > > all(my.path == out.metro$batch) [1] TRUE > > mcmc/tests/logitfun.Rout.save0000644000175100001440000001024613045741600016013 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test outfun (function) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, outfun = function(x) c(x, x^2)) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- t(apply(my.path, 1, out.metro$outfun)) > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > goom <- cbind(my.path, my.path^2) > all(dim(goom) == dim(out.metro$batch)) [1] TRUE > max(abs(goom - out.metro$batch)) < epsilon [1] TRUE > mcmc/tests/logitidx.R0000644000175100001440000000577313045741240014333 0ustar hornikusers # test outfun (positive index vector) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, outfun = c(2, 3)) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- my.path[ , out.metro$outfun] k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/temp-par.R0000644000175100001440000002221613045741240014224 0ustar hornikusers library(mcmc) set.seed(42) data(foo) attach(foo) out <- glm(y ~ x1 + x2 + x3, family = binomial, x = TRUE) summary(out) modmat <- out$x models <- cbind(rep(0:1, each = 4), rep(rep(0:1, times = 2), each = 2), rep(0:1, times = 4)) exes <- paste("x", 1:3, sep = "") betas <- NULL for (i in 1:nrow(models)) { inies <- as.logical(models[i, ]) foo <- exes[inies] bar <- paste("y ~", paste(foo, collapse = " + ")) if (! any(inies)) bar <- paste(bar, "1") baz <- glm(as.formula(bar), family = binomial) beta <- rep(0, 4) beta[c(TRUE, inies)] <- baz$coef betas <- rbind(betas, beta) } neighbors <- matrix(FALSE, nrow(models), nrow(models)) for (i in 1:nrow(neighbors)) { for (j in 1:ncol(neighbors)) { foo <- models[i, ] bar <- models[j, ] if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE } } ludfun <- function(state, ...) { stopifnot(is.numeric(state)) stopifnot(length(state) == ncol(models) + 2) stopifnot(length(state) == ncol(models) + 2) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= nrow(models)) beta <- state[-1] inies <- c(TRUE, as.logical(models[icomp, ])) beta.logl <- beta beta.logl[! inies] <- 0 eta <- as.numeric(modmat %*% beta.logl) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) val <- logl - sum(beta^2) / 2 return(val) } ludval <- NULL for (i in 1:nrow(models)) ludval <- c(ludval, ludfun(c(i, betas[i, ]))) all(is.finite(ludval)) out <- temper(ludfun, initial = betas, neighbors = neighbors, nbatch = 20, blen = 10, nspac = 5, scale = 0.56789, parallel = TRUE, debug = TRUE) names(out) ### check decision about within-component or jump/swap identical(out$unif.which < 0.5, out$which) identical(out$which, out$proposal[ , 1] == out$coproposal[ , 1]) ### check proposal and coproposal are actually current state or part thereof prop <- out$proposal coprop <- out$coproposal prop.i <- prop[ , 1] coprop.i <- coprop[ , 1] alt.prop <- prop alt.coprop <- coprop for (i in 1:nrow(prop)) { alt.prop[i, ] <- c(prop.i[i], out$state[i, prop.i[i], ]) alt.coprop[i, ] <- c(coprop.i[i], out$state[i, coprop.i[i], ]) } identical(coprop, alt.coprop) identical(prop[! out$which, ], alt.prop[! out$which, ]) identical(prop[out$which, 1], alt.prop[out$which, 1]) ### check hastings ratio calculated correctly foo <- apply(prop, 1, ludfun) fooco <- apply(coprop, 1, ludfun) prop[ , 1] <- out$coproposal[ , 1] coprop[ , 1] <- out$proposal[ , 1] foo.swap <- apply(prop, 1, ludfun) fooco.swap <- apply(coprop, 1, ludfun) log.haste <- ifelse(out$which, foo - fooco, foo.swap + fooco.swap - foo - fooco) all.equal(log.haste, out$log.hastings) ### check hastings rejection decided correctly identical(out$log.hastings >= 0, is.na(out$unif.hastings)) all(out$log.hastings < 0 | out$acceptd) identical(out$acceptd, out$log.hastings >= 0 | out$unif.hastings < exp(out$log.hastings)) ### check acceptance carried out or not (according to decision) correctly before <- out$state after <- before after[- dim(after)[1], , ] <- before[-1, , ] after[dim(after)[1], , ] <- out$final my.after <- before for (i in 1:length(out$acceptd)) { if (out$acceptd[i]) { if (out$which[i]) { j <- out$proposal[i, 1] my.after[i, j, ] <- out$proposal[i, -1] } else { j <- out$proposal[i, 1] k <- out$coproposal[i, 1] my.after[i, j, ] <- out$coproposal[i, -1] my.after[i, k, ] <- out$proposal[i, -1] } } } identical(after, my.after) ### check within-component proposal my.coproposal.within <- out$coproposal[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + out$scale * my.z all.equal(proposal.within, my.proposal.within) my.unif.choose <- out$unif.choose[out$which, 1] my.i <- floor(nrow(models) * my.unif.choose) + 1 all(1 <= my.i & my.i <= nrow(models)) identical(my.i, my.coproposal.within[ , 1]) ### check swap proposal coproposal.swap <- out$coproposal[! out$which, ] proposal.swap <- out$proposal[! out$which, ] unif.choose.swap <- out$unif.choose[! out$which, ] my.i <- floor(nrow(models) * unif.choose.swap[ , 1]) + 1 nneighbors <- apply(out$neighbors, 1, sum) my.nneighbors <- nneighbors[my.i] my.k <- floor(my.nneighbors * unif.choose.swap[ , 2]) + 1 my.j <- my.k foo <- seq(1, ncol(out$neighbors)) for (i in seq(along = my.j)) { my.j[i] <- (foo[out$neighbors[my.i[i], ]])[my.k[i]] } identical(coproposal.swap[ , 1], my.i) identical(proposal.swap[ , 1], my.j) ### check standard normal and uniform random numbers are as purported save.Random.seed <- .Random.seed .Random.seed <- out$initial.seed nx <- ncol(out$initial) niter <- out$nbatch * out$blen * out$nspac my.norm <- matrix(NA, nrow = nrow(out$norm), ncol = ncol(out$norm)) my.unif.which <- rep(NA, niter) my.unif.hastings <- rep(NA, niter) my.unif.choose <- matrix(NA, niter, 2) for (iiter in 1:niter) { my.unif.which[iiter] <- runif(1) if (out$which[iiter]) { my.unif.choose[iiter, 1] <- runif(1) my.norm[iiter, ] <- rnorm(nx) if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) } else { my.unif.choose[iiter, ] <- runif(2) if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) } } identical(my.norm, out$norm) identical(my.unif.which, out$unif.which) identical(my.unif.hastings, out$unif.hastings) identical(my.unif.choose, out$unif.choose) .Random.seed <- save.Random.seed ### check batch means foo <- after[seq(1, niter) %% out$nspac == 0, , ] foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2:3])) foo <- apply(foo, c(2, 3, 4), mean) all.equal(foo, out$batch) ### check acceptance rates accept.within <- out$acceptd[out$which] my.i.within <- out$coproposal[out$which, 1] my.acceptx <- as.vector(sapply(split(accept.within, my.i.within), mean)) identical(my.acceptx, out$acceptx) accept.swap <- out$acceptd[! out$which] my.i.swap <- out$coproposal[! out$which, 1] my.j.swap <- out$proposal[! out$which, 1] nmodel <- nrow(out$neighbors) my.accepti <- matrix(NA, nmodel, nmodel) for (i in 1:nmodel) { for (j in 1:nmodel) { if (out$neighbors[i, j]) { my.accepti[i, j] <- mean(accept.swap[my.i.swap == i & my.j.swap == j]) } } } identical(my.accepti, out$accepti) ### check scale vector nx <- ncol(models) + 1 newscale <- rnorm(nx, 0.5, 0.1) out <- temper(out, scale = newscale) my.coproposal.within <- out$coproposal[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + sweep(my.z, 2, out$scale, "*") all.equal(proposal.within, my.proposal.within) ### check scale matrix matscale <- matrix(rnorm(nx * nx, 0.0, 0.1), nx, nx) diag(matscale) <- 0.56789 out <- temper(out, scale = matscale) my.coproposal.within <- out$coproposal[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + my.z %*% t(out$scale) all.equal(proposal.within, my.proposal.within) ### check scale list lisztscale <- list(0.56789, newscale, matscale, matscale, newscale, 0.98765, 0.98765, newscale) out <- temper(out, scale = lisztscale) my.coproposal.within <- out$coproposal[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within for (iiter in 1:nrow(my.z)) { my.i <- my.coproposal.within[iiter, 1] my.scale <- out$scale[[my.i]] if (is.matrix(my.scale)) { my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + my.z[iiter, , drop = FALSE] %*% t(my.scale) } else { my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + my.z[iiter, ] * my.scale } } all.equal(proposal.within, my.proposal.within) ### check outfun outfun <- function(state, icomp, ...) { stopifnot(is.matrix(state)) stopifnot(is.numeric(state)) nx <- ncol(betas) ncomp <- nrow(betas) stopifnot(ncol(state) == nx) stopifnot(nrow(state) == ncomp) stopifnot(1 <= icomp && icomp <= ncomp) foo <- state[icomp, ] bar <- foo^2 return(c(foo, bar)) } out <- temper(out, outfun = outfun, icomp = 4) before <- out$state after <- before after[- dim(after)[1], , ] <- before[-1, , ] after[dim(after)[1], , ] <- out$final outies <- apply(after, 1, outfun, icomp = 4) outies <- t(outies) foo <- outies[seq(1, niter) %% out$nspac == 0, ] foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) foo <- apply(foo, c(2, 3), mean) all.equal(foo, out$batch) mcmc/tests/accept-batch.R0000644000175100001440000000101413062633366015016 0ustar hornikusers # new feature batching acceptance rates set.seed(42) library(mcmc) h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) out <- metrop(h, rep(0, 5), nbatch = 100, blen = 100, scale = 0.1, debug = TRUE) all.equal(out$accept, mean(out$accept.batch)) foo <- matrix(out$debug.accept, nrow = out$blen) bar <- colMeans(foo) all.equal(out$accept.batch, bar) options(digits = 4) # try to keep insanity of computer arithmetic under control out$accept t.test(out$accept.batch)$conf.int mcmc/tests/saveseed.R0000644000175100001440000000060713045741240014276 0ustar hornikusers library(mcmc) set.seed(42) h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) out <- metrop(h, initial = rep(0, 5), nbatch = 100, blen = 17, nspac = 3, scale = 0.1) save.seed <- .Random.seed out1 <- metrop(out) out2 <- metrop(out1) out3 <- metrop(out, nbatch = 2 * out$nbatch) fred <- rbind(out1$batch, out2$batch) identical(fred, out3$batch) mcmc/tests/logit.R0000644000175100001440000000520613045741600013615 0ustar hornikusers epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) options(digits = 3) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) summary(out) mlogl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(- sum(log(p[y == 1])) - sum(log(1 - p[y == 0]))) } out.nlm <- nlm(mlogl, coefficients(out), print.level = 2) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) var(out.metro$batch) olbm(out.metro$batch, 25) saveseed <- .Random.seed out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE) all(out.metro$batch[- out.metro$nbatch, ] == out.metro$current[- 1, ]) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$batch[out.metro$nbatch, ] == out.metro$final) .Random.seed <- saveseed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] all(my.path == out.metro$batch) mcmc/tests/isotropic.Rout.save0000644000175100001440000000411313045741600016173 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > library(mcmc) > isotropic <- mcmc:::isotropic > isotropic.logjacobian <- mcmc:::isotropic.logjacobian > > # create identity test function > identity <- function(x) x > d.identity <- function(x) 1 > > # check that isotropic is length preserving for vectors of lengths 1--1000 > all(sapply(1:1000, function(x) length(isotropic(identity)(rep(1, x))) == x)) [1] TRUE > > # test that isotropic(identity) is an identity function > all.equal(isotropic(identity)(1:10), 1:10) [1] TRUE > x <- seq(0, 1, length.out=200) > all.equal(isotropic(identity)(x), x) [1] TRUE > > # make sure that isotropic.logjacobian(identity, d.identity) is a 0 function > all.equal(isotropic.logjacobian(identity, d.identity)(1:10), 0) [1] TRUE > > # make sure that 0 as an input does not cause divide-by-zero errors > all.equal(isotropic(identity)(0), 0) [1] TRUE > all.equal(isotropic(identity)(0 * 1:4), rep(0, 4)) [1] TRUE > all.equal(isotropic.logjacobian(identity, d.identity)(0), 0) [1] TRUE > all.equal(isotropic.logjacobian(identity, d.identity)(0 * 1:4), 0) [1] TRUE > > # try isotropic with f(x) = x^2, then we should get > # istropic(f)(x) := |x| * x > f <- function(x) x^2 > all.equal(isotropic(f)(1), 1) [1] TRUE > all.equal(isotropic(f)(c(1, 1)), sqrt(2) * c(1, 1)) [1] TRUE > all.equal(isotropic(f)(c(1, 0, 1)), sqrt(2) * c(1, 0, 1)) [1] TRUE > > # make sure lazy-loading works properly. > g <- function(x) x^2 > g.iso <- isotropic(g) > g <- function(x) x > all.equal(g.iso(2), 2*2) [1] TRUE > mcmc/tests/logitlogidx.R0000644000175100001440000000577713045741240015041 0ustar hornikusers # test outfun (logical index vector) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, outfun = seq(1:3) > 1) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- my.path[ , out.metro$outfun] k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/logitvec.Rout.save0000644000175100001440000001014213045741600015773 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test vector (diag(foo)) scaling > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > sally <- apply(out.metro$batch, 2, sd) > > out.metro <- metrop(out.metro, scale = sally) > out.metro$accept [1] 0.398 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = sally, debug = TRUE) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- my.path > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/initseq.Rout.save0000644000175100001440000000426613062551372015651 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > n <- 1e5 > rho <- 0.99 > > x <- arima.sim(model = list(ar = rho), n = n) > gamma <- acf(x, lag.max = 1999, type = "covariance", + plot = FALSE)$acf > k <- seq(along = gamma) > Gamma <- gamma[k %% 2 == 1] + gamma[k %% 2 == 0] > k <- min(seq(along = Gamma)[Gamma < 0]) > Gamma <- Gamma[1:k] > Gamma[k] < 0 [1] TRUE > Gamma[k] <- 0 > > out <- .Call(mcmc:::C_initseq, x - mean(x)) > names(out) [1] "gamma0" "Gamma.pos" "Gamma.dec" "Gamma.con" "var.pos" "var.dec" [7] "var.con" > > all.equal(gamma[1], out$gamma0) [1] TRUE > > length(out$Gamma.pos) == length(Gamma) [1] TRUE > all.equal(out$Gamma.pos, Gamma) [1] TRUE > > Gamma.dec <- cummin(Gamma) > all.equal(out$Gamma.dec, Gamma.dec) [1] TRUE > > library(Iso) Iso 0.0-17 > Gamma.con <- Gamma.dec[1] + cumsum(c(0, pava(diff(Gamma.dec)))) > all.equal(out$Gamma.con, Gamma.con) [1] TRUE > > all.equal(0, min(out$Gamma.pos - out$Gamma.dec)) [1] TRUE > max(diff(out$Gamma.dec)) < sqrt(.Machine$double.eps) [1] TRUE > > all.equal(0, min(out$Gamma.dec - out$Gamma.con)) [1] TRUE > min(diff(diff(out$Gamma.con))) > (- sqrt(.Machine$double.eps)) [1] TRUE > > all.equal(2 * sum(out$Gamma.pos) - out$gamma0, out$var.pos) [1] TRUE > all.equal(2 * sum(out$Gamma.dec) - out$gamma0, out$var.dec) [1] TRUE > all.equal(2 * sum(out$Gamma.con) - out$gamma0, out$var.con) [1] TRUE > > rev(out$Gamma.pos)[1] == 0 [1] TRUE > rev(out$Gamma.dec)[1] == 0 [1] TRUE > all.equal(rev(out$Gamma.con)[1], 0) [1] TRUE > > > proc.time() user system elapsed 0.688 0.016 0.695 mcmc/tests/logitnegidx.R0000644000175100001440000000576713045741240015030 0ustar hornikusers # test outfun (negative index vector) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, outfun = - 2) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- my.path[ , out.metro$outfun] k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/logit.Rout.save0000644000175100001440000001140313045741600015276 0ustar hornikusers R version 3.2.1 (2015-06-18) -- "World-Famous Astronaut" Copyright (C) 2015 The R Foundation for Statistical Computing Platform: i686-pc-linux-gnu (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > options(digits = 3) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > summary(out) Call: glm(formula = y ~ x1 + x2, family = binomial()) Deviance Residuals: Min 1Q Median 3Q Max -2.064 -0.821 -0.246 0.840 2.070 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.0599 0.2477 0.24 0.80905 x1 1.3682 0.3844 3.56 0.00037 *** x2 0.4760 0.3135 1.52 0.12886 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 138.469 on 99 degrees of freedom Residual deviance: 99.293 on 97 degrees of freedom AIC: 105.3 Number of Fisher Scoring iterations: 5 > > mlogl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(- sum(log(p[y == 1])) - sum(log(1 - p[y == 0]))) + } > > out.nlm <- nlm(mlogl, coefficients(out), print.level = 2) iteration = 0 Parameter: [1] 0.0599 1.3682 0.4760 Function Value [1] 49.6 Gradient: [1] 8.24e-06 5.50e-06 6.08e-06 Relative gradient close to zero. Current iterate is probably solution. > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.0608 1.4230 0.5263 > var(out.metro$batch) [,1] [,2] [,3] [1,] 0.06755 -0.0108 0.00989 [2,] -0.01080 0.1758 -0.06155 [3,] 0.00989 -0.0615 0.10483 > olbm(out.metro$batch, 25) [,1] [,2] [,3] [1,] 4.54e-04 9.47e-05 -1.92e-05 [2,] 9.47e-05 1.84e-03 -6.45e-04 [3,] -1.92e-05 -6.45e-04 9.09e-04 > > saveseed <- .Random.seed > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE) > > all(out.metro$batch[- out.metro$nbatch, ] == out.metro$current[- 1, ]) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$batch[out.metro$nbatch, ] == out.metro$final) [1] TRUE > > .Random.seed <- saveseed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > > all(my.path == out.metro$batch) [1] TRUE > > > proc.time() user system elapsed 0.532 0.028 0.551 mcmc/tests/morph.Rout.save0000644000175100001440000001125513045741600015312 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > library(mcmc) > isotropic <- mcmc:::isotropic > isotropic.logjacobian <- mcmc:::isotropic.logjacobian > > # make sure morph identity works properly > TestMorphIdentity <- function(m.id) { + ident.func <- function(x) x + if (!all.equal(m.id$transform(1:10), 1:10)) + return(FALSE) + if (!all.equal(m.id$inverse(1:10), 1:10)) + return(FALSE) + x <- seq(-1,1, length.out=15) + if (!all.equal(sapply(x, m.id$lud(function(x) dnorm(x, log=TRUE))), + dnorm(x, log=TRUE))) + return(FALSE) + if (!all.equal(m.id$outfun(ident.func)(x), x)) + return(FALSE) + return(TRUE) + } > > TestMorphIdentity(morph()) [1] TRUE > TestMorphIdentity(morph.identity()) [1] TRUE > > TestMorphIdentityOutfun <- function(m) { + f <- m$outfun(NULL) + x <- 1:20 + if (!identical(x, f(x))) + return(FALSE) + f <- m$outfun(c(6, 8)) + if (!identical(x[c(6, 8)], f(x))) + return(FALSE) + i <- rep(FALSE, 20) + i[c(1, 3, 5)] <- TRUE + f <- m$outfun(i) + if (!identical(x[i], f(x))) + return(FALSE) + return(TRUE) + } > > TestMorphIdentityOutfun(morph()) [1] TRUE > TestMorphIdentityOutfun(morph.identity()) [1] TRUE > > # make sure that morph and morph.identity give back the same things > all.equal(sort(names(morph.identity())), sort(names(morph(b=1)))) [1] TRUE > > # test center parameter, univariate version > zero.func <- function(x) 0 > center <- 2 > x <- seq(-1,1, length.out=15) > morph.center <- morph(center=center) > all.equal(sapply(x, morph.center$transform), x-center) [1] TRUE > all.equal(sapply(x, morph.center$inverse), x+center) [1] TRUE > all.equal(sapply(x, morph.center$lud(function(y) dnorm(y, log=TRUE))), + dnorm(x, log=TRUE, mean=-2)) [1] TRUE > > # test center parameter, multivariate version > center <- 1:4 > x <- rep(0, 4) > morph.center <- morph(center=center) > lud.mult.dnorm <- function(x) prod(dnorm(x, log=TRUE)) > all.equal(morph.center$transform(x), x-center) [1] TRUE > all.equal(morph.center$inverse(x), x+center) [1] TRUE > all.equal(morph.center$lud(lud.mult.dnorm)(x), + lud.mult.dnorm(x - center)) [1] TRUE > # test 'r'. > r <- 1 > morph.r <- morph(r=r) > x <- seq(-1, 1, length.out=20) > all.equal(sapply(x, morph.r$lud(function(x) dnorm(x, log=TRUE))), + dnorm(x, log=TRUE)) [1] TRUE > x <- seq(1.1, 2, length.out=10) > all(sapply(x, morph.r$lud(function(x) dnorm(x, log=TRUE))) + != + dnorm(x, log=TRUE)) [1] TRUE > > TestExponentialEvenPWithRInverse <- function() { + r <- 0.3 + p <- 2.2 + morph.r <- morph(r=r, p=p) + x <- seq(0, r, length.out=20) + all.equal(x, sapply(x, morph.r$inverse)) + } > > TestExponentialEvenPWithRInverse() [1] TRUE > > # make sure morph$lud passes '...' arguments. > mean <- 2 > ident.morph <- morph() > dnorm.morph <- ident.morph$lud(function(x, mean=0) + dnorm(x, mean=mean, log=TRUE)) > all.equal(dnorm.morph(2, mean), dnorm(2, mean=mean, log=TRUE)) [1] TRUE > x <- seq(-3, 3, length.out=20) > m2 <- morph(r=10) > dnorm.morph <- m2$lud(function(x, mean) + dnorm(x, mean=mean, log=TRUE)) > all.equal(sapply(x, function(y) dnorm.morph(y, 2)), + dnorm(x, mean=2, log=TRUE)) [1] TRUE > > # make sure morph$outfun passes '...' arguments. > outfun.orig <- function(x, mean) x + mean > ident.morph <- morph() > mean <- 1 > outfun.morph <- ident.morph$outfun(outfun.orig) > all.equal(outfun.morph(1:10, mean), 1:10+mean) [1] TRUE > > m2 <- morph(r=10) > outfun.morph <- m2$outfun(outfun.orig) > all.equal(sapply(1:10, function(x) outfun.morph(x, mean)), 1:10+mean) [1] TRUE > > ########################################################################### > # test built-in exponential and polynomial transformations. > f <- morph(b=3) > x <- seq(0, 10, length.out=100) > all.equal(x, sapply(sapply(x, f$transform), f$inverse)) [1] TRUE > > f <- morph(p=3) > all.equal(x, sapply(sapply(x, f$transform), f$inverse)) [1] TRUE > > f <- morph(p=3, r=10) > all.equal(-10:10, Vectorize(f$transform)(-10:10)) [1] TRUE > > f <- morph(p=3, b=1) > all.equal(x, sapply(sapply(x, f$transform), f$inverse)) [1] TRUE > mcmc/tests/saveseedmorph.Rout.save0000644000175100001440000000256013045741600017031 0ustar hornikusers R version 2.15.0 (2012-03-30) Copyright (C) 2012 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: i686-pc-linux-gnu (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) > out <- morph.metrop(obj = h, initial = rep(0, 5), nbatch = 100, blen = 17, + nspac = 3, scale = 0.1) > > out1 <- morph.metrop(out) > out2 <- morph.metrop(out1) > out3 <- morph.metrop(out, nbatch = 2 * out$nbatch) > > fred <- rbind(out1$batch, out2$batch) > identical(fred, out3$batch) [1] TRUE > > out <- morph.metrop(out, morph = morph(p = 2.2, r = 0.3)) > > out1 <- morph.metrop(out) > out2 <- morph.metrop(out1) > out3 <- morph.metrop(out, nbatch = 2 * out$nbatch) > > fred <- rbind(out1$batch, out2$batch) > identical(fred, out3$batch) [1] TRUE > > > proc.time() user system elapsed 0.752 0.036 0.767 mcmc/tests/circle.R0000644000175100001440000000375113045741240013743 0ustar hornikusers epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) d <- 5 logh <- function(x) { if (! is.numeric(x)) stop("x not numeric") if (length(x) != d) stop("length(x) != d") fred <- 1 - sum(x^2) if (fred > 0) return(log(fred)) else return(-Inf) } out.metro <- metrop(logh, rep(0, d), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept out.metro <- metrop(out.metro, scale = 0.4) out.metro$accept out.metro <- metrop(out.metro, nbatch = 1e2, debug = TRUE) all(out.metro$batch[- out.metro$nbatch, ] == out.metro$current[- 1, ]) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$batch[out.metro$nbatch, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logh) my.prop.log.green <- apply(out.metro$proposal, 1, logh) all(is.na(out.metro$u) == ((my.prop.log.green == -Inf) | (my.prop.log.green > my.curr.log.green))) foo <- my.prop.log.green - my.curr.log.green blurfle <- foo - out.metro$log.green blurfle[foo == -Inf & out.metro$log.green == -Inf] <- 0 max(blurfle) < epsilon my.accept <- (my.prop.log.green > -Inf) & (is.na(my.u) | my.u < exp(foo)) sum(my.accept) == round(n * out.metro$accept) my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] all(my.path == out.metro$batch) mcmc/tests/morphtoo.R0000644000175100001440000000233713045741600014350 0ustar hornikusers library(mcmc) x <- seq(0, 10, length = 10001) ### sub-exponentially light transformation b <- 0.5 fsub <- morph(b = b) y <- unlist(Map(fsub$inverse, x)) myfsub <- function(x) ifelse(x > 1 / b, exp(b * x) - exp(1) / 3, (x * b)^3 * exp(1) / 6 + x * b * exp(1) / 2) y2 <- myfsub(x) all.equal(y, y2, tolerance = 1e-14) z <- unlist(Map(fsub$transform, y)) all.equal(z, x, tolerance = 1e-14) ### exponentially light transformation r <- 5 p <- 3 fp3 <- morph(r = r) y <- unlist(Map(fp3$inverse, x)) myfp3 <- function(x) ifelse(x < r, x, x + (x - r)^p) y2 <- myfp3(x) all.equal(y, y2, tolerance = 1e-14) z <- unlist(Map(fp3$transform, y)) all.equal(z, x, tolerance = 1e-12) ### both together fboth <- morph(b = b, r = r) y <- unlist(Map(fboth$inverse, x)) y2 <- myfsub(myfp3(x)) all.equal(y, y2, tolerance = 1e-14) z <- unlist(Map(fboth$transform, y)) all.equal(z, x, tolerance = 1e-12) ### exponentially light transformation with p != 3 r <- 5 p <- 2.2 fpo <- morph(r = r, p = p) y <- unlist(Map(fpo$inverse, x)) myfpo <- function(x) ifelse(x < r, x, x + (x - r)^p) y2 <- myfpo(x) all.equal(y, y2, tolerance = 1e-14) z <- unlist(Map(fpo$transform, y)) all.equal(z, x, tolerance = 1e-14) mcmc/tests/logitvec.R0000644000175100001440000000607513045741240014320 0ustar hornikusers # test vector (diag(foo)) scaling epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) sally <- apply(out.metro$batch, 2, sd) out.metro <- metrop(out.metro, scale = sally) out.metro$accept out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = sally, debug = TRUE) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- my.path k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/temp-ser.R0000644000175100001440000002157713045741600014244 0ustar hornikusers library(mcmc) set.seed(42) data(foo) attach(foo) out <- glm(y ~ x1 + x2 + x3, family = binomial, x = TRUE) summary(out) modmat <- out$x models <- cbind(rep(0:1, each = 4), rep(rep(0:1, times = 2), each = 2), rep(0:1, times = 4)) exes <- paste("x", 1:3, sep = "") models[nrow(models), ] beta.initial <- c(nrow(models), out$coefficients) neighbors <- matrix(FALSE, nrow(models), nrow(models)) for (i in 1:nrow(neighbors)) { for (j in 1:ncol(neighbors)) { foo <- models[i, ] bar <- models[j, ] if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE } } neighbors ludfun <- function(state, log.pseudo.prior, ...) { stopifnot(is.numeric(state)) stopifnot(length(state) == ncol(models) + 2) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= nrow(models)) stopifnot(is.numeric(log.pseudo.prior)) stopifnot(length(log.pseudo.prior) == nrow(models)) beta <- state[-1] inies <- c(TRUE, as.logical(models[icomp, ])) beta.logl <- beta beta.logl[! inies] <- 0 eta <- as.numeric(modmat %*% beta.logl) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) val <- logl - sum(beta^2) / 2 + log.pseudo.prior[icomp] return(val) } qux <- c(25.01, 5.875, 9.028, 0.6959, 11.73, 2.367, 5.864, 0.0) out <- temper(ludfun, initial = beta.initial, neighbors = neighbors, nbatch = 25, blen = 20, nspac = 5, scale = 0.56789, debug = TRUE, log.pseudo.prior = qux) names(out) apply(out$ibatch, 2, mean) ### check decision about within-component or jump/swap identical(out$unif.which < 0.5, out$which) identical(out$which, out$proposal[ , 1] == out$state[ , 1]) ### check hastings ratio calculated correctly foo <- apply(out$state, 1, ludfun, log.pseudo.prior = qux) bar <- apply(out$proposal, 1, ludfun, log.pseudo.prior = qux) all.equal(bar - foo, out$log.hastings) ### check hastings rejection decided correctly identical(out$log.hastings >= 0, is.na(out$unif.hastings)) all(out$log.hastings < 0 | out$acceptd) identical(out$acceptd, out$log.hastings >= 0 | out$unif.hastings < exp(out$log.hastings)) ### check acceptance carried out or not (according to decision) correctly before <- out$state after <- before after[- dim(after)[1], ] <- before[-1, ] after[dim(after)[1], ] <- out$final my.after <- before my.after[out$acceptd, ] <- out$proposal[out$acceptd, ] identical(after, my.after) ### check within-component proposal my.coproposal.within <- out$state[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + out$scale * my.z all.equal(proposal.within, my.proposal.within) ### check swap proposal coproposal.swap <- out$state[! out$which, ] proposal.swap <- out$proposal[! out$which, ] unif.choose.swap <- out$unif.choose[! out$which] my.i <- coproposal.swap[ , 1] nneighbors <- apply(out$neighbors, 1, sum) my.nneighbors <- nneighbors[my.i] my.k <- floor(my.nneighbors * unif.choose.swap) + 1 my.j <- my.k foo <- seq(1, ncol(out$neighbors)) for (i in seq(along = my.j)) { my.j[i] <- (foo[out$neighbors[my.i[i], ]])[my.k[i]] } identical(coproposal.swap[ , 1], my.i) identical(proposal.swap[ , 1], my.j) ### check standard normal and uniform random numbers are as purported save.Random.seed <- .Random.seed .Random.seed <- out$initial.seed nx <- length(out$initial) - 1 niter <- out$nbatch * out$blen * out$nspac my.norm <- matrix(NA, nrow = nrow(out$norm), ncol = ncol(out$norm)) my.unif.which <- rep(NA, niter) my.unif.hastings <- rep(NA, niter) my.unif.choose <- rep(NA, niter) for (iiter in 1:niter) { my.unif.which[iiter] <- runif(1) if (out$which[iiter]) { my.norm[iiter, ] <- rnorm(nx) if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) } else { my.unif.choose[iiter] <- runif(1) if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) } } identical(my.norm, out$norm) identical(my.unif.which, out$unif.which) identical(my.unif.hastings, out$unif.hastings) identical(my.unif.choose, out$unif.choose) .Random.seed <- save.Random.seed ### check batch means my.xstate <- after[ , -1] foo <- my.xstate[seq(1, niter) %% out$nspac == 0, ] foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) foo <- apply(foo, c(2, 3), mean) all.equal(foo, out$batch) ### check ibatch means my.istate <- after[ , 1] my.istate.matrix <- matrix(0, length(my.istate), nrow(models)) for (i in 1:nrow(my.istate.matrix)) my.istate.matrix[i, my.istate[i]] <- 1 foo <- my.istate.matrix[seq(1, niter) %% out$nspac == 0, ] foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) foo <- apply(foo, c(2, 3), mean) all.equal(foo, out$ibatch) ### check acceptance rates nmodel <- nrow(out$neighbors) accept.within <- out$acceptd[out$which] my.i.within <- out$state[out$which, 1] my.i.within.accept <- my.i.within[accept.within] my.acceptx.numer <- tabulate(my.i.within.accept, nbins = nmodel) my.acceptx.denom <- tabulate(my.i.within, nbins = nmodel) my.acceptx <- my.acceptx.numer / my.acceptx.denom identical(my.acceptx, out$acceptx) accept.swap <- out$acceptd[! out$which] my.i.swap <- out$state[! out$which, 1] my.j.swap <- out$proposal[! out$which, 1] my.accepti <- matrix(NA, nmodel, nmodel) for (i in 1:nmodel) { for (j in 1:nmodel) { if (out$neighbors[i, j]) { my.accepti[i, j] <- mean(accept.swap[my.i.swap == i & my.j.swap == j]) } } } identical(my.accepti, out$accepti) ### check scale vector nx <- ncol(models) + 1 newscale <- rnorm(nx, 0.5, 0.1) out <- temper(out, scale = newscale, log.pseudo.prior = qux) my.coproposal.within <- out$state[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + sweep(my.z, 2, out$scale, "*") all.equal(proposal.within, my.proposal.within) ### check scale matrix matscale <- matrix(rnorm(nx * nx, 0.0, 0.1), nx, nx) diag(matscale) <- 0.56789 out <- temper(out, scale = matscale, log.pseudo.prior = qux) my.coproposal.within <- out$state[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + my.z %*% t(out$scale) all.equal(proposal.within, my.proposal.within) ### check scale list lisztscale <- list(0.56789, newscale, matscale, matscale, newscale, 0.98765, 0.98765, newscale) out <- temper(out, scale = lisztscale, log.pseudo.prior = qux) my.coproposal.within <- out$state[out$which, ] proposal.within <- out$proposal[out$which, ] my.z <- out$norm[out$which, ] my.proposal.within <- my.coproposal.within for (iiter in 1:nrow(my.z)) { my.i <- my.coproposal.within[iiter, 1] my.scale <- out$scale[[my.i]] if (is.matrix(my.scale)) { my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + my.z[iiter, , drop = FALSE] %*% t(my.scale) } else { my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + my.z[iiter, ] * my.scale } } all.equal(proposal.within, my.proposal.within) ### check outfun outfun <- function(state, icomp) { stopifnot(is.matrix(state)) stopifnot(is.numeric(state)) nx <- ncol(initial) ncomp <- nrow(initial) stopifnot(ncol(state) == nx) stopifnot(nrow(state) == ncomp) stopifnot(1 <= icomp & icomp <= ncomp) foo <- state[icomp, ] bar <- foo^2 return(c(foo, bar)) } ncomp <- nrow(models) nx <- length(beta.initial) - 1 outfun <- function(state, icomp, ...) { stopifnot(is.numeric(state)) stopifnot(length(state) == nx + 1) istate <- state[1] stopifnot(istate == as.integer(istate)) stopifnot(1 <= istate && istate <= ncomp) stopifnot(1 <= icomp && icomp <= ncomp) if (istate == icomp) { foo <- state[-1] } else { foo <- rep(0, nx) } bar <- foo^2 return(c(foo, bar)) } out <- temper(ludfun, initial = out$final, neighbors = neighbors, nbatch = 25, blen = 20, nspac = 5, scale = 0.56789, debug = TRUE, outfun = outfun, log.pseudo.prior = qux, icomp = 4) before <- out$state after <- before after[- dim(after)[1], ] <- before[-1, ] after[dim(after)[1], ] <- out$final outies <- apply(after, 1, outfun, icomp = 4) outies <- t(outies) foo <- outies[seq(1, niter) %% out$nspac == 0, ] foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) foo <- apply(foo, c(2, 3), mean) all.equal(foo, out$batch) mcmc/tests/logitmat.R0000644000175100001440000000641613045741240014323 0ustar hornikusers # test matrix scaling epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) fred <- var(out.metro$batch) sally <- t(chol(fred)) max(abs(fred - sally %*% t(sally))) < epsilon out.metro <- metrop(out.metro, scale = sally) out.metro$accept out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = sally, debug = TRUE) names(out.metro) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) my.z <- matrix(NA, n, d) ska <- out.metro$scale for (i in 1:n) { zed <- rnorm(d) my.proposal[i, ] <- out.metro$current[i, ] + ska %*% zed if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } my.z[i, ] <- zed } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) identical(out.metro$z, my.z) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } identical(my.accept, out.metro$debug.accept) my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] fred <- my.path k <- ncol(fred) foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/saveseed.Rout.save0000644000175100001440000000205413045741600015761 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) > out <- metrop(h, initial = rep(0, 5), nbatch = 100, blen = 17, nspac = 3, + scale = 0.1) > > save.seed <- .Random.seed > > out1 <- metrop(out) > out2 <- metrop(out1) > out3 <- metrop(out, nbatch = 2 * out$nbatch) > > fred <- rbind(out1$batch, out2$batch) > identical(fred, out3$batch) [1] TRUE > > mcmc/tests/temp-ser-witch.Rout.save0000644000175100001440000001170013074144363017035 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > d <- 3 > witch.which <- 1 - (1 / 2)^(1 / d) * (1 / 4)^(seq(0, 5) / d) > witch.which [1] 0.2062995 0.5000000 0.6850197 0.8015749 0.8750000 0.9212549 > > ncomp <- length(witch.which) > > neighbors <- matrix(FALSE, ncomp, ncomp) > neighbors[row(neighbors) == col(neighbors) + 1] <- TRUE > neighbors[row(neighbors) == col(neighbors) - 1] <- TRUE > neighbors[row(neighbors) == col(neighbors) + 2] <- TRUE > neighbors[row(neighbors) == col(neighbors) - 2] <- TRUE > > ludfun <- function(state) { + stopifnot(is.numeric(state)) + stopifnot(length(state) == d + 1) + icomp <- state[1] + stopifnot(icomp == as.integer(icomp)) + stopifnot(1 <= icomp && icomp <= ncomp) + theta <- state[-1] + if (any(abs(theta) > 1.0)) return(-Inf) + bnd <- witch.which[icomp] + if(bnd >= 1.0) + stop(sprintf("witch.which[%d] >= 1.0", icomp)) + if(bnd <= 0.0) + stop(sprintf("witch.which[%d] <= 0.0", icomp)) + if (all(abs(theta) > bnd)) + return(- (d + 1) * log(2) - d * log(1 - bnd)) + return(- (d + 1) * log(2) - log1p(- (1 - bnd)^d)) + } > > initial <- c(1, rep(0, d)) > > out <- temper(ludfun, initial = initial, neighbors = neighbors, + nbatch = 50, blen = 13, nspac = 7, scale = 0.3456789) > > names(out) [1] "lud" "neighbors" "nbatch" "blen" "nspac" [6] "scale" "outfun" "debug" "parallel" "initial.seed" [11] "final.seed" "time" "batch" "acceptx" "accepti" [16] "initial" "final" "ibatch" > > out$acceptx [1] 0.6388889 0.4385246 0.3631714 0.4885246 0.4709677 0.4735516 > > out$accepti [,1] [,2] [,3] [,4] [,5] [,6] [1,] NA 0.5071770 0.2727273 NA NA NA [2,] 0.7070064 NA 0.4355828 0.4186047 NA NA [3,] 0.5816327 0.8039216 NA 0.5888889 0.5662651 NA [4,] NA 0.7415730 0.8571429 NA 0.7857143 0.6626506 [5,] NA NA 0.5204082 0.6516854 NA 0.8378378 [6,] NA NA NA 0.3515152 0.5056818 NA > > colMeans(out$ibatch) [1] 0.1830769 0.2153846 0.1630769 0.1369231 0.1353846 0.1661538 > > ### check that have prob 1 / 2 for corners > > outfun <- function(state) { + stopifnot(is.numeric(state)) + icomp <- state[1] + stopifnot(icomp == as.integer(icomp)) + stopifnot(1 <= icomp && icomp <= length(witch.which)) + theta <- state[-1] + foo <- all(abs(theta) > witch.which[icomp]) + bar <- rep(0, length(witch.which)) + baz <- rep(0, length(witch.which)) + bar[icomp] <- as.numeric(foo) + baz[icomp] <- 1 + return(c(bar, baz)) + } > > out <- temper(out, blen = 103, outfun = outfun, debug = TRUE) > > eta.batch <- out$batch[ , seq(1, ncomp)] > noo.batch <- out$batch[ , seq(ncomp + 1, ncomp + ncomp)] > eta <- colMeans(eta.batch) > noo <- colMeans(noo.batch) > mu <- eta / noo > eta [1] 0.06660194 0.06388350 0.05766990 0.06563107 0.10368932 0.22912621 > noo [1] 0.1365049 0.1258252 0.1293204 0.1370874 0.1716505 0.2996117 > mu [1] 0.4879090 0.5077160 0.4459459 0.4787535 0.6040724 0.7647440 > > eta.batch.rel <- sweep(eta.batch, 2, eta, "/") > noo.batch.rel <- sweep(noo.batch, 2, noo, "/") > mu.batch.rel <- eta.batch.rel - noo.batch.rel > > mu.mcse.rel <- apply(mu.batch.rel, 2, sd) / sqrt(out$nbatch) > mu.mcse.rel [1] 0.05010927 0.07897321 0.09678339 0.12636113 0.11261781 0.07082685 > > foo <- cbind(mu, mu * mu.mcse.rel) > colnames(foo) <- c("means", "MCSE") > foo means MCSE [1,] 0.4879090 0.02444876 [2,] 0.5077160 0.04009596 [3,] 0.4459459 0.04316016 [4,] 0.4787535 0.06049584 [5,] 0.6040724 0.06802931 [6,] 0.7647440 0.05416441 > > ### check decision about within-component or jump/swap > > identical(out$unif.which < 0.5, out$which) [1] TRUE > > identical(out$which, out$proposal[ , 1] == out$state[ , 1]) [1] TRUE > > ### check hastings ratio calculated correctly > > n <- apply(neighbors, 1, sum) > i <- out$state[ , 1] > istar <- out$proposal[ , 1] > foo <- apply(out$state, 1, ludfun) > bar <- apply(out$proposal, 1, ludfun) > my.log.hastings <- bar - foo - log(n[istar]) + log(n[i]) > all.equal(my.log.hastings, out$log.hastings) [1] TRUE > > > proc.time() user system elapsed 2.740 0.028 2.762 mcmc/tests/logitbat.R0000644000175100001440000000565713045741240014316 0ustar hornikusers # test batching (blen) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, blen = 5) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] foom <- array(as.vector(t(my.path)), c(d, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/logitsubbat.R0000644000175100001440000000572713045741240015026 0ustar hornikusers # test batching (blen) and spacing (nspac) together epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, blen = 5, nspac = 3) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] foom <- array(as.vector(t(my.path)), c(d, out.metro$blen, out.metro$nbatch)) boom <- t(apply(foom, c(1, 3), mean)) all(dim(boom) == dim(out.metro$batch)) max(abs(boom - out.metro$batch)) < epsilon mcmc/tests/morph.metrop.Rout.save0000644000175100001440000000353413045741600016620 0ustar hornikusers R version 3.2.1 (2015-06-18) -- "World-Famous Astronaut" Copyright (C) 2015 The R Foundation for Statistical Computing Platform: i686-pc-linux-gnu (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > library(mcmc) > > .morph.unmorph <- mcmc:::.morph.unmorph > > ########################################################################### > # basic functionality check, can morph.metro run? Can we change the > # transformation? > set.seed(42) > obj <- morph.metrop(function(x) dt(x, df=3, log=TRUE), + 100, 100, morph=morph(b=3)) > obj <- morph.metrop(obj, morph=morph(b=1)) > > obj <- morph.metrop(function(x) prod(dt(x, df=3, log=TRUE)), + rep(100, 3), 100, morph=morph(p=3, b=1)) > obj <- morph.metrop(obj, morph=morph(r=1, p=3, b=1)) > > all.equal(class(obj), c("mcmc", "morph.metropolis")) [1] TRUE > > ########################################################################### > # check .morph.unmorph > obj <- list(final=10) > outfun <- function(x) x > m <- morph(p=3) > obj <- .morph.unmorph(obj, m, outfun) > all.equal(class(obj), c("mcmc", "morph.metropolis")) [1] TRUE > all.equal(sort(names(obj)), + sort(c("final", "morph", "morph.final", "outfun"))) [1] TRUE > all.equal(c(obj$final, obj$morph.final), c(m$inverse(10), 10)) [1] TRUE > all.equal(obj$outfun, outfun) [1] TRUE > all.equal(obj$morph, m) [1] TRUE > > proc.time() user system elapsed 0.384 0.024 0.401 mcmc/tests/temp-par.Rout.save0000644000175100001440000002752413074144313015720 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > data(foo) > attach(foo) > > out <- glm(y ~ x1 + x2 + x3, family = binomial, x = TRUE) > summary(out) Call: glm(formula = y ~ x1 + x2 + x3, family = binomial, x = TRUE) Deviance Residuals: Min 1Q Median 3Q Max -2.0371 -0.6337 0.2394 0.6685 1.9599 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.5772 0.2766 2.087 0.036930 * x1 0.3362 0.4256 0.790 0.429672 x2 0.8475 0.4701 1.803 0.071394 . x3 1.5143 0.4426 3.422 0.000622 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 134.602 on 99 degrees of freedom Residual deviance: 86.439 on 96 degrees of freedom AIC: 94.439 Number of Fisher Scoring iterations: 5 > > modmat <- out$x > > models <- cbind(rep(0:1, each = 4), rep(rep(0:1, times = 2), each = 2), + rep(0:1, times = 4)) > > exes <- paste("x", 1:3, sep = "") > betas <- NULL > for (i in 1:nrow(models)) { + inies <- as.logical(models[i, ]) + foo <- exes[inies] + bar <- paste("y ~", paste(foo, collapse = " + ")) + if (! any(inies)) bar <- paste(bar, "1") + baz <- glm(as.formula(bar), family = binomial) + beta <- rep(0, 4) + beta[c(TRUE, inies)] <- baz$coef + betas <- rbind(betas, beta) + } > > neighbors <- matrix(FALSE, nrow(models), nrow(models)) > for (i in 1:nrow(neighbors)) { + for (j in 1:ncol(neighbors)) { + foo <- models[i, ] + bar <- models[j, ] + if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE + } + } > > ludfun <- function(state, ...) { + stopifnot(is.numeric(state)) + stopifnot(length(state) == ncol(models) + 2) + stopifnot(length(state) == ncol(models) + 2) + icomp <- state[1] + stopifnot(icomp == as.integer(icomp)) + stopifnot(1 <= icomp && icomp <= nrow(models)) + beta <- state[-1] + inies <- c(TRUE, as.logical(models[icomp, ])) + beta.logl <- beta + beta.logl[! inies] <- 0 + eta <- as.numeric(modmat %*% beta.logl) + logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) + logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) + logl <- sum(logp[y == 1]) + sum(logq[y == 0]) + val <- logl - sum(beta^2) / 2 + return(val) + } > > ludval <- NULL > for (i in 1:nrow(models)) ludval <- c(ludval, ludfun(c(i, betas[i, ]))) > all(is.finite(ludval)) [1] TRUE > > > out <- temper(ludfun, initial = betas, neighbors = neighbors, nbatch = 20, + blen = 10, nspac = 5, scale = 0.56789, parallel = TRUE, debug = TRUE) > > names(out) [1] "lud" "neighbors" "nbatch" "blen" [5] "nspac" "scale" "outfun" "debug" [9] "parallel" "initial.seed" "final.seed" "time" [13] "batch" "acceptx" "accepti" "initial" [17] "final" "which" "unif.which" "state" [21] "log.hastings" "unif.hastings" "proposal" "acceptd" [25] "norm" "unif.choose" "coproposal" > > ### check decision about within-component or jump/swap > > identical(out$unif.which < 0.5, out$which) [1] TRUE > > identical(out$which, out$proposal[ , 1] == out$coproposal[ , 1]) [1] TRUE > > ### check proposal and coproposal are actually current state or part thereof > > prop <- out$proposal > coprop <- out$coproposal > prop.i <- prop[ , 1] > coprop.i <- coprop[ , 1] > alt.prop <- prop > alt.coprop <- coprop > for (i in 1:nrow(prop)) { + alt.prop[i, ] <- c(prop.i[i], out$state[i, prop.i[i], ]) + alt.coprop[i, ] <- c(coprop.i[i], out$state[i, coprop.i[i], ]) + } > identical(coprop, alt.coprop) [1] TRUE > identical(prop[! out$which, ], alt.prop[! out$which, ]) [1] TRUE > identical(prop[out$which, 1], alt.prop[out$which, 1]) [1] TRUE > > ### check hastings ratio calculated correctly > > foo <- apply(prop, 1, ludfun) > fooco <- apply(coprop, 1, ludfun) > prop[ , 1] <- out$coproposal[ , 1] > coprop[ , 1] <- out$proposal[ , 1] > foo.swap <- apply(prop, 1, ludfun) > fooco.swap <- apply(coprop, 1, ludfun) > log.haste <- ifelse(out$which, foo - fooco, + foo.swap + fooco.swap - foo - fooco) > all.equal(log.haste, out$log.hastings) [1] TRUE > > ### check hastings rejection decided correctly > > identical(out$log.hastings >= 0, is.na(out$unif.hastings)) [1] TRUE > all(out$log.hastings < 0 | out$acceptd) [1] TRUE > identical(out$acceptd, + out$log.hastings >= 0 | out$unif.hastings < exp(out$log.hastings)) [1] TRUE > > ### check acceptance carried out or not (according to decision) correctly > > before <- out$state > after <- before > after[- dim(after)[1], , ] <- before[-1, , ] > after[dim(after)[1], , ] <- out$final > my.after <- before > for (i in 1:length(out$acceptd)) { + if (out$acceptd[i]) { + if (out$which[i]) { + j <- out$proposal[i, 1] + my.after[i, j, ] <- out$proposal[i, -1] + } else { + j <- out$proposal[i, 1] + k <- out$coproposal[i, 1] + my.after[i, j, ] <- out$coproposal[i, -1] + my.after[i, k, ] <- out$proposal[i, -1] + } + } + } > identical(after, my.after) [1] TRUE > > ### check within-component proposal > > my.coproposal.within <- out$coproposal[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + out$scale * my.z > all.equal(proposal.within, my.proposal.within) [1] TRUE > > my.unif.choose <- out$unif.choose[out$which, 1] > my.i <- floor(nrow(models) * my.unif.choose) + 1 > all(1 <= my.i & my.i <= nrow(models)) [1] TRUE > identical(my.i, my.coproposal.within[ , 1]) [1] TRUE > > ### check swap proposal > > coproposal.swap <- out$coproposal[! out$which, ] > proposal.swap <- out$proposal[! out$which, ] > unif.choose.swap <- out$unif.choose[! out$which, ] > my.i <- floor(nrow(models) * unif.choose.swap[ , 1]) + 1 > nneighbors <- apply(out$neighbors, 1, sum) > my.nneighbors <- nneighbors[my.i] > my.k <- floor(my.nneighbors * unif.choose.swap[ , 2]) + 1 > my.j <- my.k > foo <- seq(1, ncol(out$neighbors)) > for (i in seq(along = my.j)) { + my.j[i] <- (foo[out$neighbors[my.i[i], ]])[my.k[i]] + } > identical(coproposal.swap[ , 1], my.i) [1] TRUE > identical(proposal.swap[ , 1], my.j) [1] TRUE > > ### check standard normal and uniform random numbers are as purported > > save.Random.seed <- .Random.seed > .Random.seed <- out$initial.seed > > nx <- ncol(out$initial) > niter <- out$nbatch * out$blen * out$nspac > my.norm <- matrix(NA, nrow = nrow(out$norm), ncol = ncol(out$norm)) > my.unif.which <- rep(NA, niter) > my.unif.hastings <- rep(NA, niter) > my.unif.choose <- matrix(NA, niter, 2) > for (iiter in 1:niter) { + my.unif.which[iiter] <- runif(1) + if (out$which[iiter]) { + my.unif.choose[iiter, 1] <- runif(1) + my.norm[iiter, ] <- rnorm(nx) + if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) + } else { + my.unif.choose[iiter, ] <- runif(2) + if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) + } + } > identical(my.norm, out$norm) [1] TRUE > identical(my.unif.which, out$unif.which) [1] TRUE > identical(my.unif.hastings, out$unif.hastings) [1] TRUE > identical(my.unif.choose, out$unif.choose) [1] TRUE > > .Random.seed <- save.Random.seed > > ### check batch means > > foo <- after[seq(1, niter) %% out$nspac == 0, , ] > foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2:3])) > foo <- apply(foo, c(2, 3, 4), mean) > all.equal(foo, out$batch) [1] TRUE > > ### check acceptance rates > > accept.within <- out$acceptd[out$which] > my.i.within <- out$coproposal[out$which, 1] > my.acceptx <- as.vector(sapply(split(accept.within, my.i.within), mean)) > identical(my.acceptx, out$acceptx) [1] TRUE > > accept.swap <- out$acceptd[! out$which] > my.i.swap <- out$coproposal[! out$which, 1] > my.j.swap <- out$proposal[! out$which, 1] > nmodel <- nrow(out$neighbors) > my.accepti <- matrix(NA, nmodel, nmodel) > for (i in 1:nmodel) { + for (j in 1:nmodel) { + if (out$neighbors[i, j]) { + my.accepti[i, j] <- + mean(accept.swap[my.i.swap == i & my.j.swap == j]) + } + } + } > identical(my.accepti, out$accepti) [1] TRUE > > ### check scale vector > > nx <- ncol(models) + 1 > newscale <- rnorm(nx, 0.5, 0.1) > > out <- temper(out, scale = newscale) > > my.coproposal.within <- out$coproposal[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + + sweep(my.z, 2, out$scale, "*") > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check scale matrix > > matscale <- matrix(rnorm(nx * nx, 0.0, 0.1), nx, nx) > diag(matscale) <- 0.56789 > > out <- temper(out, scale = matscale) > > my.coproposal.within <- out$coproposal[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + + my.z %*% t(out$scale) > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check scale list > > lisztscale <- list(0.56789, newscale, matscale, matscale, newscale, + 0.98765, 0.98765, newscale) > > out <- temper(out, scale = lisztscale) > > my.coproposal.within <- out$coproposal[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > for (iiter in 1:nrow(my.z)) { + my.i <- my.coproposal.within[iiter, 1] + my.scale <- out$scale[[my.i]] + if (is.matrix(my.scale)) { + my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + + my.z[iiter, , drop = FALSE] %*% t(my.scale) + } else { + my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + + my.z[iiter, ] * my.scale + } + } > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check outfun > > outfun <- function(state, icomp, ...) { + stopifnot(is.matrix(state)) + stopifnot(is.numeric(state)) + nx <- ncol(betas) + ncomp <- nrow(betas) + stopifnot(ncol(state) == nx) + stopifnot(nrow(state) == ncomp) + stopifnot(1 <= icomp && icomp <= ncomp) + foo <- state[icomp, ] + bar <- foo^2 + return(c(foo, bar)) + } > > out <- temper(out, outfun = outfun, icomp = 4) > > before <- out$state > after <- before > after[- dim(after)[1], , ] <- before[-1, , ] > after[dim(after)[1], , ] <- out$final > outies <- apply(after, 1, outfun, icomp = 4) > outies <- t(outies) > > foo <- outies[seq(1, niter) %% out$nspac == 0, ] > foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) > foo <- apply(foo, c(2, 3), mean) > all.equal(foo, out$batch) [1] TRUE > > > proc.time() user system elapsed 1.376 0.008 1.376 mcmc/tests/saveseedmorph.R0000644000175100001440000000116613045741600015345 0ustar hornikusers library(mcmc) set.seed(42) h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) out <- morph.metrop(obj = h, initial = rep(0, 5), nbatch = 100, blen = 17, nspac = 3, scale = 0.1) out1 <- morph.metrop(out) out2 <- morph.metrop(out1) out3 <- morph.metrop(out, nbatch = 2 * out$nbatch) fred <- rbind(out1$batch, out2$batch) identical(fred, out3$batch) out <- morph.metrop(out, morph = morph(p = 2.2, r = 0.3)) out1 <- morph.metrop(out) out2 <- morph.metrop(out1) out3 <- morph.metrop(out, nbatch = 2 * out$nbatch) fred <- rbind(out1$batch, out2$batch) identical(fred, out3$batch) mcmc/tests/logitmat.Rout.save0000644000175100001440000001133513062634616016013 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test matrix scaling > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > fred <- var(out.metro$batch) > sally <- t(chol(fred)) > max(abs(fred - sally %*% t(sally))) < epsilon [1] TRUE > > out.metro <- metrop(out.metro, scale = sally) > out.metro$accept [1] 0.451 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = sally, debug = TRUE) > names(out.metro) [1] "accept" "batch" "initial" "final" "accept.batch" [6] "current" "proposal" "log.green" "u" "z" [11] "debug.accept" "initial.seed" "final.seed" "time" "lud" [16] "nbatch" "blen" "nspac" "scale" "debug" > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > my.z <- matrix(NA, n, d) > ska <- out.metro$scale > for (i in 1:n) { + zed <- rnorm(d) + my.proposal[i, ] <- out.metro$current[i, ] + ska %*% zed + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + my.z[i, ] <- zed + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > identical(out.metro$z, my.z) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > identical(my.accept, out.metro$debug.accept) [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- my.path > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > > proc.time() user system elapsed 0.340 0.028 0.359 mcmc/tests/temp-ser-witch.R0000644000175100001440000000526713045741240015356 0ustar hornikusers library(mcmc) set.seed(42) d <- 3 witch.which <- 1 - (1 / 2)^(1 / d) * (1 / 4)^(seq(0, 5) / d) witch.which ncomp <- length(witch.which) neighbors <- matrix(FALSE, ncomp, ncomp) neighbors[row(neighbors) == col(neighbors) + 1] <- TRUE neighbors[row(neighbors) == col(neighbors) - 1] <- TRUE neighbors[row(neighbors) == col(neighbors) + 2] <- TRUE neighbors[row(neighbors) == col(neighbors) - 2] <- TRUE ludfun <- function(state) { stopifnot(is.numeric(state)) stopifnot(length(state) == d + 1) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= ncomp) theta <- state[-1] if (any(abs(theta) > 1.0)) return(-Inf) bnd <- witch.which[icomp] if(bnd >= 1.0) stop(sprintf("witch.which[%d] >= 1.0", icomp)) if(bnd <= 0.0) stop(sprintf("witch.which[%d] <= 0.0", icomp)) if (all(abs(theta) > bnd)) return(- (d + 1) * log(2) - d * log(1 - bnd)) return(- (d + 1) * log(2) - log1p(- (1 - bnd)^d)) } initial <- c(1, rep(0, d)) out <- temper(ludfun, initial = initial, neighbors = neighbors, nbatch = 50, blen = 13, nspac = 7, scale = 0.3456789) names(out) out$acceptx out$accepti colMeans(out$ibatch) ### check that have prob 1 / 2 for corners outfun <- function(state) { stopifnot(is.numeric(state)) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= length(witch.which)) theta <- state[-1] foo <- all(abs(theta) > witch.which[icomp]) bar <- rep(0, length(witch.which)) baz <- rep(0, length(witch.which)) bar[icomp] <- as.numeric(foo) baz[icomp] <- 1 return(c(bar, baz)) } out <- temper(out, blen = 103, outfun = outfun, debug = TRUE) eta.batch <- out$batch[ , seq(1, ncomp)] noo.batch <- out$batch[ , seq(ncomp + 1, ncomp + ncomp)] eta <- colMeans(eta.batch) noo <- colMeans(noo.batch) mu <- eta / noo eta noo mu eta.batch.rel <- sweep(eta.batch, 2, eta, "/") noo.batch.rel <- sweep(noo.batch, 2, noo, "/") mu.batch.rel <- eta.batch.rel - noo.batch.rel mu.mcse.rel <- apply(mu.batch.rel, 2, sd) / sqrt(out$nbatch) mu.mcse.rel foo <- cbind(mu, mu * mu.mcse.rel) colnames(foo) <- c("means", "MCSE") foo ### check decision about within-component or jump/swap identical(out$unif.which < 0.5, out$which) identical(out$which, out$proposal[ , 1] == out$state[ , 1]) ### check hastings ratio calculated correctly n <- apply(neighbors, 1, sum) i <- out$state[ , 1] istar <- out$proposal[ , 1] foo <- apply(out$state, 1, ludfun) bar <- apply(out$proposal, 1, ludfun) my.log.hastings <- bar - foo - log(n[istar]) + log(n[i]) all.equal(my.log.hastings, out$log.hastings) mcmc/tests/morph.R0000644000175100001440000000717113045741600013627 0ustar hornikuserslibrary(mcmc) isotropic <- mcmc:::isotropic isotropic.logjacobian <- mcmc:::isotropic.logjacobian # make sure morph identity works properly TestMorphIdentity <- function(m.id) { ident.func <- function(x) x if (!all.equal(m.id$transform(1:10), 1:10)) return(FALSE) if (!all.equal(m.id$inverse(1:10), 1:10)) return(FALSE) x <- seq(-1,1, length.out=15) if (!all.equal(sapply(x, m.id$lud(function(x) dnorm(x, log=TRUE))), dnorm(x, log=TRUE))) return(FALSE) if (!all.equal(m.id$outfun(ident.func)(x), x)) return(FALSE) return(TRUE) } TestMorphIdentity(morph()) TestMorphIdentity(morph.identity()) TestMorphIdentityOutfun <- function(m) { f <- m$outfun(NULL) x <- 1:20 if (!identical(x, f(x))) return(FALSE) f <- m$outfun(c(6, 8)) if (!identical(x[c(6, 8)], f(x))) return(FALSE) i <- rep(FALSE, 20) i[c(1, 3, 5)] <- TRUE f <- m$outfun(i) if (!identical(x[i], f(x))) return(FALSE) return(TRUE) } TestMorphIdentityOutfun(morph()) TestMorphIdentityOutfun(morph.identity()) # make sure that morph and morph.identity give back the same things all.equal(sort(names(morph.identity())), sort(names(morph(b=1)))) # test center parameter, univariate version zero.func <- function(x) 0 center <- 2 x <- seq(-1,1, length.out=15) morph.center <- morph(center=center) all.equal(sapply(x, morph.center$transform), x-center) all.equal(sapply(x, morph.center$inverse), x+center) all.equal(sapply(x, morph.center$lud(function(y) dnorm(y, log=TRUE))), dnorm(x, log=TRUE, mean=-2)) # test center parameter, multivariate version center <- 1:4 x <- rep(0, 4) morph.center <- morph(center=center) lud.mult.dnorm <- function(x) prod(dnorm(x, log=TRUE)) all.equal(morph.center$transform(x), x-center) all.equal(morph.center$inverse(x), x+center) all.equal(morph.center$lud(lud.mult.dnorm)(x), lud.mult.dnorm(x - center)) # test 'r'. r <- 1 morph.r <- morph(r=r) x <- seq(-1, 1, length.out=20) all.equal(sapply(x, morph.r$lud(function(x) dnorm(x, log=TRUE))), dnorm(x, log=TRUE)) x <- seq(1.1, 2, length.out=10) all(sapply(x, morph.r$lud(function(x) dnorm(x, log=TRUE))) != dnorm(x, log=TRUE)) TestExponentialEvenPWithRInverse <- function() { r <- 0.3 p <- 2.2 morph.r <- morph(r=r, p=p) x <- seq(0, r, length.out=20) all.equal(x, sapply(x, morph.r$inverse)) } TestExponentialEvenPWithRInverse() # make sure morph$lud passes '...' arguments. mean <- 2 ident.morph <- morph() dnorm.morph <- ident.morph$lud(function(x, mean=0) dnorm(x, mean=mean, log=TRUE)) all.equal(dnorm.morph(2, mean), dnorm(2, mean=mean, log=TRUE)) x <- seq(-3, 3, length.out=20) m2 <- morph(r=10) dnorm.morph <- m2$lud(function(x, mean) dnorm(x, mean=mean, log=TRUE)) all.equal(sapply(x, function(y) dnorm.morph(y, 2)), dnorm(x, mean=2, log=TRUE)) # make sure morph$outfun passes '...' arguments. outfun.orig <- function(x, mean) x + mean ident.morph <- morph() mean <- 1 outfun.morph <- ident.morph$outfun(outfun.orig) all.equal(outfun.morph(1:10, mean), 1:10+mean) m2 <- morph(r=10) outfun.morph <- m2$outfun(outfun.orig) all.equal(sapply(1:10, function(x) outfun.morph(x, mean)), 1:10+mean) ########################################################################### # test built-in exponential and polynomial transformations. f <- morph(b=3) x <- seq(0, 10, length.out=100) all.equal(x, sapply(sapply(x, f$transform), f$inverse)) f <- morph(p=3) all.equal(x, sapply(sapply(x, f$transform), f$inverse)) f <- morph(p=3, r=10) all.equal(-10:10, Vectorize(f$transform)(-10:10)) f <- morph(p=3, b=1) all.equal(x, sapply(sapply(x, f$transform), f$inverse)) mcmc/tests/temp-par-witch.R0000644000175100001440000000366313074040453015345 0ustar hornikusers if ((! exists("DEBUG")) || (! identical(DEBUG, TRUE))) DEBUG <- FALSE library(mcmc) options(digits=4) # avoid rounding differences set.seed(42) save.initial.seed <- .Random.seed d <- 3 witch.which <- 1 - (1 / 2)^(1 / d) * (1 / 4)^(seq(0, 5) / d) witch.which ncomp <- length(witch.which) neighbors <- matrix(FALSE, ncomp, ncomp) neighbors[row(neighbors) == col(neighbors) + 1] <- TRUE neighbors[row(neighbors) == col(neighbors) - 1] <- TRUE neighbors[row(neighbors) == col(neighbors) + 2] <- TRUE neighbors[row(neighbors) == col(neighbors) - 2] <- TRUE ludfun <- function(state) { stopifnot(is.numeric(state)) stopifnot(length(state) == d + 1) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= ncomp) theta <- state[-1] if (any(abs(theta) > 1.0)) return(-Inf) bnd <- witch.which[icomp] if(bnd >= 1.0) stop(sprintf("witch.which[%d] >= 1.0", icomp)) if(bnd <= 0.0) stop(sprintf("witch.which[%d] <= 0.0", icomp)) if (all(abs(theta) > bnd)) return(- (d + 1) * log(2) - d * log(1 - bnd)) return(- (d + 1) * log(2) - log1p(- (1 - bnd)^d)) } thetas <- matrix(0, ncomp, d) out <- temper(ludfun, initial = thetas, neighbors = neighbors, nbatch = 50, blen = 13, nspac = 7, scale = 0.3456789, parallel = TRUE, debug = DEBUG) names(out) out$acceptx out$accepti ### check that have prob 1 / 2 for corners outfun <- function(state) { stopifnot(is.matrix(state)) ncomp <- nrow(state) d <- ncol(state) foo <- sweep(abs(state), 1, witch.which) bar <- apply(foo > 0, 1, all) return(as.numeric(bar)) } out2 <- temper(out, outfun = outfun) colMeans(out2$batch) apply(out2$batch, 2, sd) / sqrt(out$nbatch) ### try again out3 <- temper(out2, blen = 103) foo <- cbind(colMeans(out3$batch), apply(out3$batch, 2, sd) / sqrt(out$nbatch)) colnames(foo) <- c("means", "MCSE") foo mcmc/tests/logitsub.R0000644000175100001440000000546113045741240014332 0ustar hornikusers # test spacing (nspac) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, nspac = 3) niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac niter == nrow(out.metro$current) niter == nrow(out.metro$proposal) all(out.metro$current[1, ] == out.metro$initial) all(out.metro$current[niter, ] == out.metro$final) | all(out.metro$proposal[niter, ] == out.metro$final) .Random.seed <- out.metro$initial.seed d <- ncol(out.metro$proposal) n <- nrow(out.metro$proposal) my.proposal <- matrix(NA, n, d) my.u <- double(n) ska <- out.metro$scale for (i in 1:n) { my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) if (is.na(out.metro$u[i])) { my.u[i] <- NA } else { my.u[i] <- runif(1) } } max(abs(out.metro$proposal - my.proposal)) < epsilon all(is.na(out.metro$u) == is.na(my.u)) all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) my.curr.log.green <- apply(out.metro$current, 1, logl) my.prop.log.green <- apply(out.metro$proposal, 1, logl) all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) foo <- my.prop.log.green - my.curr.log.green max(abs(foo - out.metro$log.green)) < epsilon my.accept <- is.na(my.u) | my.u < exp(foo) sum(my.accept) == round(n * out.metro$accept) if (my.accept[niter]) { all(out.metro$proposal[niter, ] == out.metro$final) } else { all(out.metro$current[niter, ] == out.metro$final) } my.current <- out.metro$current my.current[my.accept, ] <- my.proposal[my.accept, ] my.current <- rbind(out.metro$initial, my.current[- niter, ]) max(abs(out.metro$current - my.current)) < epsilon my.path <- matrix(NA, n, d) my.path[my.accept, ] <- out.metro$proposal[my.accept, ] my.path[! my.accept, ] <- out.metro$current[! my.accept, ] nspac <- out.metro$nspac my.path <- my.path[seq(nspac, niter, by = nspac), ] all(dim(my.path) == dim(out.metro$batch)) all(my.path == out.metro$batch) mcmc/tests/logitbat.Rout.save0000644000175100001440000000767213045741600016002 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test batching (blen) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, blen = 5) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > foom <- array(as.vector(t(my.path)), c(d, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/logitidx.Rout.save0000644000175100001440000001001413045741600016000 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test outfun (positive index vector) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, outfun = c(2, 3)) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- my.path[ , out.metro$outfun] > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/logitsubbat.Rout.save0000644000175100001440000000774213045741600016512 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test batching (blen) and spacing (nspac) together > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, blen = 5, nspac = 3) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > foom <- array(as.vector(t(my.path)), c(d, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/logitlogidx.Rout.save0000644000175100001440000001002013045741600016477 0ustar hornikusers R version 2.13.1 (2011-07-08) Copyright (C) 2011 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # test outfun (logical index vector) > > epsilon <- 1e-15 > > library(mcmc) > > RNGkind("Marsaglia-Multicarry") > set.seed(42) > > n <- 100 > rho <- 0.5 > beta0 <- 0.25 > beta1 <- 1 > beta2 <- 0.5 > > x1 <- rnorm(n) > x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) > eta <- beta0 + beta1 * x1 + beta2 * x2 > p <- 1 / (1 + exp(- eta)) > y <- as.numeric(runif(n) < p) > > out <- glm(y ~ x1 + x2, family = binomial()) > > logl <- function(beta) { + if (length(beta) != 3) stop("length(beta) != 3") + beta0 <- beta[1] + beta1 <- beta[2] + beta2 <- beta[3] + eta <- beta0 + beta1 * x1 + beta2 * x2 + p <- exp(eta) / (1 + exp(eta)) + return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) + } > > out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) > out.metro$accept [1] 0.982 > > out.metro <- metrop(out.metro, scale = 0.1) > out.metro$accept [1] 0.795 > > out.metro <- metrop(out.metro, scale = 0.5) > out.metro$accept [1] 0.264 > > apply(out.metro$batch, 2, mean) [1] 0.06080257 1.42304941 0.52634149 > > out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, + scale = 0.5, debug = TRUE, outfun = seq(1:3) > 1) > > niter <- out.metro$nbatch * out.metro$blen * out.metro$nspac > niter == nrow(out.metro$current) [1] TRUE > niter == nrow(out.metro$proposal) [1] TRUE > all(out.metro$current[1, ] == out.metro$initial) [1] TRUE > all(out.metro$current[niter, ] == out.metro$final) | + all(out.metro$proposal[niter, ] == out.metro$final) [1] TRUE > > .Random.seed <- out.metro$initial.seed > d <- ncol(out.metro$proposal) > n <- nrow(out.metro$proposal) > my.proposal <- matrix(NA, n, d) > my.u <- double(n) > ska <- out.metro$scale > for (i in 1:n) { + my.proposal[i, ] <- out.metro$current[i, ] + ska * rnorm(d) + if (is.na(out.metro$u[i])) { + my.u[i] <- NA + } else { + my.u[i] <- runif(1) + } + } > max(abs(out.metro$proposal - my.proposal)) < epsilon [1] TRUE > all(is.na(out.metro$u) == is.na(my.u)) [1] TRUE > all(out.metro$u[!is.na(out.metro$u)] == my.u[!is.na(my.u)]) [1] TRUE > > my.curr.log.green <- apply(out.metro$current, 1, logl) > my.prop.log.green <- apply(out.metro$proposal, 1, logl) > all(is.na(out.metro$u) == (my.prop.log.green > my.curr.log.green)) [1] TRUE > foo <- my.prop.log.green - my.curr.log.green > max(abs(foo - out.metro$log.green)) < epsilon [1] TRUE > > my.accept <- is.na(my.u) | my.u < exp(foo) > sum(my.accept) == round(n * out.metro$accept) [1] TRUE > if (my.accept[niter]) { + all(out.metro$proposal[niter, ] == out.metro$final) + } else { + all(out.metro$current[niter, ] == out.metro$final) + } [1] TRUE > > my.current <- out.metro$current > my.current[my.accept, ] <- my.proposal[my.accept, ] > my.current <- rbind(out.metro$initial, my.current[- niter, ]) > max(abs(out.metro$current - my.current)) < epsilon [1] TRUE > > my.path <- matrix(NA, n, d) > my.path[my.accept, ] <- out.metro$proposal[my.accept, ] > my.path[! my.accept, ] <- out.metro$current[! my.accept, ] > nspac <- out.metro$nspac > > my.path <- my.path[seq(nspac, niter, by = nspac), ] > > fred <- my.path[ , out.metro$outfun] > k <- ncol(fred) > > foom <- array(as.vector(t(fred)), c(k, out.metro$blen, out.metro$nbatch)) > boom <- t(apply(foom, c(1, 3), mean)) > > all(dim(boom) == dim(out.metro$batch)) [1] TRUE > max(abs(boom - out.metro$batch)) < epsilon [1] TRUE > > mcmc/tests/temp-ser.Rout.save0000644000175100001440000003003713074144341015721 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(mcmc) > > set.seed(42) > > data(foo) > attach(foo) > > out <- glm(y ~ x1 + x2 + x3, family = binomial, x = TRUE) > summary(out) Call: glm(formula = y ~ x1 + x2 + x3, family = binomial, x = TRUE) Deviance Residuals: Min 1Q Median 3Q Max -2.0371 -0.6337 0.2394 0.6685 1.9599 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.5772 0.2766 2.087 0.036930 * x1 0.3362 0.4256 0.790 0.429672 x2 0.8475 0.4701 1.803 0.071394 . x3 1.5143 0.4426 3.422 0.000622 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 134.602 on 99 degrees of freedom Residual deviance: 86.439 on 96 degrees of freedom AIC: 94.439 Number of Fisher Scoring iterations: 5 > > modmat <- out$x > > models <- cbind(rep(0:1, each = 4), rep(rep(0:1, times = 2), each = 2), + rep(0:1, times = 4)) > > exes <- paste("x", 1:3, sep = "") > models[nrow(models), ] [1] 1 1 1 > beta.initial <- c(nrow(models), out$coefficients) > > neighbors <- matrix(FALSE, nrow(models), nrow(models)) > for (i in 1:nrow(neighbors)) { + for (j in 1:ncol(neighbors)) { + foo <- models[i, ] + bar <- models[j, ] + if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE + } + } > neighbors [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE [2,] TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE [3,] TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE [4,] FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE [5,] TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE [6,] FALSE TRUE FALSE FALSE TRUE FALSE FALSE TRUE [7,] FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE [8,] FALSE FALSE FALSE TRUE FALSE TRUE TRUE FALSE > > ludfun <- function(state, log.pseudo.prior, ...) { + stopifnot(is.numeric(state)) + stopifnot(length(state) == ncol(models) + 2) + icomp <- state[1] + stopifnot(icomp == as.integer(icomp)) + stopifnot(1 <= icomp && icomp <= nrow(models)) + stopifnot(is.numeric(log.pseudo.prior)) + stopifnot(length(log.pseudo.prior) == nrow(models)) + beta <- state[-1] + inies <- c(TRUE, as.logical(models[icomp, ])) + beta.logl <- beta + beta.logl[! inies] <- 0 + eta <- as.numeric(modmat %*% beta.logl) + logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) + logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) + logl <- sum(logp[y == 1]) + sum(logq[y == 0]) + val <- logl - sum(beta^2) / 2 + log.pseudo.prior[icomp] + return(val) + } > > qux <- c(25.01, 5.875, 9.028, 0.6959, 11.73, 2.367, 5.864, 0.0) > > out <- temper(ludfun, initial = beta.initial, neighbors = neighbors, + nbatch = 25, blen = 20, nspac = 5, scale = 0.56789, debug = TRUE, + log.pseudo.prior = qux) > > names(out) [1] "lud" "neighbors" "nbatch" "blen" [5] "nspac" "scale" "outfun" "debug" [9] "parallel" "initial.seed" "final.seed" "time" [13] "batch" "acceptx" "accepti" "initial" [17] "final" "ibatch" "which" "unif.which" [21] "state" "log.hastings" "unif.hastings" "proposal" [25] "acceptd" "norm" "unif.choose" > > apply(out$ibatch, 2, mean) [1] 0.776 0.170 0.000 0.006 0.024 0.010 0.004 0.010 > > ### check decision about within-component or jump/swap > > identical(out$unif.which < 0.5, out$which) [1] TRUE > > identical(out$which, out$proposal[ , 1] == out$state[ , 1]) [1] TRUE > > ### check hastings ratio calculated correctly > > foo <- apply(out$state, 1, ludfun, log.pseudo.prior = qux) > bar <- apply(out$proposal, 1, ludfun, log.pseudo.prior = qux) > all.equal(bar - foo, out$log.hastings) [1] TRUE > > ### check hastings rejection decided correctly > > identical(out$log.hastings >= 0, is.na(out$unif.hastings)) [1] TRUE > all(out$log.hastings < 0 | out$acceptd) [1] TRUE > identical(out$acceptd, + out$log.hastings >= 0 | out$unif.hastings < exp(out$log.hastings)) [1] TRUE > > ### check acceptance carried out or not (according to decision) correctly > > before <- out$state > after <- before > after[- dim(after)[1], ] <- before[-1, ] > after[dim(after)[1], ] <- out$final > my.after <- before > my.after[out$acceptd, ] <- out$proposal[out$acceptd, ] > identical(after, my.after) [1] TRUE > > ### check within-component proposal > > my.coproposal.within <- out$state[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + out$scale * my.z > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check swap proposal > > coproposal.swap <- out$state[! out$which, ] > proposal.swap <- out$proposal[! out$which, ] > unif.choose.swap <- out$unif.choose[! out$which] > my.i <- coproposal.swap[ , 1] > nneighbors <- apply(out$neighbors, 1, sum) > my.nneighbors <- nneighbors[my.i] > my.k <- floor(my.nneighbors * unif.choose.swap) + 1 > my.j <- my.k > foo <- seq(1, ncol(out$neighbors)) > for (i in seq(along = my.j)) { + my.j[i] <- (foo[out$neighbors[my.i[i], ]])[my.k[i]] + } > identical(coproposal.swap[ , 1], my.i) [1] TRUE > identical(proposal.swap[ , 1], my.j) [1] TRUE > > ### check standard normal and uniform random numbers are as purported > > save.Random.seed <- .Random.seed > .Random.seed <- out$initial.seed > > nx <- length(out$initial) - 1 > niter <- out$nbatch * out$blen * out$nspac > my.norm <- matrix(NA, nrow = nrow(out$norm), ncol = ncol(out$norm)) > my.unif.which <- rep(NA, niter) > my.unif.hastings <- rep(NA, niter) > my.unif.choose <- rep(NA, niter) > for (iiter in 1:niter) { + my.unif.which[iiter] <- runif(1) + if (out$which[iiter]) { + my.norm[iiter, ] <- rnorm(nx) + if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) + } else { + my.unif.choose[iiter] <- runif(1) + if (out$log.hastings[iiter] < 0) my.unif.hastings[iiter] <- runif(1) + } + } > identical(my.norm, out$norm) [1] TRUE > identical(my.unif.which, out$unif.which) [1] TRUE > identical(my.unif.hastings, out$unif.hastings) [1] TRUE > identical(my.unif.choose, out$unif.choose) [1] TRUE > > .Random.seed <- save.Random.seed > > ### check batch means > > my.xstate <- after[ , -1] > foo <- my.xstate[seq(1, niter) %% out$nspac == 0, ] > foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) > foo <- apply(foo, c(2, 3), mean) > all.equal(foo, out$batch) [1] TRUE > > ### check ibatch means > > my.istate <- after[ , 1] > my.istate.matrix <- matrix(0, length(my.istate), nrow(models)) > for (i in 1:nrow(my.istate.matrix)) + my.istate.matrix[i, my.istate[i]] <- 1 > foo <- my.istate.matrix[seq(1, niter) %% out$nspac == 0, ] > foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) > foo <- apply(foo, c(2, 3), mean) > all.equal(foo, out$ibatch) [1] TRUE > > ### check acceptance rates > > nmodel <- nrow(out$neighbors) > > accept.within <- out$acceptd[out$which] > my.i.within <- out$state[out$which, 1] > my.i.within.accept <- my.i.within[accept.within] > my.acceptx.numer <- tabulate(my.i.within.accept, nbins = nmodel) > my.acceptx.denom <- tabulate(my.i.within, nbins = nmodel) > my.acceptx <- my.acceptx.numer / my.acceptx.denom > identical(my.acceptx, out$acceptx) [1] TRUE > > accept.swap <- out$acceptd[! out$which] > my.i.swap <- out$state[! out$which, 1] > my.j.swap <- out$proposal[! out$which, 1] > my.accepti <- matrix(NA, nmodel, nmodel) > for (i in 1:nmodel) { + for (j in 1:nmodel) { + if (out$neighbors[i, j]) { + my.accepti[i, j] <- + mean(accept.swap[my.i.swap == i & my.j.swap == j]) + } + } + } > identical(my.accepti, out$accepti) [1] TRUE > > ### check scale vector > > nx <- ncol(models) + 1 > newscale <- rnorm(nx, 0.5, 0.1) > > out <- temper(out, scale = newscale, log.pseudo.prior = qux) > > my.coproposal.within <- out$state[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + + sweep(my.z, 2, out$scale, "*") > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check scale matrix > > matscale <- matrix(rnorm(nx * nx, 0.0, 0.1), nx, nx) > diag(matscale) <- 0.56789 > > out <- temper(out, scale = matscale, log.pseudo.prior = qux) > > my.coproposal.within <- out$state[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > my.proposal.within[ , -1] <- my.coproposal.within[ , -1] + + my.z %*% t(out$scale) > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check scale list > > lisztscale <- list(0.56789, newscale, matscale, matscale, newscale, + 0.98765, 0.98765, newscale) > > out <- temper(out, scale = lisztscale, log.pseudo.prior = qux) > > my.coproposal.within <- out$state[out$which, ] > proposal.within <- out$proposal[out$which, ] > my.z <- out$norm[out$which, ] > my.proposal.within <- my.coproposal.within > for (iiter in 1:nrow(my.z)) { + my.i <- my.coproposal.within[iiter, 1] + my.scale <- out$scale[[my.i]] + if (is.matrix(my.scale)) { + my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + + my.z[iiter, , drop = FALSE] %*% t(my.scale) + } else { + my.proposal.within[iiter, -1] <- my.coproposal.within[iiter, -1] + + my.z[iiter, ] * my.scale + } + } > all.equal(proposal.within, my.proposal.within) [1] TRUE > > ### check outfun > > outfun <- function(state, icomp) { + stopifnot(is.matrix(state)) + stopifnot(is.numeric(state)) + nx <- ncol(initial) + ncomp <- nrow(initial) + stopifnot(ncol(state) == nx) + stopifnot(nrow(state) == ncomp) + stopifnot(1 <= icomp & icomp <= ncomp) + foo <- state[icomp, ] + bar <- foo^2 + return(c(foo, bar)) + } > > ncomp <- nrow(models) > nx <- length(beta.initial) - 1 > > outfun <- function(state, icomp, ...) { + stopifnot(is.numeric(state)) + stopifnot(length(state) == nx + 1) + istate <- state[1] + stopifnot(istate == as.integer(istate)) + stopifnot(1 <= istate && istate <= ncomp) + stopifnot(1 <= icomp && icomp <= ncomp) + if (istate == icomp) { + foo <- state[-1] + } else { + foo <- rep(0, nx) + } + bar <- foo^2 + return(c(foo, bar)) + } > > out <- temper(ludfun, initial = out$final, neighbors = neighbors, + nbatch = 25, blen = 20, nspac = 5, scale = 0.56789, debug = TRUE, + outfun = outfun, log.pseudo.prior = qux, icomp = 4) > > before <- out$state > after <- before > after[- dim(after)[1], ] <- before[-1, ] > after[dim(after)[1], ] <- out$final > outies <- apply(after, 1, outfun, icomp = 4) > outies <- t(outies) > > foo <- outies[seq(1, niter) %% out$nspac == 0, ] > foo <- array(as.vector(foo), dim = c(out$blen, out$nbatch, dim(foo)[2])) > foo <- apply(foo, c(2, 3), mean) > all.equal(foo, out$batch) [1] TRUE > > > proc.time() user system elapsed 1.808 0.020 1.820 mcmc/tests/logitfunarg.R0000644000175100001440000000313213045741240015014 0ustar hornikusers # test outfun (function) epsilon <- 1e-15 library(mcmc) RNGkind("Marsaglia-Multicarry") set.seed(42) n <- 100 rho <- 0.5 beta0 <- 0.25 beta1 <- 1 beta2 <- 0.5 x1 <- rnorm(n) x2 <- rho * x1 + sqrt(1 - rho^2) * rnorm(n) eta <- beta0 + beta1 * x1 + beta2 * x2 p <- 1 / (1 + exp(- eta)) y <- as.numeric(runif(n) < p) out <- glm(y ~ x1 + x2, family = binomial()) logl <- function(beta) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, coefficients(out), 1e3, scale = 0.01) out.metro$accept out.metro <- metrop(out.metro, scale = 0.1) out.metro$accept out.metro <- metrop(out.metro, scale = 0.5) out.metro$accept apply(out.metro$batch, 2, mean) out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, outfun = function(x) c(x, x^2)) out.metro <- metrop(out.metro) out.metro$outfun dim(out.metro$batch) logl <- function(beta, x1, x2, y) { if (length(beta) != 3) stop("length(beta) != 3") beta0 <- beta[1] beta1 <- beta[2] beta2 <- beta[3] eta <- beta0 + beta1 * x1 + beta2 * x2 p <- exp(eta) / (1 + exp(eta)) return(sum(log(p[y == 1])) + sum(log(1 - p[y == 0]))) } out.metro <- metrop(logl, as.numeric(coefficients(out)), 1e2, scale = 0.5, debug = TRUE, x1 = x1, x2 = x2, y = y) out.metro$lud out.metro <- metrop(out.metro, x1 = x1, x2 = x2, y = y) out.metro$lud mcmc/tests/morph.metrop.R0000644000175100001440000000205413045741600015127 0ustar hornikuserslibrary(mcmc) .morph.unmorph <- mcmc:::.morph.unmorph ########################################################################### # basic functionality check, can morph.metro run? Can we change the # transformation? set.seed(42) obj <- morph.metrop(function(x) dt(x, df=3, log=TRUE), 100, 100, morph=morph(b=3)) obj <- morph.metrop(obj, morph=morph(b=1)) obj <- morph.metrop(function(x) prod(dt(x, df=3, log=TRUE)), rep(100, 3), 100, morph=morph(p=3, b=1)) obj <- morph.metrop(obj, morph=morph(r=1, p=3, b=1)) all.equal(class(obj), c("mcmc", "morph.metropolis")) ########################################################################### # check .morph.unmorph obj <- list(final=10) outfun <- function(x) x m <- morph(p=3) obj <- .morph.unmorph(obj, m, outfun) all.equal(class(obj), c("mcmc", "morph.metropolis")) all.equal(sort(names(obj)), sort(c("final", "morph", "morph.final", "outfun"))) all.equal(c(obj$final, obj$morph.final), c(m$inverse(10), 10)) all.equal(obj$outfun, outfun) all.equal(obj$morph, m) mcmc/tests/accept-batch.Rout.save0000644000175100001440000000247313062634500016504 0ustar hornikusers R version 3.3.3 (2017-03-06) -- "Another Canoe" Copyright (C) 2017 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > # new feature batching acceptance rates > > set.seed(42) > > library(mcmc) > > h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) > out <- metrop(h, rep(0, 5), nbatch = 100, blen = 100, scale = 0.1, + debug = TRUE) > > all.equal(out$accept, mean(out$accept.batch)) [1] TRUE > > foo <- matrix(out$debug.accept, nrow = out$blen) > bar <- colMeans(foo) > all.equal(out$accept.batch, bar) [1] TRUE > > options(digits = 4) # try to keep insanity of computer arithmetic under control > > out$accept [1] 0.2257 > t.test(out$accept.batch)$conf.int [1] 0.2124 0.2390 attr(,"conf.level") [1] 0.95 > > > proc.time() user system elapsed 0.168 0.020 0.184 mcmc/tests/initseq.R0000644000175100001440000000225213062544417014157 0ustar hornikusers library(mcmc) set.seed(42) n <- 1e5 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) gamma <- acf(x, lag.max = 1999, type = "covariance", plot = FALSE)$acf k <- seq(along = gamma) Gamma <- gamma[k %% 2 == 1] + gamma[k %% 2 == 0] k <- min(seq(along = Gamma)[Gamma < 0]) Gamma <- Gamma[1:k] Gamma[k] < 0 Gamma[k] <- 0 out <- .Call(mcmc:::C_initseq, x - mean(x)) names(out) all.equal(gamma[1], out$gamma0) length(out$Gamma.pos) == length(Gamma) all.equal(out$Gamma.pos, Gamma) Gamma.dec <- cummin(Gamma) all.equal(out$Gamma.dec, Gamma.dec) library(Iso) Gamma.con <- Gamma.dec[1] + cumsum(c(0, pava(diff(Gamma.dec)))) all.equal(out$Gamma.con, Gamma.con) all.equal(0, min(out$Gamma.pos - out$Gamma.dec)) max(diff(out$Gamma.dec)) < sqrt(.Machine$double.eps) all.equal(0, min(out$Gamma.dec - out$Gamma.con)) min(diff(diff(out$Gamma.con))) > (- sqrt(.Machine$double.eps)) all.equal(2 * sum(out$Gamma.pos) - out$gamma0, out$var.pos) all.equal(2 * sum(out$Gamma.dec) - out$gamma0, out$var.dec) all.equal(2 * sum(out$Gamma.con) - out$gamma0, out$var.con) rev(out$Gamma.pos)[1] == 0 rev(out$Gamma.dec)[1] == 0 all.equal(rev(out$Gamma.con)[1], 0) mcmc/src/0000755000175100001440000000000013074514645012007 5ustar hornikusersmcmc/src/Makevars0000644000175100001440000000003313074514645013477 0ustar hornikusersPKG_CFLAGS=$(C_VISIBILITY) mcmc/src/temper.c0000644000175100001440000012020213074514645013444 0ustar hornikusers #include #include #include #include "myutil.h" #include "mcmc.h" #ifdef BLEAT #include #endif /* BLEAT */ static void propose(SEXP coproposal, SEXP proposal, SEXP scale, double *z); static double logh(SEXP func, SEXP state, SEXP rho); static SEXP outfun(SEXP func, SEXP state, SEXP rho); static void check_valid_scale(SEXP scale, int i, int ncomp, int nx); SEXP temper(SEXP func1, SEXP initial, SEXP neighbors, SEXP nbatch, SEXP blen, SEXP nspac, SEXP scale, SEXP func2, SEXP debug, SEXP parallel, SEXP rho1, SEXP rho2) { if (! isFunction(func1)) error("argument \"func1\" must be function"); if (! isEnvironment(rho1)) error("argument \"rho1\" must be environment"); int is_parallel = getScalarLogical(parallel, "parallel"); int is_debug = getScalarLogical(debug, "debug"); if (! isLogical(neighbors)) error("argument \"neighbors\" must be logical"); if (! isMatrix(neighbors)) error("argument \"neighbors\" must be matrix"); if (nrows(neighbors) != ncols(neighbors)) error("argument \"neighbors\" must have same row and column dimension"); int ncomp = nrows(neighbors); if (ncomp <= 1) error("must have at least two components"); for (int i = 0; i < ncomp; i++) for (int j = 0; j < ncomp; j++) if (LOGICAL(neighbors)[i + ncomp * j] != LOGICAL(neighbors)[j + ncomp * i]) error("argument \"neighbors\" must be symmetric matrix"); if (! isReal(initial)) error("argument \"initial\" must be real"); if (! isAllFinite(initial)) error("argument \"initial\" must have all elements finite"); int nx; if (is_parallel) { if (! isMatrix(initial)) error("argument \"initial\" must be matrix in parallel case"); if (nrows(initial) != ncomp) error("row dim of args \"initial\" and \"neighbors\" must be same in parallel case"); nx = ncols(initial); } else /* serial */ { if (! (LENGTH(initial) > 1)) error("argument \"initial\" must have length > 1 in serial case"); double reali = REAL(initial)[0]; int i = reali; if (i != reali) error("1st elem of argument \"initial\" must be integer in serial case"); if (i <= 0 || i > ncomp) error("1st elem of argument \"initial\" must be in 1, ..., k in serial case"); nx = LENGTH(initial) - 1; } int int_nbatch = getScalarInteger(nbatch, "nbatch"); if (int_nbatch <= 0) error("argument \"nbatch\" must be positive"); int int_blen = getScalarInteger(blen, "blen"); if (int_blen <= 0) error("argument \"blen\" must be positive"); int int_nspac = getScalarInteger(nspac, "nspac"); if (int_nspac <= 0) error("argument \"nspac\" must be positive"); if (isNewList(scale)) { if (LENGTH(scale) != ncomp) error("argument \"scale\" must have length k if list"); for (int i = 0; i < ncomp; i++) { SEXP fred = VECTOR_ELT(scale, i); check_valid_scale(fred, i, ncomp, nx); } } else /* scale not list */ { check_valid_scale(scale, -1, ncomp, nx); } int no_outfun = isNull(func2); if (! no_outfun) { if (! isFunction(func2)) error("argument \"outfun\" must be function"); if (! isEnvironment(rho2)) error("argument \"rho2\" must be environment"); } double current_log_dens[ncomp]; if (is_parallel) { SEXP fred; PROTECT(fred = allocVector(REALSXP, nx + 1)); for (int i = 0; i < ncomp; i++) { REAL(fred)[0] = i + 1; for (int j = 0; j < nx; j++) REAL(fred)[j + 1] = REAL(initial)[i + ncomp * j]; current_log_dens[i] = logh(func1, fred, rho1); #ifdef BLATHER fprintf(stderr, "current_log_dens[%d] = %e\n", i, current_log_dens[i]); for (int j = 0; j < nx; j++) fprintf(stderr, " state[%d, %d] = %e\n", i, j, REAL(initial)[i + ncomp * j]); for (int j = 0; j <= nx; j++) fprintf(stderr, " fred[%d] = %e\n", j, REAL(fred)[j]); fprintf(stderr, " logh(func1, fred, rho1)) = %e\n", logh(func1, fred, rho1)); #endif /* BLATHER */ if (current_log_dens[i] == R_NegInf) error("log unnormalized density -Inf at initial state"); } UNPROTECT(1); } else /* serial */ { for (int i = 0; i < ncomp; i++) current_log_dens[i] = R_NaN; int i = REAL(initial)[0] - 1; current_log_dens[i] = logh(func1, initial, rho1); if (current_log_dens[i] == R_NegInf) error("log unnormalized density -Inf at initial state"); } /* at this point, all arguments have been checked for validity */ // current_log_dens saves info that cuts in half // the number of invocations of log unnormalized density SEXP state, proposal, coproposal; PROTECT(state = duplicate(initial)); PROTECT(proposal = allocVector(REALSXP, nx + 1)); PROTECT(coproposal = allocVector(REALSXP, nx + 1)); int nout; if (no_outfun) { nout = is_parallel ? nx * ncomp : nx; } else /* has outfun */ { nout = LENGTH(outfun(func2, state, rho2)); } int niter = int_nbatch * int_blen * int_nspac; // TO DO LIST // // regular output (if serial) // // batch nbatch x nout matrix // ibatch nbatch x ncomp matrix // acceptx vector of length ncomp // accepti ncomp x ncomp matrix // initial copy of initial state // final final state // // regular output (if parallel) // // batch (if outfun) nbatch x nout matrix // (if no outfun) nbatch x ncomp x nx array // acceptx vector of length ncomp // accepti ncomp x ncomp matrix // initial copy of initial state // final final state // // debug output (if not parallel) // // which vector of length niter (TRUE if within-component) // unif_which vector of length niter (uniform for deciding which) // state niter x (nx + 1) matrix (state before) // proposal niter x (nx + 1) matrix // log_hastings niter vector // unif_hastings niter vector (uniform for deciding acceptance) // acceptd niter vector (TRUE if accept) // norm niter x nx matrix (std normals) // unif_choose niter vector (uniform for choosing neighbor) // // debug output (if parallel) // // which vector of length niter (TRUE if within-component) // unif_which vector of length niter (uniform for deciding which) // state niter x ncomp x nx array (state before) // coproposal niter x (nx + 1) matrix // proposal niter x (nx + 1) matrix // log_hastings niter vector // unif_hastings niter vector (uniform for deciding acceptance) // acceptd niter vector (TRUE if accept) // norm niter x nx matrix (std normals) // unif_choose niter x 2 matrix (uniforms for choosing components // to update) // // for within-component move coproposal and proposal have natural // meaning // for swap move we have 2 coproposals (i, x_i) and (j, x_j) // and 2 proposals (i, x_j) and (j, x_i) but since the information // here is quite redundant we just store (i, x_i) in "coproposal" // and (j, x_j) in "proposal" -- the checker can figure it out int len_result_regular = is_parallel ? 5 : 6; int len_result_debug = is_parallel ? 10 : 9; int len_result = len_result_regular; len_result += is_debug ? len_result_debug : 0; #ifdef BLEAT fprintf(stderr, "len_result = %d\n", len_result); #endif /* BLEAT */ SEXP result, resultnames, acceptx, accepti, batch, ibatch, save_initial, save_final, debug_which, debug_unif_which, debug_state, debug_coproposal, debug_proposal, debug_log_hastings, debug_unif_hastings, debug_acceptd, debug_norm, debug_unif_choose; PROTECT(result = allocVector(VECSXP, len_result)); PROTECT(resultnames = allocVector(STRSXP, len_result)); namesgets(result, resultnames); UNPROTECT(1); if (no_outfun && is_parallel) PROTECT(batch = alloc3DArray(REALSXP, int_nbatch, ncomp, nx)); else PROTECT(batch = allocMatrix(REALSXP, int_nbatch, nout)); SET_VECTOR_ELT(result, 0, batch); SET_STRING_ELT(resultnames, 0, mkChar("batch")); UNPROTECT(1); PROTECT(acceptx = allocVector(REALSXP, ncomp)); SET_VECTOR_ELT(result, 1, acceptx); SET_STRING_ELT(resultnames, 1, mkChar("acceptx")); UNPROTECT(1); PROTECT(accepti = allocMatrix(REALSXP, ncomp, ncomp)); SET_VECTOR_ELT(result, 2, accepti); SET_STRING_ELT(resultnames, 2, mkChar("accepti")); UNPROTECT(1); PROTECT(save_initial = duplicate(initial)); SET_VECTOR_ELT(result, 3, save_initial); SET_STRING_ELT(resultnames, 3, mkChar("initial")); UNPROTECT(1); SET_STRING_ELT(resultnames, 4, mkChar("final")); // at end need to duplicate state as save_final and copy to result[4] if (! is_parallel) { PROTECT(ibatch = allocMatrix(REALSXP, int_nbatch, ncomp)); SET_VECTOR_ELT(result, 5, ibatch); SET_STRING_ELT(resultnames, 5, mkChar("ibatch")); UNPROTECT(1); } if (is_debug) { PROTECT(debug_which = allocVector(LGLSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 0, debug_which); SET_STRING_ELT(resultnames, len_result_regular + 0, mkChar("which")); UNPROTECT(1); PROTECT(debug_unif_which = allocVector(REALSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 1, debug_unif_which); SET_STRING_ELT(resultnames, len_result_regular + 1, mkChar("unif.which")); UNPROTECT(1); if (is_parallel) PROTECT(debug_state = alloc3DArray(REALSXP, niter, ncomp, nx)); else PROTECT(debug_state = allocMatrix(REALSXP, niter, nx + 1)); SET_VECTOR_ELT(result, len_result_regular + 2, debug_state); SET_STRING_ELT(resultnames, len_result_regular + 2, mkChar("state")); UNPROTECT(1); PROTECT(debug_log_hastings = allocVector(REALSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 3, debug_log_hastings); SET_STRING_ELT(resultnames, len_result_regular + 3, mkChar("log.hastings")); UNPROTECT(1); PROTECT(debug_unif_hastings = allocVector(REALSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 4, debug_unif_hastings); SET_STRING_ELT(resultnames, len_result_regular + 4, mkChar("unif.hastings")); UNPROTECT(1); PROTECT(debug_proposal = allocMatrix(REALSXP, niter, nx + 1)); SET_VECTOR_ELT(result, len_result_regular + 5, debug_proposal); SET_STRING_ELT(resultnames, len_result_regular + 5, mkChar("proposal")); UNPROTECT(1); PROTECT(debug_acceptd = allocVector(LGLSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 6, debug_acceptd); SET_STRING_ELT(resultnames, len_result_regular + 6, mkChar("acceptd")); UNPROTECT(1); PROTECT(debug_norm = allocMatrix(REALSXP, niter, nx)); SET_VECTOR_ELT(result, len_result_regular + 7, debug_norm); SET_STRING_ELT(resultnames, len_result_regular + 7, mkChar("norm")); UNPROTECT(1); if (is_parallel) PROTECT(debug_unif_choose = allocMatrix(REALSXP, niter, 2)); else PROTECT(debug_unif_choose = allocVector(REALSXP, niter)); SET_VECTOR_ELT(result, len_result_regular + 8, debug_unif_choose); SET_STRING_ELT(resultnames, len_result_regular + 8, mkChar("unif.choose")); UNPROTECT(1); if (is_parallel) { PROTECT(debug_coproposal = allocMatrix(REALSXP, niter, nx + 1)); SET_VECTOR_ELT(result, len_result_regular + 9, debug_coproposal); SET_STRING_ELT(resultnames, len_result_regular + 9, mkChar("coproposal")); UNPROTECT(1); } } // at this point, entire output structure (SEXP result) is set up, except // for aforementioned need to duplicate final state and put in result[4] GetRNGstate(); // need buffers for acceptance rate(s) double acceptx_numer[ncomp]; double acceptx_denom[ncomp]; double accepti_numer[ncomp][ncomp]; double accepti_denom[ncomp][ncomp]; for (int i = 0; i < ncomp; i++) { acceptx_numer[i] = 0; acceptx_denom[i] = 0; for (int j = 0; j < ncomp; j++) { accepti_numer[i][j] = 0; accepti_denom[i][j] = 0; } } // need neighbor counts and neighbors // note: the_neighbors uses zero-origin indexing both for indexing // and values double n_neighbors[ncomp]; for (int i = 0; i < ncomp; i++) { n_neighbors[i] = 0; for (int j = 0; j < ncomp; j++) n_neighbors[i] += LOGICAL(neighbors)[i + ncomp * j]; } double the_neighbors[ncomp][ncomp]; for (int i = 0; i < ncomp; i++) { for (int j = 0, k = 0; j < ncomp; j++) { if (LOGICAL(neighbors)[i + ncomp * j]) the_neighbors[i][k++] = j; } } // need buffers for batch means double batch_buff[nout]; double ibatch_buff[ncomp]; for (int kbatch = 0, iiter = 0; kbatch < int_nbatch; kbatch++) { for (int i = 0; i < nout; i++) batch_buff[i] = 0.0; for (int i = 0; i < ncomp; i++) ibatch_buff[i] = 0.0; for (int jbatch = 0; jbatch < int_blen; jbatch++) { for (int ispac = 0; ispac < int_nspac; ispac++, iiter++) { #ifdef EXTRA_CHECK #ifdef WOOF fprintf(stderr, "Check for validity of current_log_dens at top of inner loop\n"); #endif /* WOOF */ if (is_parallel) { for (int i = 0; i < ncomp; i++) { REAL(proposal)[0] = i + 1; for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(state)[i + ncomp * j]; #ifdef BLATHER fprintf(stderr, "current_log_dens[%d] = %e\n", i, current_log_dens[i]); for (int j = 0; j < nx; j++) fprintf(stderr, " state[%d, %d] = %e\n", i, j, REAL(state)[i + ncomp * j]); for (int j = 0; j <= nx; j++) fprintf(stderr, " proposal[%d] = %e\n", j, REAL(proposal)[j]); fprintf(stderr, " logh(func1, proposal, rho1)) = %e\n", logh(func1, proposal, rho1)); #endif /* BLATHER */ if (current_log_dens[i] != logh(func1, proposal, rho1)) error("current_log_dens[%d] bogus\n", i); } } else /* serial */ { for (int j = 0; j <= nx; j++) REAL(proposal)[j] = REAL(state)[j]; int i = REAL(proposal)[0] - 1; double my_actual_logh = logh(func1, proposal, rho1); double my_stored_logh = current_log_dens[i]; #ifdef WOOF fprintf(stderr, "icomp = %d, stored logh = %e, actual logh = %e\n", i + 1, my_stored_logh, my_actual_logh); #endif /* WOOF */ if (my_stored_logh != my_actual_logh) error("current_log_dens[%d] bogus\n", i); } #endif /* EXTRA_CHECK */ if (is_debug) { int len_state = is_parallel ? ncomp * nx : nx + 1; for (int j = 0; j < len_state; j++) REAL(debug_state)[iiter + niter * j] = REAL(state)[j]; } double my_unif_which = unif_rand(); int my_which = my_unif_which < 0.5; if (is_debug) { LOGICAL(debug_which)[iiter] = my_which; REAL(debug_unif_which)[iiter] = my_unif_which; } if (my_which) /* within-component update */ { if (is_parallel) { // note: my_i and my_j are 1-origin indexing (for R) // go from 1, ..., ncomp // everything else 0-origin indexing (for C) double unif_choose = unif_rand(); int my_i = trunc(ncomp * unif_choose) + 1; if (my_i > ncomp) my_i--; if (my_i <= 0 || my_i > ncomp) error("Can't happen: my_i out of range"); REAL(coproposal)[0] = my_i; for (int j = 0; j < nx; j++) REAL(coproposal)[j + 1] = REAL(state)[(my_i - 1) + ncomp * j]; double z[nx]; propose(coproposal, proposal, scale, z); double my_coproposal_log_dens = current_log_dens[my_i - 1]; #ifdef EXTRA_CHECK if (my_coproposal_log_dens != logh(func1, coproposal, rho1)) { fprintf(stderr, "with-in component update (parallel)\n"); error("saving logh didn't work right (coproposal)"); } #endif /* EXTRA_CHECK */ #ifdef BLEAT if (my_coproposal_log_dens == R_NegInf) { fprintf(stderr, "Oopsie #1!\n"); fprintf(stderr, " my_i = %d\n", my_i); fprintf(stderr, " current_log_dens[my_i - 1] = %e\n", current_log_dens[my_i - 1]); fprintf(stderr, " my_coproposal_log_dens = %e\n", my_coproposal_log_dens); for (int j = 0; j <= nx; j++) fprintf(stderr, " coproposal[%d] = %e\n", j + 1, REAL(coproposal)[j]); fprintf(stderr, " logh(coproposal) = %e\n", logh(func1, coproposal, rho1)); } #endif /* BLEAT */ if (my_coproposal_log_dens == R_NegInf) error("Can't happen: log density -Inf at current state"); double my_new_log_dens = logh(func1, proposal, rho1); double my_log_hastings = my_new_log_dens - my_coproposal_log_dens; if (isnan(my_log_hastings) || (isinf(my_log_hastings) && my_log_hastings > 0)) error("Can't happen: log hastings ratio +Inf or NaN\n"); int my_accept = 1; double my_unif_hastings = R_NaReal; if (my_log_hastings < 0.0) { my_unif_hastings = unif_rand(); my_accept = my_unif_hastings < exp(my_log_hastings); } if (is_debug) { for (int j = 0; j <= nx; j++) REAL(debug_proposal)[iiter + niter * j] = REAL(proposal)[j]; for (int j = 0; j <= nx; j++) REAL(debug_coproposal)[iiter + niter * j] = REAL(coproposal)[j]; REAL(debug_log_hastings)[iiter] = my_log_hastings; REAL(debug_unif_hastings)[iiter] = my_unif_hastings; LOGICAL(debug_acceptd)[iiter] = my_accept; for (int j = 0; j < nx; j++) REAL(debug_norm)[iiter + niter * j] = z[j]; REAL(debug_unif_choose)[iiter] = unif_choose; REAL(debug_unif_choose)[iiter + niter] = R_NaReal; } if (my_accept) { for (int j = 0; j < nx; j++) REAL(state)[(my_i - 1) + ncomp * j] = REAL(proposal)[j + 1]; current_log_dens[my_i - 1] = my_new_log_dens; acceptx_numer[my_i - 1]++; } acceptx_denom[my_i - 1]++; } else /* serial */ { int my_i = REAL(state)[0]; if (my_i <= 0 || my_i > ncomp) error("Can't happen: my_i out of range"); REAL(coproposal)[0] = my_i; for (int j = 0; j < nx; j++) REAL(coproposal)[j + 1] = REAL(state)[j + 1]; double z[nx]; propose(coproposal, proposal, scale, z); double my_new_log_dens = logh(func1, proposal, rho1); double my_old_log_dens = current_log_dens[my_i - 1]; #ifdef BLEAT if (my_old_log_dens == R_NegInf) { fprintf(stderr, "Oopsie #2!\n"); } #endif /* BLEAT */ if (my_old_log_dens == R_NegInf) error("Can't happen: log density -Inf at current state"); #ifdef EXTRA_CHECK if (my_old_log_dens != logh(func1, coproposal, rho1)) { fprintf(stderr, "with-in component update (serial)\n"); error("saving logh didn't work right (coproposal)"); } #endif /* EXTRA_CHECK */ double my_log_hastings = my_new_log_dens - my_old_log_dens; if (isnan(my_log_hastings) || (isinf(my_log_hastings) && my_log_hastings > 0)) { #ifdef WOOF fprintf(stderr, "my_old_log_dens = %e\n", my_old_log_dens); fprintf(stderr, "my_new_log_dens = %e\n", my_old_log_dens); fprintf(stderr, "my_i = %d\n", my_i); #endif /* WOOF */ error("Can't happen: log hastings ratio +Inf or NaN\n"); } int my_accept = 1; double my_unif_hastings = R_NaReal; if (my_log_hastings < 0.0) { my_unif_hastings = unif_rand(); my_accept = my_unif_hastings < exp(my_log_hastings); } if (is_debug) { for (int j = 0; j <= nx; j++) REAL(debug_proposal)[iiter + niter * j] = REAL(proposal)[j]; REAL(debug_log_hastings)[iiter] = my_log_hastings; REAL(debug_unif_hastings)[iiter] = my_unif_hastings; LOGICAL(debug_acceptd)[iiter] = my_accept; for (int j = 0; j < nx; j++) REAL(debug_norm)[iiter + niter * j] = z[j]; REAL(debug_unif_choose)[iiter] = R_NaReal; } if (my_accept) { for (int j = 0; j <= nx; j++) REAL(state)[j] = REAL(proposal)[j]; current_log_dens[my_i - 1] = my_new_log_dens; acceptx_numer[my_i - 1]++; } acceptx_denom[my_i - 1]++; } } else /* jump/swap update */ { if (is_parallel) { double unif_choose_one = unif_rand(); double unif_choose_two = unif_rand(); int my_i = trunc(ncomp * unif_choose_one) + 1; if (my_i > ncomp) my_i--; if (my_i <= 0 || my_i > ncomp) error("Can't happen: my_i out of range"); REAL(coproposal)[0] = my_i; for (int j = 0; j < nx; j++) REAL(coproposal)[j + 1] = REAL(state)[(my_i - 1) + ncomp * j]; int my_i_neighbors = n_neighbors[my_i - 1]; int foo = trunc(my_i_neighbors * unif_choose_two) + 1; if (foo > my_i_neighbors) foo--; int my_j = the_neighbors[my_i - 1][foo - 1] + 1; #ifdef BLEAT fprintf(stderr, "my_i = %d, my_i_neighbors = %d, foo = %d\n", my_i, my_i_neighbors, foo); fprintf(stderr, "(parallel) ncomp = %d, my_j = %d\n", ncomp, my_j); #endif /* BLEAT */ if (my_j <= 0 || my_j > ncomp) error("Can't happen: my_j out of range"); REAL(proposal)[0] = my_j; for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(state)[(my_j - 1) + ncomp * j]; double my_coproposal_log_dens = current_log_dens[my_i - 1]; #ifdef EXTRA_CHECK if (my_coproposal_log_dens != logh(func1, coproposal, rho1)) { fprintf(stderr, "swap component update (parallel)\n"); error("saving logh didn't work right (coproposal)"); } #endif /* EXTRA_CHECK */ #ifdef BLEAT if (my_coproposal_log_dens == R_NegInf) { fprintf(stderr, "Oopsie #3!\n"); fprintf(stderr, " my_i = %d\n", my_i); fprintf(stderr, " current_log_dens[my_i - 1] = %e\n", current_log_dens[my_i - 1]); fprintf(stderr, " my_coproposal_log_dens = %e\n", my_coproposal_log_dens); for (int j = 0; j <= nx; j++) fprintf(stderr, " coproposal[%d] = %e\n", j + 1, REAL(coproposal)[j]); fprintf(stderr, " logh(coproposal) = %e\n", logh(func1, coproposal, rho1)); } #endif /* BLEAT */ if (my_coproposal_log_dens == R_NegInf) error("Can't happen: log density -Inf at current state"); double my_proposal_log_dens = current_log_dens[my_j - 1]; #ifdef EXTRA_CHECK if (my_proposal_log_dens != logh(func1, proposal, rho1)) { fprintf(stderr, "swap component update (parallel)\n"); error("saving logh didn't work right (proposal)"); } #endif /* EXTRA_CHECK */ #ifdef BLEAT if (my_proposal_log_dens == R_NegInf) { fprintf(stderr, "Oopsie #4!\n"); } #endif /* BLEAT */ if (my_proposal_log_dens == R_NegInf) error("Can't happen: log density -Inf at current state"); if (is_debug) { for (int j = 0; j <= nx; j++) REAL(debug_proposal)[iiter + niter * j] = REAL(proposal)[j]; for (int j = 0; j <= nx; j++) REAL(debug_coproposal)[iiter + niter * j] = REAL(coproposal)[j]; } // proposal and coproposal now saved and logh evaluated // for them, can clobber to evaluate for swap REAL(proposal)[0] = my_i; REAL(coproposal)[0] = my_j; double my_swapped_coproposal_log_dens = logh(func1, coproposal, rho1); double my_swapped_proposal_log_dens = logh(func1, proposal, rho1); double my_log_hastings = my_swapped_proposal_log_dens + my_swapped_coproposal_log_dens - my_proposal_log_dens - my_coproposal_log_dens; if (isnan(my_log_hastings) || (isinf(my_log_hastings) && my_log_hastings > 0)) error("Can't happen: log hastings ratio +Inf or NaN\n"); int my_accept = 1; double my_unif_hastings = R_NaReal; if (my_log_hastings < 0.0) { my_unif_hastings = unif_rand(); my_accept = my_unif_hastings < exp(my_log_hastings); } if (is_debug) { REAL(debug_log_hastings)[iiter] = my_log_hastings; REAL(debug_unif_hastings)[iiter] = my_unif_hastings; LOGICAL(debug_acceptd)[iiter] = my_accept; for (int j = 0; j < nx; j++) REAL(debug_norm)[iiter + niter * j] = R_NaReal; REAL(debug_unif_choose)[iiter] = unif_choose_one; REAL(debug_unif_choose)[iiter + niter] = unif_choose_two; } if (my_accept) { for (int j = 0; j < nx; j++) REAL(state)[(my_j - 1) + ncomp * j] = REAL(coproposal)[j + 1]; for (int j = 0; j < nx; j++) REAL(state)[(my_i - 1) + ncomp * j] = REAL(proposal)[j + 1]; current_log_dens[my_i - 1] = my_swapped_proposal_log_dens; current_log_dens[my_j - 1] = my_swapped_coproposal_log_dens; accepti_numer[my_i - 1][my_j - 1]++; } accepti_denom[my_i - 1][my_j - 1]++; } else /* serial */ { int my_i = REAL(state)[0]; if (my_i <= 0 || my_i > ncomp) error("Can't happen: my_i out of range"); int my_i_neighbors = n_neighbors[my_i - 1]; double unif_choose = unif_rand(); int foo = trunc(my_i_neighbors * unif_choose) + 1; if (foo > my_i_neighbors) foo--; int my_j = the_neighbors[my_i - 1][foo - 1] + 1; int my_j_neighbors = n_neighbors[my_j - 1]; #ifdef BLEAT fprintf(stderr, "(serial) ncomp = %d, my_j = %d, my_i_neighbors = %d, foo = %d\n", ncomp, my_j, my_i_neighbors, foo); fprintf(stderr, " unif_choose = %f\n", unif_choose); for (int j = 0; j < ncomp; j++) fprintf(stderr, " LOGICAL(neighbors)[(my_i - 1) + ncomp * %d] = %d\n", j, LOGICAL(neighbors)[(my_i - 1) + ncomp * j]); #endif /* BLEAT */ if (my_j <= 0 || my_j > ncomp) error("Can't happen: my_j out of range"); REAL(proposal)[0] = my_j; for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(state)[j + 1]; #ifdef WOOF fprintf(stderr, "got to here, about to call logh\n"); fprintf(stderr, " REAL(proposal)[0] = %e\n", REAL(proposal)[0]); #endif /* WOOF */ double my_new_log_dens = logh(func1, proposal, rho1); double my_old_log_dens = current_log_dens[my_i - 1]; #ifdef BLEAT if (my_old_log_dens == R_NegInf) { fprintf(stderr, "Oopsie #5!\n"); } #endif /* BLEAT */ #ifdef EXTRA_CHECK for (int j = 0; j <= nx; j++) REAL(coproposal)[j] = REAL(state)[j]; if (my_old_log_dens != logh(func1, coproposal, rho1)) { fprintf(stderr, "swap component update (serial)\n"); error("saving logh didn't work right (coproposal)"); } #endif /* EXTRA_CHECK */ if (my_old_log_dens == R_NegInf) error("Can't happen: log density -Inf at current state"); double my_log_hastings = my_new_log_dens - my_old_log_dens + log(my_i_neighbors) - log(my_j_neighbors); if (isnan(my_log_hastings) || (isinf(my_log_hastings) && my_log_hastings > 0)) error("Can't happen: log hastings ratio +Inf or NaN\n"); int my_accept = 1; double my_unif_hastings = R_NaReal; if (my_log_hastings < 0.0) { my_unif_hastings = unif_rand(); my_accept = my_unif_hastings < exp(my_log_hastings); } if (is_debug) { for (int j = 0; j <= nx; j++) REAL(debug_proposal)[iiter + niter * j] = REAL(proposal)[j]; REAL(debug_log_hastings)[iiter] = my_log_hastings; REAL(debug_unif_hastings)[iiter] = my_unif_hastings; LOGICAL(debug_acceptd)[iiter] = my_accept; for (int j = 0; j < nx; j++) REAL(debug_norm)[iiter + niter * j] = R_NaReal; REAL(debug_unif_choose)[iiter] = unif_choose; } #ifdef WOOF_WOOF fprintf(stderr, "unif_choose = %f, ", unif_choose); fprintf(stderr, "REAL(debug_unif_choose)[iiter] = %f\n", REAL(debug_unif_choose)[iiter]); #endif /* WOOF_WOOF */ if (my_accept) { for (int j = 0; j <= nx; j++) REAL(state)[j] = REAL(proposal)[j]; current_log_dens[my_j - 1] = my_new_log_dens; accepti_numer[my_i - 1][my_j - 1]++; } accepti_denom[my_i - 1][my_j - 1]++; } } } /* end of inner loop (one iteration) */ if (no_outfun) { if (is_parallel) for (int i = 0; i < nout; i++) batch_buff[i] += REAL(state)[i]; else for (int i = 0; i < nout; i++) batch_buff[i] += REAL(state)[i + 1]; } else /* has outfun */ { SEXP fred = outfun(func2, state, rho2); if (LENGTH(fred) != nout) error("function outfun returns results of different lengths"); for (int i = 0; i < nout; i++) batch_buff[i] += REAL(fred)[i]; } if (! is_parallel) ibatch_buff[((int) REAL(state)[0]) - 1]++; } /* end of middle loop (one batch) */ if (no_outfun && is_parallel) for (int i = 0; i < ncomp; i++) for (int j = 0; j < nx; j++) REAL(batch)[kbatch + int_nbatch * (i + ncomp * j)] = batch_buff[i + ncomp * j] / int_blen; else for (int i = 0; i < nout; i++) REAL(batch)[kbatch + int_nbatch * i] = batch_buff[i] / int_blen; if (! is_parallel) for (int i = 0; i < ncomp; i++) REAL(ibatch)[kbatch + int_nbatch * i] = ibatch_buff[i] / int_blen; } /* end of outer loop */ for (int i = 0; i < ncomp; i++) REAL(acceptx)[i] = acceptx_numer[i] / acceptx_denom[i]; for (int i = 0; i < ncomp; i++) for (int j = 0; j < ncomp; j++) if (LOGICAL(neighbors)[i + ncomp * j]) REAL(accepti)[i + ncomp * j] = accepti_numer[i][j] / accepti_denom[i][j]; else REAL(accepti)[i + ncomp * j] = R_NaReal; PutRNGstate(); PROTECT(save_final = duplicate(state)); SET_VECTOR_ELT(result, 4, save_final); UNPROTECT(5); return result; } static void check_valid_scale(SEXP scale, int i, int ncomp, int nx) { if (i > ncomp) error("check_valid_scale: i = %d, ncomp = %d, invalid\n", i, ncomp); if (! isReal(scale)) { if (i >= 0) error("component %d of scale not type double", i + 1); else error("scale not type double"); } if (! isAllFinite(scale)) { if (i >= 0) error("component %d of scale has non-finite element", i + 1); else error("scale has non-finite element"); } if (isMatrix(scale)) { if (nrows(scale) != nx) { if (i >= 0) error("component %d of scale matrix with wrong row dim", i + 1); else error("scale matrix with wrong row dim"); } if (ncols(scale) != nx) { if (i >= 0) error("component %d of scale matrix with wrong col dim", i + 1); else error("scale matrix with wrong col dim"); } } else /* scale not matrix */ { if (! (LENGTH(scale) == 1 || LENGTH(scale) == nx)) { if (i >= 0) error("component %d of scale not matrix, scalar, or vector of length k", i + 1); else error("scale not matrix, scalar, or vector of length k"); } } } static double logh(SEXP func, SEXP state, SEXP rho) { SEXP call, result, foo; double bar; PROTECT(call = lang2(func, state)); PROTECT(result = eval(call, rho)); if (! isNumeric(result)) error("log unnormalized density function returned non-numeric"); if (LENGTH(result) != 1) error("log unnormalized density function returned non-scalar"); PROTECT(foo = coerceVector(result, REALSXP)); bar = REAL(foo)[0]; UNPROTECT(3); if (bar == R_PosInf) error("log unnormalized density function returned +Inf"); if (R_IsNaN(bar) || R_IsNA(bar)) error("log unnormalized density function returned NA or NaN"); /* Note: -Inf is allowed */ return bar; } static SEXP outfun(SEXP func, SEXP state, SEXP rho) { SEXP call, result, foo; PROTECT(call = lang2(func, state)); PROTECT(result = eval(call, rho)); if (! isNumeric(result)) error("outfun returned non-numeric"); PROTECT(foo = coerceVector(result, REALSXP)); UNPROTECT(3); return foo; } static void propose(SEXP coproposal, SEXP proposal, SEXP scale, double *z) { int my_i = REAL(coproposal)[0]; int nx = LENGTH(coproposal) - 1; for (int j = 0; j < nx; j++) z[j] = norm_rand(); if (isNewList(scale)) scale = VECTOR_ELT(scale, my_i - 1); REAL(proposal)[0] = my_i; if (LENGTH(scale) == 1) { for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(coproposal)[j + 1] + REAL(scale)[0] * z[j]; } else if (LENGTH(scale) == nx) { for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(coproposal)[j + 1] + REAL(scale)[j] * z[j]; } else /* scale is nx by nx matrix */ { for (int j = 0; j < nx; j++) REAL(proposal)[j + 1] = REAL(coproposal)[j + 1]; for (int j = 0, m = 0; j < nx; j++) { double u = z[j]; for (int k = 0; k < nx; k++) REAL(proposal)[k + 1] += REAL(scale)[m++] * u; } } } mcmc/src/getScalarInteger.c0000644000175100001440000000066513074514645015405 0ustar hornikusers #include #include #include "myutil.h" int getScalarInteger(SEXP foo, char *argname) { if (! isNumeric(foo)) error("argument \"%s\" must be numeric", argname); if (LENGTH(foo) != 1) error("argument \"%s\" must be scalar", argname); if (isInteger(foo)) { return INTEGER(foo)[0]; } else { SEXP bar = coerceVector(foo, INTSXP); return INTEGER(bar)[0]; } } mcmc/src/olbm.c0000644000175100001440000000324313074514645013106 0ustar hornikusers #include #include "mcmc.h" /* overlapping batch means for vector time series * * input: * * x time series, n x p matrix, n time points and p components * len batch length * * output: * * mean sample mean, a p vector * var estimated variance of sample mean, a p x p matrix * */ #define X(I,J) x[(I) + n * (J)] #define VAR(I,J) var[(I) + p * (J)] void olbm(double *x, int *nin, int *pin, int *lin, double *mean, double *var, int *nocalcin) { int n = nin[0]; int p = pin[0]; int len = lin[0]; double nbatch = n - len + 1; int nocalc = nocalcin[0]; double *work = (double *) R_alloc(p, sizeof(double)); int i, j, k, l; if (len > n) error("len > n\n"); if (! nocalc) for (i=0; i=0; j--) VAR(i,j) = (work[i] - mean[i]) * (work[j] - mean[j]); } for (k=0, l=len; l=0; j--) VAR(i,j) += (work[i] - mean[i]) * (work[j] - mean[j]); } /* fix up means and variances, divide out factors of len and len^2 */ for (i=0; i #include SEXP metrop(SEXP func1, SEXP initial, SEXP nbatch, SEXP blen, SEXP nspac, SEXP scale, SEXP func2, SEXP debug, SEXP rho1, SEXP rho2); SEXP temper(SEXP func1, SEXP initial, SEXP neighbors, SEXP nbatch, SEXP blen, SEXP nspac, SEXP scale, SEXP func2, SEXP debug, SEXP parallel, SEXP rho1, SEXP rho2); SEXP initseq(SEXP x); void olbm(double *x, int *nin, int *pin, int *lin, double *mean, double *var, int *nocalcin); #endif /* MCMC_MCMC_H */ mcmc/src/isAllFinite.c0000644000175100001440000000046013074514645014356 0ustar hornikusers #include #include #include "myutil.h" int isAllFinite(SEXP foo) { int d, i; int result = TRUE; if (! isReal(foo)) error("argument must be real"); d = LENGTH(foo); for (i = 0; i < d; i++) result &= R_finite(REAL(foo)[i]); return result; } mcmc/src/metrop.c0000644000175100001440000004017213074514645013465 0ustar hornikusers #include #include #include #include "myutil.h" #include "mcmc.h" static void proposal_setup(SEXP scale, int d); static void propose(SEXP state, SEXP proposal, double *z); static double logh(SEXP func, SEXP state, SEXP rho); static int out_setup(SEXP func, SEXP rho, SEXP state); static void outfun(SEXP state, SEXP buffer); SEXP metrop(SEXP func1, SEXP initial, SEXP nbatch, SEXP blen, SEXP nspac, SEXP scale, SEXP func2, SEXP debug, SEXP rho1, SEXP rho2) { int int_nbatch, int_blen, int_nspac, int_debug; SEXP state, proposal; int dim_state, dim_out; SEXP result, resultnames, acceptance_rate, path, save_initial, save_final, acceptance_rate_batches; double *batch_buffer; SEXP out_buffer; double acceptances = 0.0; double tries = 0.0; double current_log_dens; if (! isFunction(func1)) error("argument \"func1\" must be function"); if (! isEnvironment(rho1)) error("argument \"rho1\" must be environment"); if (! isNumeric(initial)) error("argument \"initial\" must be numeric"); if (! isNumeric(nbatch)) error("argument \"nbatch\" must be numeric"); if (! isNumeric(blen)) error("argument \"blen\" must be numeric"); if (! isNumeric(nspac)) error("argument \"nspac\" must be numeric"); if (! isNumeric(scale)) error("argument \"scale\" must be numeric"); if (! isLogical(debug)) error("argument \"debug\" must be logical"); int_nbatch = getScalarInteger(nbatch, "nbatch"); int_blen = getScalarInteger(blen, "blen"); int_nspac = getScalarInteger(nspac, "nspac"); int_debug = getScalarLogical(debug, "debug"); if (int_nbatch <= 0) error("argument \"nbatch\" must be positive"); if (int_blen <= 0) error("argument \"blen\" must be positive"); if (int_nspac <= 0) error("argument \"nspac\" must be positive"); PROTECT(state = coerceVector(duplicate(initial), REALSXP)); if (! isAllFinite(state)) error("all elements of \"state\" must be finite"); dim_state = LENGTH(state); PROTECT(proposal = allocVector(REALSXP, dim_state)); proposal_setup(scale, dim_state); dim_out = out_setup(func2, rho2, state); batch_buffer = (double *) R_alloc(dim_out, sizeof(double)); PROTECT(out_buffer = allocVector(REALSXP, dim_out)); if (! int_debug) { PROTECT(result = allocVector(VECSXP, 5)); PROTECT(resultnames = allocVector(STRSXP, 5)); } else { PROTECT(result = allocVector(VECSXP, 11)); PROTECT(resultnames = allocVector(STRSXP, 11)); } PROTECT(acceptance_rate = allocVector(REALSXP, 1)); SET_VECTOR_ELT(result, 0, acceptance_rate); PROTECT(path = allocMatrix(REALSXP, dim_out, int_nbatch)); SET_VECTOR_ELT(result, 1, path); PROTECT(save_initial = duplicate(state)); SET_VECTOR_ELT(result, 2, save_initial); /* cannot set final yet because we haven't got it yet (final value at end of run). See third to last statement of this function. */ PROTECT(acceptance_rate_batches = allocVector(REALSXP, int_nbatch)); SET_VECTOR_ELT(result, 4, acceptance_rate_batches); UNPROTECT(4); SET_STRING_ELT(resultnames, 0, mkChar("accept")); SET_STRING_ELT(resultnames, 1, mkChar("batch")); SET_STRING_ELT(resultnames, 2, mkChar("initial")); SET_STRING_ELT(resultnames, 3, mkChar("final")); SET_STRING_ELT(resultnames, 4, mkChar("accept.batch")); if (int_debug) { SEXP spath, ppath, gpath, upath, zpath, apath; int nn = int_nbatch * int_blen * int_nspac; PROTECT(spath = allocMatrix(REALSXP, dim_state, nn)); SET_VECTOR_ELT(result, 5, spath); PROTECT(ppath = allocMatrix(REALSXP, dim_state, nn)); SET_VECTOR_ELT(result, 6, ppath); PROTECT(gpath = allocVector(REALSXP, nn)); SET_VECTOR_ELT(result, 7, gpath); PROTECT(upath = allocVector(REALSXP, nn)); SET_VECTOR_ELT(result, 8, upath); PROTECT(zpath = allocMatrix(REALSXP, dim_state, nn)); SET_VECTOR_ELT(result, 9, zpath); PROTECT(apath = allocVector(LGLSXP, nn)); SET_VECTOR_ELT(result, 10, apath); UNPROTECT(6); SET_STRING_ELT(resultnames, 5, mkChar("current")); SET_STRING_ELT(resultnames, 6, mkChar("proposal")); SET_STRING_ELT(resultnames, 7, mkChar("log.green")); SET_STRING_ELT(resultnames, 8, mkChar("u")); SET_STRING_ELT(resultnames, 9, mkChar("z")); SET_STRING_ELT(resultnames, 10, mkChar("debug.accept")); } namesgets(result, resultnames); UNPROTECT(1); GetRNGstate(); current_log_dens = logh(func1, state, rho1); if (current_log_dens == R_NegInf) error("log unnormalized density -Inf at initial state"); for (int ibatch = 0, k = 0; ibatch < int_nbatch; ibatch++) { double acceptances_this_batch = 0.0; double tries_this_batch = 0.0; for (int i = 0; i < dim_out; i++) batch_buffer[i] = 0.0; for (int jbatch = 0; jbatch < int_blen; jbatch++) { double proposal_log_dens; for (int ispac = 0; ispac < int_nspac; ispac++) { int accept; double u = -1.0; /* impossible return from unif_rand() */ double z[dim_state]; /* buffer for output of norm_rand() */ /* Note: should never happen! */ if (current_log_dens == R_NegInf) error("log density -Inf at current state"); propose(state, proposal, z); proposal_log_dens = logh(func1, proposal, rho1); accept = FALSE; if (proposal_log_dens != R_NegInf) { if (proposal_log_dens > current_log_dens) { accept = TRUE; } else { double green = exp(proposal_log_dens - current_log_dens); u = unif_rand(); accept = u < green; } } if (int_debug) { int l = ispac + int_nspac * (jbatch + int_blen * ibatch); int lbase = l * dim_state; SEXP spath = VECTOR_ELT(result, 5); SEXP ppath = VECTOR_ELT(result, 6); SEXP gpath = VECTOR_ELT(result, 7); SEXP upath = VECTOR_ELT(result, 8); SEXP zpath = VECTOR_ELT(result, 9); SEXP apath = VECTOR_ELT(result, 10); for (int lj = 0; lj < dim_state; lj++) { REAL(spath)[lbase + lj] = REAL(state)[lj]; REAL(ppath)[lbase + lj] = REAL(proposal)[lj]; REAL(zpath)[lbase + lj] = z[lj]; } REAL(gpath)[l] = proposal_log_dens - current_log_dens; if (u == -1.0) REAL(upath)[l] = NA_REAL; else REAL(upath)[l] = u; LOGICAL(apath)[l] = accept; } if (accept) { for (int jj = 0; jj < dim_state; jj++) REAL(state)[jj] = REAL(proposal)[jj]; current_log_dens = proposal_log_dens; acceptances++; acceptances_this_batch++; } tries++; tries_this_batch++; } /* end of inner loop (one iteration) */ outfun(state, out_buffer); for (int j = 0; j < dim_out; j++) batch_buffer[j] += REAL(out_buffer)[j]; } /* end of middle loop (one batch) */ for (int j = 0; j < dim_out; j++, k++) REAL(path)[k] = batch_buffer[j] / int_blen; REAL(acceptance_rate_batches)[ibatch] = acceptances_this_batch / tries_this_batch; } /* end of outer loop */ PutRNGstate(); REAL(acceptance_rate)[0] = acceptances / tries; PROTECT(save_final = coerceVector(state, REALSXP)); SET_VECTOR_ELT(result, 3, save_final); UNPROTECT(5); return result; } static double logh(SEXP func, SEXP state, SEXP rho) { SEXP call, result, foo; double bar; PROTECT(call = lang2(func, state)); PROTECT(result = eval(call, rho)); if (! isNumeric(result)) error("logh: result of function call must be numeric"); if (LENGTH(result) != 1) error("logh: result of function call must be scalar"); PROTECT(foo = coerceVector(result, REALSXP)); bar = REAL(foo)[0]; UNPROTECT(3); if (bar == R_PosInf) error("logh: func returned +Inf"); if (R_IsNaN(bar) || R_IsNA(bar)) error("logh: func returned NA or NaN"); /* Note: -Inf is allowed */ return bar; } static double *scale_factor; static double scale_factor_buffer; static int scale_option; static int state_dimension; #define CONSTANT 1 #define DIAGONAL 2 #define FULL 3 static void proposal_setup(SEXP scale, int d) { SEXP foo; state_dimension = d; PROTECT(foo = coerceVector(scale, REALSXP)); if (isMatrix(scale)) { SEXP bar; PROTECT(bar = getAttrib(scale, R_DimSymbol)); if (INTEGER(bar)[0] == d && INTEGER(bar)[1] == d) { scale_factor = (double *) R_alloc(d * d, sizeof(double)); for (int i = 0; i < d * d; i++) scale_factor[i] = REAL(foo)[i]; scale_option = FULL; } else { error("dimensions of \"scale\" matrix not d by d"); } UNPROTECT(1); } else if (LENGTH(foo) == d) { scale_factor = (double *) R_alloc(d, sizeof(double)); for (int i = 0; i < d; i++) scale_factor[i] = REAL(foo)[i]; scale_option = DIAGONAL; } else if (LENGTH(foo) == 1) { scale_factor = &scale_factor_buffer; scale_factor[0] = REAL(foo)[0]; scale_option = CONSTANT; } else { error("length of \"scale\" vector not d or 1"); } UNPROTECT(1); } static void propose(SEXP state, SEXP proposal, double *z) { int d = state_dimension; if (scale_option == 0) error("attempt to call propose without setup"); if (LENGTH(state) != d || LENGTH(proposal) != d) error("State or proposal length different from initialization\n"); for (int j = 0; j < d; j++) z[j] = norm_rand(); switch (scale_option) { case CONSTANT: for (int j = 0; j < d; j++) REAL(proposal)[j] = REAL(state)[j] + scale_factor[0] * z[j]; break; case DIAGONAL: for (int j = 0; j < d; j++) REAL(proposal)[j] = REAL(state)[j] + scale_factor[j] * z[j]; break; case FULL: for (int j = 0; j < d; j++) REAL(proposal)[j] = REAL(state)[j]; for (int i = 0, k = 0; i < d; i++) { double u = z[i]; for (int j = 0; j < d; j++) REAL(proposal)[j] += scale_factor[k++] * u; } break; default: error("bogus scaling option\n"); } } static SEXP out_func; static SEXP out_env; static int *out_index; static int out_option; static int out_dimension; static int out_state_dimension; #define OUT_FUNCTION 1 #define OUT_INDEX 2 #define OUT_IDENTITY 3 static int out_setup(SEXP func, SEXP rho, SEXP state) { out_state_dimension = LENGTH(state); if (func == R_NilValue) { out_option = OUT_IDENTITY; out_dimension = out_state_dimension; out_func = R_NilValue; out_env = R_NilValue; } else if (isFunction(func)) { if (! isEnvironment(rho)) error("out_setup: argument \"rho\" must be environment"); out_option = OUT_FUNCTION; out_func = func; out_env = rho; out_dimension = LENGTH(eval(lang2(func, state), rho)); } else if (isLogical(func)) { if (LENGTH(func) != out_state_dimension) error("is.logical(outfun) & (length(outfun) != length(initial))"); out_option = OUT_INDEX; out_index = (int *) R_alloc(out_state_dimension, sizeof(int)); out_dimension = 0; for (int i = 0; i < out_state_dimension; i++) { out_index[i] = LOGICAL(func)[i]; out_dimension += out_index[i]; } } else if (isNumeric(func)) { SEXP foo; int foopos = 0; int fooneg = 0; PROTECT(foo = coerceVector(func, REALSXP)); int foolen = LENGTH(foo); for (int i = 0; i < foolen; i++) { double foodble = REAL(foo)[i]; if (ISNAN(foodble)) error("NA or NaN index for outfun"); if (! R_FINITE(foodble)) error("-Inf or Inf index for outfun"); int fooint = foodble; int fooabs = fooint >= 0 ? fooint : (- fooint); if (fooint == 0) error("is.numeric(outfun) & any(outfun == 0)"); if (foodble != fooint) error("is.numeric(outfun) & any(outfun != as.integer(outfun))"); if (fooabs > out_state_dimension) error("is.numeric(outfun) & any(abs(outfun) > length(initial)"); if (foodble > 0) foopos++; else if (foodble < 0) fooneg++; } if ((foopos > 0) && (fooneg > 0)) error("is.numeric(outfun) & any(outfun > 0) & any(outfun < 0)"); out_option = OUT_INDEX; out_index = (int *) R_alloc(out_state_dimension, sizeof(int)); if (foopos > 0) { for (int i = 0; i < out_state_dimension; i++) out_index[i] = FALSE; for (int i = 0; i < foolen; i++) { int fooint = REAL(foo)[i]; out_index[fooint - 1] = TRUE; } } else /* (fooneg > 0) */ { for (int i = 0; i < out_state_dimension; i++) out_index[i] = TRUE; for (int i = 0; i < foolen; i++) { int fooint = REAL(foo)[i]; int fooabs = (- fooint); out_index[fooabs - 1] = FALSE; } } out_dimension = 0; for (int i = 0; i < out_state_dimension; i++) out_dimension += out_index[i]; UNPROTECT(1); } else { error("outfun must be NULL, a function, a numeric vector," " or a logical vector"); } return out_dimension; } static void outfun(SEXP state, SEXP buffer) { if (out_option == 0) error("attempt to call outfun without setup"); if (LENGTH(state) != out_state_dimension) error("outfun: state length different from initialization"); if (! isReal(buffer)) error("outfun: buffer must be real"); if (LENGTH(buffer) != out_dimension) error("outfun: buffer length different from initialization"); switch (out_option) { case OUT_IDENTITY: for (int j = 0; j < out_state_dimension; j++) REAL(buffer)[j] = REAL(state)[j]; break; case OUT_INDEX: for (int j = 0, k = 0; j < out_state_dimension; j++) if (out_index[j]) REAL(buffer)[k++] = REAL(state)[j]; break; case OUT_FUNCTION: { SEXP call, result, foo; PROTECT(call = lang2(out_func, state)); PROTECT(result = eval(call, out_env)); if (! isNumeric(result)) error("outfun: result of function call must be numeric"); PROTECT(foo = coerceVector(result, REALSXP)); if (! isAllFinite(foo)) error("outfun returned vector with non-finite element"); if (LENGTH(foo) != out_dimension) error("outfun return vector length changed from initial"); for (int k = 0; k < out_dimension; k++) REAL(buffer)[k] = REAL(foo)[k]; UNPROTECT(3); } break; default: error("bogus out option\n"); } } mcmc/src/initseq.c0000644000175100001440000000716213074514645013635 0ustar hornikusers #include #include #include "myutil.h" #include "mcmc.h" SEXP initseq(SEXP x) { SEXP xreal; if (! isNumeric(x)) error("argument must be numeric"); PROTECT(xreal = coerceVector(x, REALSXP)); if (! isAllFinite(x)) error("all elements of argument must be finite"); int len = LENGTH(xreal); double *buff = (double *) R_alloc(len / 2, sizeof(double)); int i; double gamma_zero = 0.0; /* for gcc -Wall -Wextra */ for (i = 0; i < len / 2; ++i) { int lag1 = 2 * i; double gam1 = 0.0; for (int j = 0; j + lag1 < len; ++j) gam1 += REAL(xreal)[j] * REAL(xreal)[j + lag1]; gam1 /= len; if (i == 0) gamma_zero = gam1; int lag2 = lag1 + 1; double gam2 = 0.0; for (int j = 0; j + lag2 < len; ++j) gam2 += REAL(xreal)[j] * REAL(xreal)[j + lag2]; gam2 /= len; buff[i] = gam1 + gam2; if (buff[i] < 0.0) { buff[i] = 0.0; ++i; break; } } SEXP gamma_pos, gamma_dec, gamma_con; PROTECT(gamma_pos = allocVector(REALSXP, i)); for (int j = 0; j < i; ++j) REAL(gamma_pos)[j] = buff[j]; for (int j = 1; j < i; ++j) if (buff[j] > buff[j - 1]) buff[j] = buff[j - 1]; PROTECT(gamma_dec = allocVector(REALSXP, i)); for (int j = 0; j < i; ++j) REAL(gamma_dec)[j] = buff[j]; for (int j = i - 1; j > 0; --j) buff[j] -= buff[j - 1]; /* Pool Adjacent Violators Algorithm (PAVA) */ double *puff = (double *) R_alloc(i, sizeof(double)); int *nuff = (int *) R_alloc(i, sizeof(int)); int nstep = 0; for (int j = 1; j < i; ++j) { puff[nstep] = buff[j]; nuff[nstep] = 1; ++nstep; while(nstep > 1 && puff[nstep - 1] / nuff[nstep - 1] < puff[nstep - 2] / nuff[nstep - 2]) { puff[nstep - 2] += puff[nstep - 1]; nuff[nstep - 2] += nuff[nstep - 1]; --nstep; } } for (int jstep = 0, j = 1; jstep < nstep; ++jstep) { double muff = puff[jstep] / nuff[jstep]; for (int k = 0; k < nuff[jstep]; ++j, ++k) buff[j] = buff[j - 1] + muff; } PROTECT(gamma_con = allocVector(REALSXP, i)); for (int j = 0; j < i; ++j) REAL(gamma_con)[j] = buff[j]; double var_pos = 0.0; double var_dec = 0.0; double var_con = 0.0; for (int j = 0; j < i; ++j) { var_pos += REAL(gamma_pos)[j]; var_dec += REAL(gamma_dec)[j]; var_con += REAL(gamma_con)[j]; } var_pos *= 2.0; var_dec *= 2.0; var_con *= 2.0; var_pos -= gamma_zero; var_dec -= gamma_zero; var_con -= gamma_zero; SEXP result, resultnames; PROTECT(result = allocVector(VECSXP, 7)); PROTECT(resultnames = allocVector(STRSXP, 7)); SET_VECTOR_ELT(result, 0, ScalarReal(gamma_zero)); SET_STRING_ELT(resultnames, 0, mkChar("gamma0")); SET_VECTOR_ELT(result, 1, gamma_pos); SET_STRING_ELT(resultnames, 1, mkChar("Gamma.pos")); SET_VECTOR_ELT(result, 2, gamma_dec); SET_STRING_ELT(resultnames, 2, mkChar("Gamma.dec")); SET_VECTOR_ELT(result, 3, gamma_con); SET_STRING_ELT(resultnames, 3, mkChar("Gamma.con")); SET_VECTOR_ELT(result, 4, ScalarReal(var_pos)); SET_STRING_ELT(resultnames, 4, mkChar("var.pos")); SET_VECTOR_ELT(result, 5, ScalarReal(var_dec)); SET_STRING_ELT(resultnames, 5, mkChar("var.dec")); SET_VECTOR_ELT(result, 6, ScalarReal(var_con)); SET_STRING_ELT(resultnames, 6, mkChar("var.con")); namesgets(result, resultnames); UNPROTECT(6); return result; } mcmc/src/getScalarLogical.c0000644000175100001440000000046613074514645015361 0ustar hornikusers #include #include #include "myutil.h" int getScalarLogical(SEXP foo, char *argname) { if (! isLogical(foo)) error("argument \"%s\" must be logical", argname); if (LENGTH(foo) != 1) error("argument \"%s\" must be scalar", argname); return LOGICAL(foo)[0]; } mcmc/src/init.c0000644000175100001440000000127313074514645013121 0ustar hornikusers #include #include #include #include "mcmc.h" static R_NativePrimitiveArgType olbm_types[7] = {REALSXP, INTSXP, INTSXP, INTSXP, REALSXP, REALSXP, LGLSXP}; static R_CMethodDef cMethods[] = { {"olbm", (DL_FUNC) &olbm, 7, olbm_types}, {NULL, NULL, 0, NULL} }; static R_CallMethodDef callMethods[] = { {"metrop", (DL_FUNC) &metrop, 10}, {"temper", (DL_FUNC) &temper, 12}, {"initseq", (DL_FUNC) &initseq, 1}, {NULL, NULL, 0} }; void attribute_visible R_init_mcmc(DllInfo *info) { R_registerRoutines(info, cMethods, callMethods, NULL, NULL); R_useDynamicSymbols(info, FALSE); R_forceSymbols(info, TRUE); } mcmc/src/getListElement.c0000644000175100001440000000066613074514645015110 0ustar hornikusers #include #include #include "myutil.h" SEXP getListElement(SEXP list, char *str) { SEXP elmt = R_NilValue; SEXP names = getAttrib(list, R_NamesSymbol); int i; if (names == R_NilValue) return R_NilValue; for (i = 0; i < length(list); i++) if(strcmp(CHAR(STRING_ELT(names, i)), str) == 0) { elmt = VECTOR_ELT(list, i); break; } return elmt; } mcmc/src/myutil.h0000644000175100001440000000042713074514645013506 0ustar hornikusers #ifndef MCMC_MYUTIL_H #define MCMC_MYUTIL_H #include #include SEXP getListElement(SEXP list, char *str); int getScalarInteger(SEXP foo, char *argname); int getScalarLogical(SEXP foo, char *argname); int isAllFinite(SEXP foo); #endif /* MCMC_MYUTIL_H */ mcmc/NAMESPACE0000644000175100001440000000066113062534650012435 0ustar hornikusers useDynLib(mcmc, .registration = TRUE, .fixes = "C_") export(metrop) export(morph) export(olbm) export(initseq) export(temper) export(morph.metrop) export(morph) export(morph.identity) S3method(metrop, metropolis) S3method(metrop, "function") S3method(morph.metrop, morph.metropolis) S3method(morph.metrop, "function") S3method(temper, tempering) S3method(temper, "function") importFrom(stats, runif) importFrom(compiler, cmpfun) mcmc/data/0000755000175100001440000000000013074514645012131 5ustar hornikusersmcmc/data/foo.txt.gz0000644000175100001440000000542113074514645014076 0ustar hornikusers=[r 9 E{8 vֳ9•NI_?gݧVg/Yߛ79uNwZk]Y?tz徶+mV>Ӟ[٬Otin;o%~X{B^!ڝNoIbNNjnHIpR2lÐy " $Ь ЙIZwD]P gllf$^}aap,v`s [xx16<3sH3^&'QoQ$~9A[Mn)PJX E&9h/;Ժp.a Hb!J~6턮|R\L"ˢ{~@ sFIDJQ neϴ'mɳM@!ΙSL%ˌGW;lR}T?Y%KJrSD HTLN\OƁ4.f?; l(Qa,yUWHNggwC?ėd̉$BJ(mqtI.WB^֎ѐL Ѥ)ɃH5)VOC!E =Ks HފdS.cj0Rp&-lt4 j]=\cLyi2fbP(JgS%D9${=}87em Q $A ?x{I5j)P 4ug}]/!J$ۣoR3o}P5NN^x8,\Bj([ M&=bLy_D,yˍ`E-`n,.hc~&kBW*U5j& TcM܄}d$gL@Ɂ5_נ%4MS e!%]'9KD)"0KytHx>|_CxM!Nu4Q!Y9}y¨~d8 A-Z[u 7w# lp,bO,Dvh{_z_ B<˾(A 3Rc "!ҌK D>726p$I#zW\=QoQxjuqS\_C$ NAs~-VIEϔگI!ԧlU'(| IsƢ13Ͻl(C+V %Eїo,BC95]pNF\ǹRz>1mcmc/data/logit.txt.gz0000644000175100001440000000210613074514645014426 0ustar hornikusersUV]$7{S27>F7A~}~w_eq'-K?y*TY"=L 5/LYT{XCw{ \f)8\&};5g1{ƞ,r{xS>x fqim9>&^"3f)`T%"fE'[թ-9yAgGr u9+Fᰞ2Er~Jn5ypju\rJ/7eqexRi kB>vk７CYvhWڵM퓦/ #W0}tr}IĤ aGЩLjT}),C.:aߵU@PK9݋ ~^(`̿)Q DIJ9G7t>hK8Es[t7rDܣ܄EaetYm+OʫQYAsEMLObzlp|7^W)29^%`HW$Wߡ;9u_g: mcmc/R/0000755000175100001440000000000013074143477011422 5ustar hornikusersmcmc/R/olbm.R0000644000175100001440000000113113062534551012464 0ustar hornikusers olbm <- function(x, batch.length, demean = TRUE) { x <- as.matrix(x) n <- nrow(x) p <- ncol(x) storage.mode(x) <- "double" if (batch.length > n) stop("batch.length must be <= nrow(x)") if (demean) { mean <- apply(x, 2, mean) no.calc.mean <- TRUE } else { mean <- double(p) no.calc.mean <- FALSE } out <- .C(C_olbm, x=x, n=as.integer(n), p=as.integer(p), batch.length=as.integer(batch.length), mean=as.double(mean), var=matrix(as.double(0), p, p), no.calc.mean=as.logical(no.calc.mean)) return(out$var) } mcmc/R/morph.R0000644000175100001440000001211213045741600012655 0ustar hornikuserseuclid.norm <- function(x) { sqrt(sum(x * x)) } isotropic <- function(f) { force(f) function(x) { x.norm <- euclid.norm(x) if (x.norm == 0) rep(0, length(x)) else f(x.norm) * x / x.norm } } isotropic.logjacobian <- function(f, d.f) { force(f) force(d.f) function(x) { x.norm <- euclid.norm(x) k <- length(x) if (x.norm == 0) { k * log(d.f(x.norm)) } else { log(d.f(x.norm)) + (k - 1) * (log(f(x.norm)) - log(x.norm)) } } } # rearranged by charlie # add starting point so we can use a better starting point for # the current actual use # also do one more Newton step after error is < sqrt(machine.eps) # so ultimate error is about machine.eps newton.raphson <- function(f, df, x, r, x0 = 2 * r) { f.err <- function(cur) f(cur) - x step <- function(cur, err=NULL) cur - ifelse(is.null(err), f.err(cur), err) / df(cur) err <- f.err(x0) while(err >= sqrt(.Machine$double.eps)) { x0 <- step(x0, err) err <- f.err(x0) } # if you don't want to use an extra-step, replace this return with # return(x0) return(step(x0, err)) } subexponential <- function(b=1) { if (missing(b) | is.null(b)) b <- 1 stopifnot(b > 0) force(b) f.inv <- function(x) ifelse(x > 1/b, exp(b * x) - exp(1)/3, (x * b)^3 * exp(1) / 6 + x * b * exp(1) / 2) d.f.inv <- function(x) ifelse(x > 1/b, b * exp(b * x), b * (x * b)^2 * exp(1) / 2 + b * exp(1) / 2) f <- function(x) { # x > exp(b * 1 / b) - exp(1) / 3 if (x > 2 * exp(1) / 3) { log(x + exp(1)/3) / b } else { poly.inv <- exp(1/3) * (sqrt(b^12 * (9 * x^2 + exp(2))) - 3 * b^6 * x)^(-1/3) poly.inv * b - 1 / (poly.inv * b^3) } } return(list(f=f, f.inv=f.inv, d.f.inv=d.f.inv)) } exponential <- function(r=1, p=3) { if (missing(p) || is.null(p)) p <- 3 if (missing(r) || is.null(r)) r <- 0 stopifnot(p > 2) stopifnot(r >= 0) f.inv <- function(x) ifelse(x <= r, x, x + (x-r)^p) d.f.inv <- function(x) ifelse(x <= r, 1, 1 + p * (x-r)^(p-1)) if (p == 3) { g <- function(x) { n <- sqrt((27*r-27*x)^2 + 108) + 27 * (r - x) r + (2/n)^(1/3) - (n/2)^(1/3)/3 } f <- function(x) ifelse(x < r, x, g(x)) } else { # No general closed form solution exists. However, since the # transformation has polynomial form, using the Newton-Raphson method # should work well. f <- function(x) ifelse(x < r, x, newton.raphson(f.inv, d.f.inv, x, r, x0 = r + x^(1 / p))) } return(list(f=f, f.inv=f.inv, d.f.inv=d.f.inv)) } .make.outfun <- function(out) { force(out) function(f) { force(f) if (is.null(f)) return(out$inverse) else if (is.function(f)) return(function(state, ...) f(out$inverse(state), ...)) else return(function(state) out$inverse(state)[f]) } } identity.func <- function(x) x morph.identity <- function() { out <- list(transform=identity.func, inverse=identity.func, lud=function(f) function(x, ...) f(x, ...), log.jacobian=function(x) 0, center=0, f=identity.func, f.inv=identity.func) out$outfun <- .make.outfun(out) return(out) } morph <- function(b, r, p, center) { if (all(missing(b), missing(r), missing(p), missing(center))) return(morph.identity()) if (missing(center)) center <- 0 use.subexpo <- !missing(b) use.expo <- !(missing(r) && missing(p)) if (!use.expo && !use.subexpo) { f <- function(x) x f.inv <- function(x) x log.jacobian <- function(x) 0 } else { if (use.expo && !use.subexpo) { expo <- exponential(r, p) f <- expo$f f.inv <- expo$f.inv d.f.inv <- expo$d.f.inv } else if (!use.expo && use.subexpo) { subexpo <- subexponential(b) f <- subexpo$f f.inv <- subexpo$f.inv d.f.inv <- subexpo$d.f.inv } else { #use.expo && use.subexpo expo <- exponential(r, p) subexpo <- subexponential(b) f <- function(x) expo$f(subexpo$f(x)) f.inv <- function(x) subexpo$f.inv(expo$f.inv(x)) d.f.inv <- function(x) expo$d.f.inv(x) * subexpo$d.f.inv(expo$f.inv(x)) } f <- isotropic(f) f.inv <- isotropic(f.inv) log.jacobian <- isotropic.logjacobian(f.inv, d.f.inv) } out <- list(f=f, f.inv=f.inv, log.jacobian=log.jacobian, center=center) out$transform <- function(state) out$f(state - out$center) out$inverse <- function(state) out$f.inv(state) + out$center out$outfun <- .make.outfun(out) out$lud <- function(lud) { force(lud) function(state, ...) { foo <- lud(out$inverse(state), ...) if (length(foo) != 1) stop("log unnormalized density function returned vector not scalar") if (is.na(foo)) stop("log unnormalized density function returned NA or NaN") if (foo == -Inf) return(foo) if (! is.finite(foo)) stop("log unnormalized density function returned +Inf") foo + out$log.jacobian(state) } } return(out) } mcmc/R/temper.R0000644000175100001440000000557713074142671013052 0ustar hornikusers temper <- function(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, ...) UseMethod("temper") temper.tempering <- function(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, ...) { # like metrop, ignore initial argument initial <- obj$final # makes no (at least little?) sense to change neighbor structure neighbors <- obj$neighbors if (missing(nbatch)) nbatch <- obj$nbatch if (missing(blen)) blen <- obj$blen if (missing(nspac)) nspac <- obj$nspac if (missing(scale)) scale <- obj$scale if (missing(debug)) debug <- obj$debug # makes no sense to change from parallel to serial or vice versa # size and shape of state wouldn't even be the same parallel <- obj$parallel assign(".Random.seed", obj$final.seed, .GlobalEnv) if (missing(outfun)) { if (is.null(obj$outfun)) { temper.function(obj$lud, initial, neighbors, nbatch, blen, nspac, scale, debug = debug, parallel = parallel, ...) } else { temper.function(obj$lud, initial, neighbors, nbatch, blen, nspac, scale, obj$outfun, debug = debug, parallel = parallel, ...) } } else { temper.function(obj$lud, initial, neighbors, nbatch, blen, nspac, scale, outfun, debug = debug, parallel = parallel, ...) } } temper.function <- function(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, ...) { if (! exists(".Random.seed")) runif(1) saveseed <- .Random.seed obj <- cmpfun(obj) func1 <- function(state) obj(state, ...) func1 <- cmpfun(func1) env1 <- environment(fun = func1) if (missing(outfun)) { func2 <- NULL env2 <- NULL outfun <- NULL } else if (is.function(outfun)) { outfun <- cmpfun(outfun) func2 <- function(state) outfun(state, ...) func2 <- cmpfun(func2) env2 <- environment(fun = func2) } stopifnot(is.numeric(initial)) storage.mode(initial) <- "double" if (is.list(scale)) { for (i in 1:length(scale)) { stopifnot(is.numeric(scale[[i]])) storage.mode(scale[[i]]) <- "double" } } else { stopifnot(is.numeric(scale)) storage.mode(scale) <- "double" } out.time <- system.time( out <- .Call(C_temper, func1, initial, neighbors, nbatch, blen, nspac, scale, func2, debug, parallel, env1, env2) ) result <- structure(c(list(lud = obj, neighbors = neighbors, nbatch = nbatch, blen = blen, nspac = nspac, scale = scale, outfun = outfun, debug = debug, parallel = parallel, initial.seed = saveseed, final.seed = .Random.seed, time = out.time), out), class = c("mcmc", "tempering")) return(result) } mcmc/R/metrop.R0000644000175100001440000000435513062616565013062 0ustar hornikusers metrop <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) UseMethod("metrop") metrop.metropolis <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) { if (missing(nbatch)) nbatch <- obj$nbatch if (missing(blen)) blen <- obj$blen if (missing(nspac)) nspac <- obj$nspac if (missing(scale)) scale <- obj$scale if (missing(debug)) debug <- obj$debug assign(".Random.seed", obj$final.seed, .GlobalEnv) if (missing(outfun)) { if (is.null(obj$outfun)) { metrop.function(obj$lud, obj$final, nbatch, blen, nspac, scale, debug = debug, ...) } else { metrop.function(obj$lud, obj$final, nbatch, blen, nspac, scale, obj$outfun, debug, ...) } } else { metrop.function(obj$lud, obj$final, nbatch, blen, nspac, scale, outfun, debug, ...) } } metrop.function <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) { if (! exists(".Random.seed")) runif(1) saveseed <- .Random.seed obj <- cmpfun(obj) func1 <- function(state) obj(state, ...) func1 <- cmpfun(func1) env1 <- environment(fun = func1) if (missing(outfun)) { func2 <- NULL env2 <- NULL outfun <- NULL } else if (is.function(outfun)) { outfun <- cmpfun(outfun) func2 <- function(state) outfun(state, ...) func2 <- cmpfun(func2) env2 <- environment(fun = func2) } else { func2 <- outfun env2 <- NULL } out.time <- system.time( out <- .Call(C_metrop, func1, initial, nbatch, blen, nspac, scale, func2, debug, env1, env2) ) out$initial.seed <- saveseed out$final.seed <- .Random.seed out$time <- out.time out$lud <- obj out$nbatch <- nbatch out$blen <- blen out$nspac <- nspac out$scale <- scale out$outfun <- outfun out$batch <- t(out$batch) out$debug <- debug if (! is.null(out$current)) out$current <- t(out$current) if (! is.null(out$proposal)) out$proposal <- t(out$proposal) if (! is.null(out$z)) out$z <- t(out$z) class(out) <- c("mcmc", "metropolis") return(out) } mcmc/R/morph.metrop.R0000644000175100001440000000421113045741600014163 0ustar hornikusersmorph.metrop <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, morph, ...) UseMethod("morph.metrop") morph.metrop.morph.metropolis <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, morph, ...) { if (missing(morph)) { morph <- obj$morph obj$final <- obj$morph.final } else { # if the transformation was changed, transform the last state from the # original space to be the initial state. obj$final <- morph$transform(obj$final) } if (missing(outfun)) outfun <- obj$outfun if (missing(blen)) blen <- obj$blen if (missing(nspac)) nspac <- obj$nspac if (missing(debug)) debug <- obj$debug if (missing(scale)) scale <- obj$scale morphed.obj <- metrop.metropolis(obj, nbatch=nbatch, blen=blen, nspac=nspac, scale=scale, outfun=morph$outfun(outfun), debug=debug, ...) unmorphed.obj <- .morph.unmorph(morphed.obj, morph, outfun) return(unmorphed.obj) } morph.metrop.function <- function(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, morph, ...) { if (missing(morph)) morph <- morph.identity() if (missing(outfun)) outfun <- NULL morphed.obj <- metrop.function(morph$lud(obj), initial=morph$transform(initial), nbatch=nbatch, blen=blen, scale=scale, outfun=morph$outfun(outfun), debug=debug, ...) unmorphed.obj <- .morph.unmorph(morphed.obj, morph, outfun) return(unmorphed.obj) } .morph.unmorph <- function(obj, morph, outfun) { obj$morph <- morph obj$morph.final <- obj$final obj$final <- morph$inverse(obj$final) obj$outfun <- outfun class(obj) <- c("mcmc", "morph.metropolis") return(obj) } mcmc/R/initseq.R0000644000175100001440000000016613062534425013216 0ustar hornikusersinitseq <- function(x) { stopifnot(is.numeric(x)) stopifnot(is.finite(x)) .Call(C_initseq, x - mean(x)) } mcmc/vignettes/0000755000175100001440000000000013074514644013227 5ustar hornikusersmcmc/vignettes/debug.Rnw0000644000175100001440000003135013045741600014777 0ustar hornikusers \documentclass{article} \usepackage{amstext} % \VignetteIndexEntry{Debugging MCMC Code} \begin{document} <>= options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") @ \title{Debugging MCMC Code} \author{Charles J. Geyer} \maketitle \section{Introduction} This document discusses debugging Markov chain Monte Carlo code using the R contributed package \texttt{mcmc} (Version \Sexpr{foo$Version}) for examples. It also documents the debugging output of the functions \texttt{mcmc} and \texttt{temper}. Debugging MCMC code if the code is taken as a black box is basically impossible. In interesting examples, the only thing one knows about the equilibrium distribution of an MCMC sampler is what one learns from the samples. This obviously doesn't help with testing. If the sampler is buggy, then the only thing you know about the equilibrium distribution is wrong, but if you don't know it is buggy, then you don't know it is wrong. So you don't know anything. There is no way to tell whether random output has the correct distribution when you don't know anything about the distribution it is supposed to have. The secret to debugging MCMC code lies in two principles: \begin{itemize} \item take the randomness out, and \item expose the innards. \end{itemize} The first slogan means consider the algorithm a deterministic function of the elementary pseudo-random numbers that are trusted (for example, the outputs of the R random number generators, which you aren't responsible for debugging and are also well tested). The second slogan means output, at least for debugging purposes, enough intermediate state so that testing is straightforward. For a Gibbs sampler, this means outputting all of the trusted elementary pseudo-random numbers used, the state before and after each elementary Gibbs update, and which update is being done if a random scan is used. Also one needs to output the initial seeds of the pseudo-random number generator (this is true for all situations and will not be mentioned again). For a Metropolis-Hastings sampler, this means outputting all of the trusted elementary pseudo-random numbers used, the state before and after each elementary Metropolis-Hastings update, the proposal for that update, the Hastings ratio for that update, decision (accept or reject) in that update. For more complicated MCMC samplers, there is more ``innards'' to ``expose'' (see the discussion of the \texttt{temper} function below), but you get the idea. You can't output too much debugging information. \section{The Metrop Function} The R function \texttt{metrop} in the \texttt{mcmc} package has an argument \verb@debug = FALSE@ that when \verb@TRUE@ causes extra debugging information to be output. Let \texttt{niter} be the number of iterations \verb@nbatch * blen * nspac@, and let \texttt{d} be the dimension of the state vector. The result of invoking \texttt{metrop} is a list. When \verb@debug = TRUE@ it has the following additional components \begin{itemize} \item \texttt{current}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@current[i, ]@ \item \texttt{proposal}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ \item \texttt{z}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ \item \texttt{log.green}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{u}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{debug.accept}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \end{itemize} (The components \texttt{z} and \texttt{debug.accept} were added in version 0.7-3 of the \texttt{mcmc} package. Before that only the others were output.) Two components of the list returned by the \texttt{metrop} function always (whether \verb@debug = TRUE@ or \verb@debug = FALSE@) are also necessary for debugging. They are \begin{itemize} \item \texttt{initial.seed} the value of the variable \texttt{.Random.seed} that contains the seeds of the R random number generator system before invocation of the \texttt{metrop} function \item \texttt{final}, a vector of length \texttt{d} and mode \verb@"numeric"@, the state after the last iteration \end{itemize} All of the files in the \texttt{tests} directory of the source for the package (not installed but found in the source tarball on CRAN) test the \texttt{metrop} function except those beginning \texttt{temp}, which test the \texttt{temper} function. Since these tests were written many years ago, are spread out over many files, and are not commented, we will not describe them in detail. Suffice it to say that they check every aspect of the functioning of the \texttt{metrop} function. \section{The Temper Function} The R function \texttt{temper} in the \texttt{mcmc} package has an argument \verb@debug = FALSE@ that when \verb@TRUE@ causes extra debugging information to be output. Let \texttt{niter} be the number of iterations \verb@nbatch * blen * nspac@, let \texttt{d} be the dimension of the state vector, and let \texttt{ncomp} be the number of components of the tempering mixture. The result of invoking \texttt{temper} is a list. When \verb@debug = TRUE@ and \verb@parallel = TRUE@ it has the following additional components % which % unif.which % state % log.hastings % unif.hastings % proposal % acceptd % norm % unif.choose % coproposal \begin{itemize} \item \texttt{which}, a vector of length \texttt{niter} and mode \verb@"logical"@ the type of update for each iteration, within component (\texttt{TRUE}) or swap components (\texttt{FALSE}). \item \texttt{unif.which}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate used to decide which type of update is done. \item \texttt{state}, an \texttt{niter} by \texttt{ncomp} by \texttt{d} array of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@state[i, , ]@ \item \texttt{proposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ (explanation below) \item \texttt{coproposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@coproposal[i, ]@ (explanation below) \item \texttt{log.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{unif.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{acceptd}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \item \texttt{norm}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ unless none are needed (for swap updates) when it is \texttt{NA} \item \texttt{unif.choose}, an \texttt{niter} by 2 matrix of mode \verb@"numeric"@, the vector of $\text{Uniform}(0, 1)$ random variates used to choose the components to update in iteration \texttt{i} is \verb@unif.choose[i, ]@; in a swap update two are used; in a within-component update only one is used and the second is \texttt{NA} \end{itemize} In a within-component update, one component say \texttt{j} is chosen for update. The \emph{coproposal} is the current value of the state for this component, which is a vector of length \verb@d + 1@, the first component of which is \texttt{j} and the rest of which is \verb@state[i, j, ]@ if we are in iteration \texttt{i}. The \emph{proposal} is a similar vector, the first component of which is again \texttt{j} and the rest of which is a multivariate normal random vector centered at \verb@state[i, j, ]@. The coproposal is the current state; the proposal is the possible value (if accepted) of the state at the next time. In a swap update, two components say \texttt{j1} and \texttt{j2} are chosen for update. Strictly, speaking the coproposal is the pair of vectors \verb@c(j1, state[i, j1, ])@ and \verb@c(j2, state[i, j2, ])@ and the proposal is these swapped, that is, the pair of vectors \verb@c(j2, state[i, j1, ])@ and \verb@c(j1, state[i, j2, ])@ if we are in iteration \texttt{i}. Since, however, there is a lot of redundant information here, the vector \verb@c(j1, state[i, j1, ])@ is output as \verb@coproposal[i, ]@ and the vector \verb@c(j2, state[i, j2, ])@ is output as \verb@proposal[i, ]@. When \verb@debug = TRUE@ and \verb@parallel = FALSE@ the result of invoking \texttt{temper} is a list having the following additional components % which % unif.which % state % log.hastings % unif.hastings % proposal % acceptd % norm % unif.choose \begin{itemize} \item \texttt{which}, a vector of length \texttt{niter} and mode \verb@"logical"@ the type of update for each iteration, within component (\texttt{TRUE}) or jump from one component to another (\texttt{FALSE}). \item \texttt{unif.which}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate used to decide which type of update is done. \item \texttt{state}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the state before iteration \texttt{i} is \verb@state[i, ]@ \item \texttt{proposal}, an \texttt{niter} by \verb@d + 1@ matrix of mode \verb@"numeric"@, the proposal for iteration \texttt{i} is \verb@proposal[i, ]@ \item \texttt{log.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the logarithm of the Hastings ratio for each iteration \item \texttt{unif.hastings}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variate compared to the Hastings ratio for each iteration or \texttt{NA} if none is needed (when the log Hastings ratio is nonnegative) \item \texttt{acceptd}, a vector of length \texttt{niter} and mode \verb@"logical"@, the decision for each iteration, accept the proposal (\texttt{TRUE}) or reject it (\texttt{FALSE}) \item \texttt{norm}, an \texttt{niter} by \texttt{d} matrix of mode \verb@"numeric"@, the vector of standard normal random variates used to generate the proposal for iteration \texttt{i} is \verb@z[i, ]@ unless none are needed (for jump updates) when it is \texttt{NA} \item \texttt{unif.choose}, a vector of length \texttt{niter} and mode \verb@"numeric"@, the $\text{Uniform}(0, 1)$ random variates used to choose the component to update in iteration \texttt{i} is \verb@unif.choose[i, ]@; in a jump update one is used; in a within-component update none is used and \texttt{NA} is output \end{itemize} All of the files in the \texttt{tests} directory of the source for the package (not installed but found in the source tarball on CRAN) beginning \texttt{temp} test the \texttt{temper} function. They check every aspect of the functioning of the \texttt{temper} function. In the file \texttt{temp-par.R} in the \texttt{tests} directory, the following checks are made according to the comments in that file \begin{enumerate} \item check decision about within-component or jump/swap \item check proposal and coproposal are actually current state or part thereof \item check hastings ratio calculated correctly \item check hastings rejection decided correctly \item check acceptance carried out or not (according to decision) correctly \item check within-component proposal \item check swap proposal \item check standard normal and uniform random numbers are as purported \item check batch means \item check acceptance rates \item check scale vector \item check scale matrix \item check scale list \item check outfun \end{enumerate} In the file \texttt{temp-ser.R} in the \texttt{tests} directory, the all of the same checks are made according to the comments in that file except for check number 2 above, which would make no sense because there is no \texttt{coproposal} component in the serial (\verb@parallel = FALSE@) case. \end{document} mcmc/vignettes/morph2.rda0000644000175100001440000004260413045741600015124 0ustar hornikusers} \Mmߵk!EZ0B*)Em-m}W$HbHDTPBd)տˍU}{?ow9g,̙jRQ#b1=d_R?DRtH-@0OGO?K0EGڵwyI^n B;'o\8;y/lw;k{_C7v !i SOoL TlE-=:<Ȋyssu|pqٵ @lMHhTo<~t.;]PZ =tId7H)w@T4L<<`^^)_K[GBZSJW] ffG:n 5!ĠW +K;+@&IsEE)8 |.Xsؿ JI\a: =zFІeu0NNRa9>Chj45-WaAټ}'a ^7(?lMrA3u*d[ 5O> 8hF PDJ bf"8'I zsyNDX1_٠ќaAN 5k0@ޛ;8,I|=L S~_>@q3[#qi& I2vCTI?W$,Sdzl 9끞'E[rLHo-g@H-yl%P+ᤷfUPB zVNAk@[ÏaK腄)`u$j%&DUE|w.WҀ Ԋa (/rKDKd,]x7jkR=@D=ur>L9o] }gLʚG}pkpNR/P6^Uq>IaSTA/kmI\G|_/;wѺXg! 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Geyer} \maketitle \section{The Problem} This is an example of using the \verb@mcmc@ package in R. The problem comes from a take-home question on a (take-home) PhD qualifying exam (School of Statistics, University of Minnesota). Simulated data for the problem are in the dataset \verb@logit@. There are five variables in the data set, the response \verb@y@ and four predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. A frequentist analysis for the problem is done by the following R statements <>= library(mcmc) data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial(), x = TRUE) summary(out) @ But this problem isn't about that frequentist analysis, we want a Bayesian analysis. For our Bayesian analysis we assume the same data model as the frequentist, and we assume the prior distribution of the five parameters (the regression coefficients) makes them independent and identically normally distributed with mean 0 and standard deviation 2. The log unnormalized posterior (log likelihood plus log prior) density for this model is calculated by the following R function (given the preceding data definitions) <>= x <- out$x y <- out$y lupost <- function(beta, x, y) { eta <- as.numeric(x %*% beta) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) return(logl - sum(beta^2) / 8) } @ The tricky calculation of the log likelihood avoids overflow and catastrophic cancellation in calculation of $\log(p)$ and $\log(q)$ where \begin{align*} p & = \frac{\exp(\eta)}{1 + \exp(\eta)} = \frac{1}{1 + \exp(- \eta)} \\ q & = \frac{1}{1 + \exp(\eta)} = \frac{\exp(- \eta)}{1 + \exp(- \eta)} \end{align*} so taking logs gives \begin{align*} \log(p) & = \eta - \log(1 + \exp(\eta)) = - \log(1 + \exp(- \eta)) \\ \log(q) & = - \log(1 + \exp(\eta)) = - \eta - \log(1 + \exp(- \eta)) \end{align*} To avoid overflow, we always chose the case where the argument of $\exp$ is negative. We have also avoided catastrophic cancellation when $\lvert\eta\rvert$ is large. If $\eta$ is large and positive, then \begin{align*} p & \approx 1 \\ q & \approx 0 \\ \log(p) & \approx - \exp(- \eta) \\ \log(q) & \approx - \eta - \exp(- \eta) \end{align*} and our use of the R function \texttt{log1p}, which calculates the function $x \mapsto \log(1 + x)$ correctly for small $x$ avoids all problems. The case where $\eta$ is large and negative is similar. \section{Beginning MCMC} With those definitions in place, the following code runs the Metropolis algorithm to simulate the posterior. <>= set.seed(42) # to get reproducible results beta.init <- as.numeric(coefficients(out)) out <- metrop(lupost, beta.init, 1e3, x = x, y = y) names(out) out$accept @ The arguments to the \verb@metrop@ function used here (there are others we don't use) are \begin{itemize} \item an R function (here \verb@lupost@) that evaluates the log unnormalized density of the desired stationary distribution of the Markov chain (here a posterior distribution). Note that (although this example does not exhibit the phenomenon) that the unnormalized density may be zero, in which case the log unnormalized density is \verb@-Inf@. \item an initial state (here \verb@beta.init@) of the Markov chain. \item a number of batches (here \verb@1e3@) for the Markov chain. This combines with batch length and spacing (both 1 by default) to determine the number of iterations done. \item additional arguments (here \verb@x@ and \verb@y@) supplied to provided functions (here \verb@lupost@). \item there is no ``burn-in'' argument, although burn-in is easily accomplished, if desired (more on this below). \end{itemize} The output is in the component \verb@out$batch@ returned by the \verb@metrop@ function. We'll look at it presently, but first we need to adjust the proposal to get a higher acceptance rate (\verb@out$accept@). It is generally accepted \citep*{grg} that an acceptance rate of about 20\% is right, although this recommendation is based on the asymptotic analysis of a toy problem (simulating a multivariate normal distribution) for which one would never use MCMC and is very unrepresentative of difficult MCMC applications. \citet{geyer-temp} came to a similar conclusion, that a 20\% acceptance rate is about right, in a very different situation. But they also warned that a 20\% acceptance rate could be very wrong and produced an example where a 20\% acceptance rate was impossible and attempting to reduce the acceptance rate below 70\% would keep the sampler from ever visiting part of the state space. So the 20\% magic number must be considered like other rules of thumb we teach in intro courses (like $n > 30$ means means normal approximation is valid). We know these rules of thumb can fail. There are examples in the literature where they do fail. We keep repeating them because we want something simple to tell beginners, and they are all right for some problems. Be that as it may, we try for 20\%. <>= out <- metrop(out, scale = 0.1, x = x, y = y) out$accept out <- metrop(out, scale = 0.3, x = x, y = y) out$accept out <- metrop(out, scale = 0.5, x = x, y = y) out$accept out <- metrop(out, scale = 0.4, x = x, y = y) out$accept @ Here the first argument to each instance of the \verb@metrop@ function is the output of a previous invocation. The Markov chain continues where the previous run stopped, doing just what it would have done if it had kept going, the initial state and random seed being the final state and random seed of the previous invocation. Everything stays the same except for the arguments supplied (here \verb@scale@). \begin{itemize} \item The argument \verb@scale@ controls the size of the Metropolis ``normal random walk'' proposal. The default is \verb@scale = 1@. Big steps give lower acceptance rates. Small steps give higher. We want something about 20\%. It is also possible to make \verb@scale@ a vector or a matrix. See \verb@help(metrop)@. \end{itemize} Because each run starts where the last one stopped (when the first argument to \verb@metrop@ is the output of the previous invocation), each run serves as ``burn-in'' for its successor (assuming that any part of that run was worth anything at all). \section{Diagnostics} O.~K. That does it for the acceptance rate. So let's do a longer run and look at the results. <>= out <- metrop(out, nbatch = 1e4, x = x, y = y) out$accept out$time @ Figure~\ref{fig:fig1} (page~\pageref{fig:fig1}) shows the time series plot made by the R statement <>= plot(ts(out$batch)) @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Time series plot of MCMC output.} \label{fig:fig1} \end{figure} Another way to look at the output is an autocorrelation plot. Figure~\ref{fig:fig2} (page~\pageref{fig:fig2}) shows the time series plot made by the R statement <>= acf(out$batch) @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot of MCMC output.} \label{fig:fig2} \end{figure} As with any multiplot plot, these are a bit hard to read. Readers are invited to make the separate plots to get a better picture. As with all ``diagnostic'' plots in MCMC, these don't ``diagnose'' subtle problems. As \begin{verbatim} http://www.stat.umn.edu/~charlie/mcmc/diag.html \end{verbatim} says \begin{quotation} The purpose of regression diagnostics is to find obvious, gross, embarrassing problems that jump out of simple plots. \end{quotation} The time series plots will show \emph{obvious} nonstationarity. They will not show \emph{nonobvious} nonstationarity. They provide no guarantee whatsoever that your Markov chain is sampling anything remotely resembling the correct stationary distribution (with log unnormalized density \verb@lupost@). In this very easy problem, we do not expect any convergence difficulties and so believe what the diagnostics seem to show, but one is a fool to trust such diagnostics in difficult problems. The autocorrelation plots seem to show that the the autocorrelations are negligible after about lag 25. This diagnostic inference is reliable if the sampler is actually working (has nearly reached equilibrium) and worthless otherwise. Thus batches of length 25 should be sufficient, but let's use length 100 to be safe. \section{Monte Carlo Estimates and Standard Errors} <>= out <- metrop(out, nbatch = 1e2, blen = 100, outfun = function(z, ...) c(z, z^2), x = x, y = y) out$accept out$time @ We have added an argument \verb@outfun@ that gives the ``functional'' of the state we want to average. For this problem we are interested in both posterior mean and variance. Mean is easy, just average the variables in question. But variance is a little tricky. We need to use the identity $$ \var(X) = E(X^2) - E(X)^2 $$ to write variance as a function of two things that can be estimated by simple averages. Hence we want to average the state itself and the squares of each component. Hence our \verb@outfun@ returns \verb@c(z, z^2)@ for an argument (the state vector) \verb@z@. The \verb@...@ argument to \verb@outfun@ is required, since the function is also passed the other arguments (here \verb@x@ and \verb@y@) to \verb@metrop@. \subsection{Simple Means} The grand means (means of batch means) are <>= apply(out$batch, 2, mean) @ The first 5 numbers are the Monte Carlo estimates of the posterior means. The second 5 numbers are the Monte Carlo estimates of the posterior ordinary second moments. We get the posterior variances by <>= foo <- apply(out$batch, 2, mean) mu <- foo[1:5] sigmasq <- foo[6:10] - mu^2 mu sigmasq @ Monte Carlo standard errors (MCSE) are calculated from the batch means. This is simplest for the means. <>= mu.mcse <- apply(out$batch[ , 1:5], 2, sd) / sqrt(out$nbatch) mu.mcse @ The extra factor \verb@sqrt(out$nbatch)@ arises because the batch means have variance $\sigma^2 / b$ where $b$ is the batch length, which is \verb@out$blen@, whereas the overall means \verb@mu@ have variance $\sigma^2 / n$ where $n$ is the total number of iterations, which is \verb@out$blen * out$nbatch@. \subsection{Functions of Means} To get the MCSE for the posterior variances we apply the delta method. Let $u_i$ denote the sequence of batch means of the first kind for one parameter and $\bar{u}$ the grand mean (the estimate of the posterior mean of that parameter), let $v_i$ denote the sequence of batch means of the second kind for the same parameter and $\bar{v}$ the grand mean (the estimate of the posterior second absolute moment of that parameter), and let $\mu = E(\bar{u})$ and $\nu = E(\bar{v})$. Then the delta method linearizes the nonlinear function $$ g(\mu, \nu) = \nu - \mu^2 $$ as $$ \Delta g(\mu, \nu) = \Delta \nu - 2 \mu \Delta \mu $$ saying that $$ g(\bar{u}, \bar{v}) - g(\mu, \nu) $$ has the same asymptotic normal distribution as $$ (\bar{v} - \nu) - 2 \mu (\bar{u} - \mu) $$ which, of course, has variance \verb@1 / nbatch@ times that of $$ (v_i - \nu) - 2 \mu (u_i - \mu) $$ and this variance is estimated by $$ \frac{1}{n_{\text{batch}}} \sum_{i = 1}^{n_{\text{batch}}} \bigl[ (v_i - \bar{v}) - 2 \bar{u} (u_i - \bar{u}) \bigr]^2 $$ So <>= u <- out$batch[ , 1:5] v <- out$batch[ , 6:10] ubar <- apply(u, 2, mean) vbar <- apply(v, 2, mean) deltau <- sweep(u, 2, ubar) deltav <- sweep(v, 2, vbar) foo <- sweep(deltau, 2, ubar, "*") sigmasq.mcse <- sqrt(apply((deltav - 2 * foo)^2, 2, mean) / out$nbatch) sigmasq.mcse @ does the MCSE for the posterior variance. Let's just check that this complicated \verb@sweep@ and \verb@apply@ stuff does do the right thing. <>= sqrt(mean(((v[ , 2] - vbar[2]) - 2 * ubar[2] * (u[ , 2] - ubar[2]))^2) / out$nbatch) @ \paragraph{Comment} Through version 0.5 of this vignette it contained an incorrect procedure for calculating this MCSE, justified by a handwave (which was incorrect). Essentially, it said to use the standard deviation of the batch means called \verb@v@ here, which appears to be very conservative. \subsection{Functions of Functions of Means} If we are also interested in the posterior standard deviation (a natural question, although not asked on the exam problem), the delta method gives its standard error in terms of that for the variance <>= sigma <- sqrt(sigmasq) sigma.mcse <- sigmasq.mcse / (2 * sigma) sigma sigma.mcse @ \section{A Final Run} So that's it. The only thing left to do is a little more precision (the exam problem directed ``use a long enough run of your Markov chain sampler so that the MCSE are less than 0.01'') <>= out <- metrop(out, nbatch = 5e2, blen = 400, x = x, y = y) out$accept out$time <> <> <> <> @ and some nicer output, which is presented in three tables constructed from the R variables defined above using the R \verb@xtable@ command in the \verb@xtable@ library. First the posterior means, \begin{table}[ht] \caption{Posterior Means} \label{tab:mu} \begin{center} <>= foo <- rbind(mu, mu.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} then the posterior variances (table on page~\pageref{tab:sigmasq}), \begin{table}[ht] \caption{Posterior Variances} \label{tab:sigmasq} \begin{center} <>= foo <- rbind(sigmasq, sigmasq.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} and finally the posterior standard deviations (table on page~\pageref{tab:sigma}). \begin{table}[ht] \caption{Posterior Standard Deviations} \label{tab:sigma} \begin{center} <>= foo <- rbind(sigma, sigma.mcse) dimnames(foo) <- list(c("estimate", "MCSE"), c("constant", paste("$x_", 1:4, "$", sep = ""))) library(xtable) print(xtable(foo, digits = rep(4, 6), align = c("l", rep("c", 5))), floating = FALSE, caption.placement = "top", sanitize.colnames.function = function(x) return(x)) @ \end{center} \end{table} Note for the record that the all the results presented in the tables are from ``one long run'' where long here took only <>= cat(out$time[1], "\n") @ seconds (on whatever computer it was run on). \section{New Variance Estimation Functions} A new function \texttt{initseq} estimates variances in the Markov chain central limit theorem (CLT) following the methodology introduced by \citet[Section~3.3]{practical}. These methods only apply to scalar-valued functionals of reversible Markov chains, but the Markov chains produced by the \texttt{metrop} function satisfy this condition, even, as we shall see below, when batching is used. Rather than redo the Markov chains in the preceding material, we just look at a toy problem, an AR(1) time series, which can be simulated in one line of R. This is the example on the help page for \texttt{initseq}. <>= n <- 2e4 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) @ The time series \texttt{x} is a reversible Markov chain and trivially a scalar-valued functional of a Markov chain. Define \begin{equation} \label{eq:little} \gamma_k = \cov(X_i, X_{i + k}) \end{equation} where the covariances refer to the stationary Markov chain having the same transition probabilities as \texttt{x}. Then the variance in the CLT is $$ \sigma^2 = \gamma_0 + 2 \sum_{k = 1}^\infty \gamma_k $$ \citep[Theorem~2.1]{practical}, that is, $$ \bar{x}_n \approx \text{Normal}\left(\mu, \frac{\sigma^2}{n}\right), $$ where $\mu = E(X_i)$ is the quantity being estimated by MCMC (in this toy problem we know $\mu = 0$). Naive estimates of $\sigma^2$ obtained by plugging in empirical estimates of the gammas do not provide consistent estimation \citep[Section~3.1]{practical}. Thus the scheme implemented by the R function \texttt{initseq}. Define \begin{equation} \label{eq:big} \Gamma_k = \gamma_{2 k} + \gamma_{2 k + 1} \end{equation} \citet[Theorem~3.1]{practical} says that $\Gamma_k$ considered as a function of $k$ is strictly positive, strictly decreasing, and strictly convex (provided we are, as stated above, working with a reversible Markov chain). Thus it makes sense to use estimators that use these properties. The estimators implemented by the R function \texttt{initseq} and described by \citet[Section~3.3]{practical} are conservative-consistent in the sense of Theorem~3.2 of that section. Figure~\ref{fig:gamma} (page~\pageref{fig:gamma}) shows the time series plot made by the R statement <>= out <- initseq(x) plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l") lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, lty = "dotted") lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, lty = "dashed") @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Plot ``Big Gamma'' defined by \eqref{eq:little} and \eqref{eq:big}. Solid line, initial positive sequence estimator. Dotted line, initial monotone sequence estimator. Dashed line, initial convex sequence estimator.} \label{fig:gamma} \end{figure} One can use whichever curve one chooses, but now that the \texttt{initseq} function makes the computation trivial, it makes sense to use the initial convex sequence. Of course, one is not interested in Figure~\ref{fig:gamma}, except perhaps when explaining the methodology. What is actually important is the estimate of $\sigma^2$, which is given by <>= out$var.con (1 + rho) / (1 - rho) * 1 / (1 - rho^2) @ where for comparison we have given the exact theoretical value of $\sigma^2$, which, of course, is never available in a non-toy problem. These initial sequence estimators seem, at first sight to be a competitor for the method of batch means. However, appearances can be deceiving. The two methods are complementary. The sequence of batch means is itself a scalar-valued functional of a reversible Markov chain. Hence the initial sequence estimators can be applied to it. <>= blen <- 5 x.batch <- apply(matrix(x, nrow = blen), 2, mean) bout <- initseq(x.batch) @ Because the batch length is too short, the variance of the batch means does not estimate $\sigma^2$. We must account for the autocorrelation of the batches, shown in Figure~\ref{fig:gambat}. <>= plot(seq(along = bout$Gamma.con) - 1, bout$Gamma.con, xlab = "k", ylab = expression(Gamma[k]), type = "l") @ \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Plot ``Big Gamma'' defined by \eqref{eq:little} and \eqref{eq:big} for the sequence of batch means (batch length \Sexpr{blen}). Only initial convex sequence estimator is shown.} \label{fig:gambat} \end{figure} Because the the variance is proportional to one over the batch length, we need to multiply by the batch length to estimate the $\sigma^2$ for the original series. <>= out$var.con bout$var.con * blen @ Another way to look at this is that the MCMC estimator of $\mu$ is either \texttt{mean(x)} or \texttt{mean(x.batch)}. And the variance must be divided by the sample size to give standard errors. So either <>= mean(x) + c(-1, 1) * qnorm(0.975) * sqrt(out$var.con / length(x)) mean(x.batch) + c(-1, 1) * qnorm(0.975) * sqrt(bout$var.con / length(x.batch)) @ is an asymptotic 95\% confidence interval for $\mu$. Just divide by the relevant sample size. \begin{thebibliography}{} \bibitem[Gelman et al.(1996)Gelman, Roberts, and Gilks]{grg} Gelman, A., G.~O. Roberts, and W.~R. Gilks (1996). \newblock Efficient Metropolis jumping rules. \newblock In \emph{Bayesian Statistics, 5 (Alicante, 1994)}, pp.~599--607. Oxford University Press. \bibitem[Geyer(1992)]{practical} Geyer, C.~J. (1992). \newblock Practical Markov chain Monte Carlo (with discussion). \newblock \emph{Statistical Science}, 7, 473--511. \bibitem[Geyer and Thompson(1995)]{geyer-temp} Geyer, C.~J. and E.~A. Thompson (1995). \newblock Annealing Markov chain Monte Carlo with applications to ancestral inference. \newblock \emph{Journal of the American Statistical Association}, 90, 909--920. \end{thebibliography} \end{document} mcmc/vignettes/bfst.Rnw0000644000175100001440000007725213045741600014662 0ustar hornikusers \documentclass[11pt]{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{indentfirst} \usepackage{natbib} \usepackage{url} \usepackage[utf8]{inputenc} \newcommand{\real}{\mathbb{R}} \DeclareMathOperator{\prior}{pri} \DeclareMathOperator{\posterior}{post} \DeclareMathOperator{\indicator}{ind} \newcommand{\fatdot}{\,\cdot\,} % \VignetteIndexEntry{Bayes Factors via Serial Tempering} \begin{document} \title{Bayes Factors via Serial Tempering} \author{Charles J. Geyer} \maketitle <>= options(keep.source = TRUE, width = 65) @ \section{Introduction} \subsection{Bayes Factors} \label{sec:bayes-factors} Let $\mathcal{M}$ be a finite or countable set of models (here we only deal with finite $\mathcal{M}$ but Bayes factors make sense for countable $\mathcal{M}$). For each model $m \in \mathcal{M}$ we have the prior probability of the model $\prior(m)$. It does not matter if this prior on models is unnormalized. Each model $m$ has a parameter space $\Theta_m$ and a prior $$ g(\theta \mid m), \qquad \theta \in \Theta_m $$ The spaces $\Theta_m$ can and usually do have different dimensions. That's the point. These within model priors must be normalized proper priors. The calculations to follow make no sense if these priors are unnormalized or improper. Each model $m$ has a data distribution $$ f(y \mid \theta, m) $$ and the observed data $y$ may be either discrete or continuous (it makes no difference to the Bayesian who treats $y$ as fixed after it is observed and treats only $\theta$ and $m$ as random). The unnormalized posterior for everything (for models and parameters within models) is $$ f(y \mid \theta, m) g(\theta \mid m) \prior(m) $$ To obtain the conditional distribution of $y$ given $m$, we must integrate out the nuisance parameter $\theta$ \begin{align*} q(y \mid m) & = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \prior(m) \, d \theta \\ & = \prior(m) \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta \end{align*} These are the unnormalized posterior probabilities of the models. The normalized posterior probabilities are $$ \posterior(m \mid y) = \frac{ q(y \mid m) }{ \sum_{m \in \mathcal{M}} q(y \mid m) } $$ It is considered useful to define $$ b(y \mid m) = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta $$ so $$ q(y \mid m) = b(y \mid m) \prior(m) $$ Then the ratio of posterior probabilities of models $m_1$ and $m_2$ is $$ \frac{\posterior(m_1 \mid y)}{\posterior(m_2 \mid y)} = \frac{q(y \mid m_1)}{q(y \mid m_2)} = \frac{b(y \mid m_1)}{b(y \mid m_2)} \cdot \frac{\prior(m_1)}{\prior(m_2)} $$ This ratio is called the \emph{posterior odds} of the models (a ratio of probabilities is called an \emph{odds}) of these models. The \emph{prior odds} is $$ \frac{\prior(m_1)}{\prior(m_2)} $$ The term we have not yet named in $$ \frac{\posterior(m_1 \mid y)}{\posterior(m_2 \mid y)} = \frac{b(y \mid m_1)}{b(y \mid m_2)} \cdot \frac{\prior(m_1)}{\prior(m_2)} $$ is called the \emph{Bayes factor} \begin{equation} \label{eq:factor} \frac{b(y \mid m_1)}{b(y \mid m_2)} \end{equation} the ratio of posterior odds to prior odds. The prior odds tells how the prior compares the probability of the models. The Bayes factor tells us how the data shifts that comparison going from prior to posterior via Bayes rule. Bayes factors are the primary tool Bayesians use for model comparison, the competitor for frequentist $P$-values in frequentist hypothesis tests of model comparison. Note that our clumsy multiple letter notation for priors and posteriors $\prior(m)$ and $\posterior(m \mid y)$ does not matter because neither is involved in the actual calculation of Bayes factors \eqref{eq:factor}. Priors and posteriors are involved in motivating Bayes factors but not in calculating them. \subsection{Tempering} \label{sec:temper} Simulated tempering \citep{marinari-parisi,geyer-thompson} is a method of Markov chain Monte Carlo (MCMC) simulation of many distributions at once. It was originally invented with the primary aim of speeding up MCMC convergence, but was also recognized to be useful for sampling multiple distributions \citep{geyer-thompson}. In the latter role it is sometimes referred to as ``umbrella sampling'' which is a term coined by \citet{torrie-valleau} for sampling multiple distributions via MCMC. We have a finite set of unnormalized distributions we want to sample, all related in some way. The R function \texttt{temper} in the CRAN package \texttt{mcmc} requires all to have continuous distributions for random vectors of the same dimension (all distributions have the same domain $\real^p$). Let $h_i$, $i \in \mathcal{I}$ denote the unnormalized densities of these distributions. Simulated tempering (called ``serial tempering'' by the \texttt{temper} function to distinguish from a related scheme not used in this document called ``parallel tempering'' and in either case abbreviated ST) runs a Markov chain whose state is a pair $(i, x)$ where $i \in \mathcal{I}$ and $x \in \real^p$. The unnormalized density of stationary distribution of the ST chain is \begin{equation} \label{eq:st-joint} h(i, x) = h_i(x) c_i \end{equation} where the $c_i$ are arbitrary constants chosen by the user (more on this later). The equilibrium distribution of the ST state $(I, X)$ --- both bits random --- is such that conditional distribution of $X$ given $I = i$ is the distribution with unnormalized density $h_i$. This is obvious from $h(i, x)$ being the unnormalized conditional density --- the same function thought of as a function of both variables is the unnormalized joint density and thought of as a function of just one of the variables is an unnormalized conditional density --- and $h(i, x)$ thought of as a function of $x$ for fixed $i$ being proportional to $h_i$. The equilibrium unnormalized marginal distribution of $I$ is \begin{equation} \label{eq:margin} \int h(i, x) \, d x = c_i \int h_i(x) \, d x = c_i d_i \end{equation} where $$ d_i = \int h_i(x) \, d x $$ is the normalizing constant for $h_i$, that is, $h_i / d_i$ is a normalized distribution. It is clear from \eqref{eq:margin} being the unnormalized marginal distribution that in order for the marginal distribution to be uniform we must choose the tuning constants $c_i$ to be proportional to $1 / d_i$. It is not important that the marginal distribution be exactly uniform, but unless it is approximately uniform, the sampler will not visit each distribution frequently. Thus we do need to have the $c_i$ to be approximately proportional to $1 / d_i$. This is accomplished by trial and error (one example is done in this document) and is easy for easy problems and hard for hard problems \citep[have much to say about adjusting the $c_i$]{geyer-thompson}. For the rest of this section we will assume the tuning constants $c_i$ have been so adjusted: we do not have the $c_i$ exactly proportional to $1 / d_i$ but do have them approximately proportional to $1 / d_i$. \subsection{Tempering and Bayes Factors} Bayes factors are very important in Bayesian inference and many methods have been invented to calculate them. No method except the one described here using ST is anywhere near as accurate and straightforward. Thus no competitors will be discussed. In using ST for Bayes factors we identify the index set $\mathcal{I}$ with the model set $\mathcal{M}$ and use the integers 1, $\ldots$, $k$ for both. We would like to identify the within model parameter vector $\theta$ with the vector $x$ that is the continuous part of the state of the ST Markov chain, but cannot because the dimension of $\theta$ depends on $m$ and this is not allowed. Thus we have to do something a bit more complicated. We ``pad'' $\theta$ so that it always has the same dimension, doing so in a way that does not interfere with the Bayes factor calculation. Write $\theta = (\theta_{\text{actual}}, \theta_{\text{pad}})$, the dimension of both parts depending on the model $m$. Then we insist on the following conditions: $$ f(y \mid \theta, m) = f(y \mid \theta_{\text{actual}}, m) $$ so the data distribution does not depend on the ``padding'' and $$ g(\theta \mid m) = g_{\text{actual}}(\theta_{\text{actual}} \mid m) \cdot g_{\text{pad}}(\theta_{\text{pad}} \mid m) $$ so the two parts are \emph{a priori} independent and both parts of the prior are normalized proper priors. This assures that \begin{equation} \label{eq:unnormalized-bayes-factors} \begin{split} b(y \mid m) & = \int_{\Theta_m} f(y \mid \theta, m) g(\theta \mid m) \, d \theta \\ & = \iint f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) g_{\text{pad}}(\theta_{\text{pad}} \mid m) \, d \theta_{\text{actual}} \, d \theta_{\text{pad}} \\ & = \int_{\Theta_m} f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) \, d \theta_{\text{actual}} \end{split} \end{equation} so the calculation of the unnormalized Bayes factors is the same whether or not we ``pad'' $\theta$, and we may then take \begin{align*} h_m(\theta) & = f(y \mid \theta, m) g(\theta \mid m) \\ & = f(y \mid \theta_{\text{actual}}, m) g_{\text{actual}}(\theta_{\text{actual}} \mid m) g_{\text{pad}}(\theta_{\text{pad}} \mid m) \end{align*} to be the unnormalized densities for the component distributions of the ST chain, in which case the unnormalized Bayes factors are proportional to the normalizing constants $d_i$ in Section~\ref{sec:temper}. \subsection{Tempering and Normalizing Constants} Let $d$ be the normalizing constant for the joint equilibrium distribution of the ST chain \eqref{eq:st-joint}. When we are running the ST chain we know neither $d$ nor the $d_i$ but we do know the $c_i$, which are constants we have chosen based on the results of previous runs but are fixed known numbers for the current run. Let $(I_t, X_t)$, $t = 1$, 2, $\ldots$ be the sample path of the ST chain. Recall that (somewhat annoyingly) we are using the notation $(i, x)$ for the state vector of a general ST chain and the notation $(m, \theta)$ for ST chains used to calculate Bayes factors, identifying $i = m$ and $x = \theta$. Let $\indicator(\fatdot)$ denote the function that maps logical values to numerical values, false to zero and true to one. Normalizing constants are estimated by averaging the time spent in each model \begin{equation} \label{eq:st-estimates} \hat{\delta}_n(m) = \frac{1}{n} \sum_{t = 1}^n \indicator(I_t = m) \end{equation} For the purposes of approximating Bayes factors the $X_t$ are ignored. The $X_t$ may be useful for other purposes, such as Bayesian model averaging \citep*{bma}, but this is not discussed here. The Monte Carlo approximations \eqref{eq:st-estimates} converge to their expected values under the equilibrium distribution \begin{equation} \label{eq:st-expectations} E\{ \indicator(I_t = m) \} = \int \frac{h(m, x)}{d} \, d x = \frac{c_m d_m}{d} = \delta(m) \end{equation} We want to estimate the unnormalized Bayes factors \eqref{eq:unnormalized-bayes-factors}, which are in this context proportional to the $d_m$. The $c_m$ are known, $d$ is unknown but does not matter since we only need to estimate the $d_m = b(m \mid y)$ up to an overall unknown constant of proportionality, which cancels out of Bayes factors \eqref{eq:factor}. Note that our discussion here applies unchanged to the general problem of estimating normalizing constants up to an unknown constant of proportionality, which has applications other than Bayes factors, for example, missing data maximum likelihood \citep{thompson-guo,geyer,sung-geyer}. The ST method approximates normalizing constants up to an overall constant of proportionality with high accuracy regardless of how large or small they are (whether they are $10^{100}$ or $10^{-100}$), and no other method that does not use essentially the same idea can do this. The key is what seems at first sight to be a weakness of ST, the need to adjust the tuning constants $c_i$ by trial and error. In this context the weakness is actually a strength: the adjusted $c_i$ contain most of the information about the size of the normalizing constants $d_i$ and the Monte Carlo averages \eqref{eq:st-estimates} add only the finishing touch. Thus multiple runs of the ST chain with different choices of the $c_i$ used in each run are needed (the ``trial and error''), but the information from all are incorporated in the final run used for final approximation of the normalizing constants (Bayes factors). It is perhaps surprising that the Monte Carlo error approximation is trivial. In the context of the last run of the ST chain the $c_i$ are known constants and contribute no error. The Monte Carlo error of the averages \eqref{eq:st-estimates} is straightforwardly estimated by batch means or competing methods. \citet{geyer-thompson} note that the $c_i$ enter formally like a prior: one can think of $h_i(x) c_i$ as likelihood times prior. But one should not think of the $c_i$ as representing prior information, informative, non-informative, or in between. The $c_i$ are adjusted to make the ST distribution sample all the models $h_i$, and that is the only criterion for the adjustment. For this reason \citet{geyer-thompson} call the $c_i$ the \emph{pseudoprior}. This is a special case of a general principle of MCMC. When doing MCMC one should forget the statistical motivation (in this case Bayes factors). One should set up a Markov chain that does a good job of simulating the required equilibrium distribution, whatever it is. Thinking about the statistical motivation of the equilibrium does not help and can hurt (if one thinks of the pseudoprior as an actual prior, one may be tempted to adjust it to represent prior information). \section{R Package MCMC} We use the R statistical computing environment \citep{rcore} in our analysis. It is free software and can be obtained from \url{http://cran.r-project.org}. Precompiled binaries are available for Windows, Macintosh, and popular Linux distributions. We use the contributed package \verb@mcmc@ \citep{mcmc-R-package} If R has been installed, but this package has not yet been installed, do \begin{verbatim} install.packages("mcmc") \end{verbatim} from the R command line (or do the equivalent using the GUI menus if on Apple Macintosh or Microsoft Windows). This may require root or administrator privileges. Assuming the \verb@mcmc@ package has been installed, we load it <>= library(mcmc) @ <>= baz <- library(help = "mcmc") baz <- baz$info[[1]] baz <- baz[grep("Version", baz)] baz <- sub("^Version: *", "", baz) bazzer <- paste(R.version$major, R.version$minor, sep = ".") @ The version of the package used to make this document is \Sexpr{baz} (which is available on CRAN). The version of R used to make this document is \Sexpr{bazzer}. We also set the random number generator seed so that the results are reproducible. <>= set.seed(42) @ To get different results, change the setting or don't set the seed at all. \section{Logistic Regression Example} We use the same logistic regression example used in the \texttt{mcmc} package vignette for the \texttt{metrop} function (file \texttt{demo.pdf}. Simulated data for the problem are in the data set \verb@logit@. There are five variables in the data set, the response \verb@y@ and four predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. A frequentist analysis for the problem is done by the following R statements <>= data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, data = logit, family = binomial, x = TRUE) summary(out) @ But this example isn't about frequentist analysis, we want a Bayesian analysis. For our Bayesian analysis we assume the same data model as the frequentist, and we assume the prior distribution of the five parameters (the regression coefficients) makes them independent and identically normally distributed with mean 0 and standard deviation 2. Moreover, we wish to calculate Bayes factors for the $16 = 2^4$ possible submodels that include or exclude each of the predictors, \verb@x1@, \verb@x2@, \verb@x3@, and \verb@x4@. \subsection{Setup} We set up a matrix that indicates these models. <>= varnam <- names(coefficients(out)) varnam <- varnam[varnam != "(Intercept)"] nvar <- length(varnam) models <- NULL foo <- seq(0, 2^nvar - 1) for (i in 1:nvar) { bar <- foo %/% 2^(i - 1) bar <- bar %% 2 models <- cbind(bar, models, deparse.level = 0) } colnames(models) <- varnam models @ In each row, 1 indicates the predictor is in the model and 0 indicates it is out. The function \texttt{temper} in the \text{mcmc} package that does tempering requires a notion of neighbors among models. It attempts jumps only between neighboring models. Here we choose models to be neighbors if they differ only by one predictor. <>= neighbors <- matrix(FALSE, nrow(models), nrow(models)) for (i in 1:nrow(neighbors)) { for (j in 1:ncol(neighbors)) { foo <- models[i, ] bar <- models[j, ] if (sum(foo != bar) == 1) neighbors[i, j] <- TRUE } } @ Now we specify the equilibrium distribution of the ST chain. Its state vector is $(i, x)$ or $(m, \theta)$ in our alternative notations, where $i$ is an integer between $1$ and \verb@nrow(models)@ = \Sexpr{nrow(models)} and $\theta$ is the parameter vector ``padded'' to always be the same length, so we take it to be the length of the parameter vector of the full model which is \verb@length(out$coefficients)@ or \verb@ncol(models) + 1@ which makes the length of the state of the ST chain \verb@ncol(models) + 2@. We take the within model priors for the ``padded'' components of the parameter vector to be the same as those for the ``actual'' components, normal with mean 0 and standard deviation 2 for all cases. As is seen in \eqref{eq:unnormalized-bayes-factors} the priors for the ``padded'' components (parameters not in the model for the current state) do not matter because they drop out of the Bayes factor calculation. The choice does not matter much for this toy example. See the discussion section for more on this issue. It is important that we use normalized log priors, the term \verb@dnorm(beta, 0, 2, log = TRUE)@ in the function, unlike when we are simulating only one model as in the \texttt{mcmc} package vignette where it would be o.~k.\ to use unnormalized log priors \verb@- beta^2 / 8@. The \texttt{temper} function wants the log unnormalized density of the equilibrium distribution. We include an additional argument \texttt{log.pseudo.prior}, which is $\log(c_i)$ in our mathematical development, because this changes from run to run as we adjust it by trial and error. Other ``arguments'' are the model matrix of the full model \texttt{modmat}, the matrix \texttt{models} relating integer indices (the first component of the state vector of the ST chain) to which predictors are in or out of the model, and the data vector \texttt{y}, but these are not passed as arguments to our function and instead are found in the R global environment. <>= modmat <- out$x y <- logit$y ludfun <- function(state, log.pseudo.prior) { stopifnot(is.numeric(state)) stopifnot(length(state) == ncol(models) + 2) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= nrow(models)) stopifnot(is.numeric(log.pseudo.prior)) stopifnot(length(log.pseudo.prior) == nrow(models)) beta <- state[-1] inies <- c(TRUE, as.logical(models[icomp, ])) beta.logl <- beta beta.logl[! inies] <- 0 eta <- as.numeric(modmat %*% beta.logl) logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta))) logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta))) logl <- sum(logp[y == 1]) + sum(logq[y == 0]) logl + sum(dnorm(beta, 0, 2, log = TRUE)) + log.pseudo.prior[icomp] } @ \subsection{Trial and Error} Now we are ready to try it out. We start in the full model at its MLE, and we initialize \texttt{log.pseudo.prior} at all zeros, having no idea \emph{a priori} what it should be. <>= state.initial <- c(nrow(models), out$coefficients) qux <- rep(0, nrow(models)) out <- temper(ludfun, initial = state.initial, neighbors = neighbors, nbatch = 1000, blen = 100, log.pseudo.prior = qux) names(out) out$time @ So what happened? <>= ibar <- colMeans(out$ibatch) ibar @ The ST chain did not mix well, several models not being visited even once. So we adjust the pseudo priors to get uniform distribution. <>= qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux @ The new pseudoprior should be proportional to \verb@1 / ibar@ if \texttt{ibar} is an accurate estimate of \eqref{eq:st-expectations}, but this makes no sense when the estimates are bad, in particular, when the are exactly zero. Thus we put an upper bound, chosen arbitrarily (here 10) on the maximum increase of the log pseudoprior. The statement \begin{verbatim} qux <- qux - min(qux) \end{verbatim} is unnecessary. An overall arbitrary constant can be added to the log pseudoprior without changing the equilibrium distribution of the ST chain. We do this only to make \texttt{qux} more comparable from run to run. Now we repeat this until the log pseudoprior ``converges'' roughly. Because this loop takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("bfst1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { qux.save <- qux time.save <- out$time repeat{ out <- temper(out, log.pseudo.prior = qux) ibar <- colMeans(out$ibatch) qux <- qux + pmin(log(max(ibar) / ibar), 10) qux <- qux - min(qux) qux.save <- rbind(qux.save, qux, deparse.level = 0) time.save <- rbind(time.save, out$time, deparse.level = 0) if (max(ibar) / min(ibar) < 2) break } save(out, qux, qux.save, time.save, file = "bfst1.rda") } else { .Random.seed <- out$final.seed } print(qux.save, digits = 3) print(qux, digits = 3) apply(time.save, 2, sum) @ Now that the pseudoprior is adjusted well enough, we need to perhaps make other adjustments to get acceptance rates near 20\%. <>= print(out$accepti, digits = 3) print(out$acceptx, digits = 3) @ The acceptance rates for swaps seem o. k. <>= min(as.vector(out$accepti), na.rm = TRUE) @ and there is nothing simple we can do to adjust them (adjustment is possible, see the discussion section for more on this issue). We adjust the acceptance rates for within model moves by adjusting the scaling. <>= out <- temper(out, scale = 0.5, log.pseudo.prior = qux) time.save <- rbind(time.save, out$time, deparse.level = 0) print(out$acceptx, digits = 3) @ Looks o.~k.\ now. Inspection of autocorrelation functions for components of \verb@out$ibatch@ (not shown) says batch length needs to be at least 4 times longer. We make it 10 times longer for safety. Because this run takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("bfst2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out <- temper(out, blen = 10 * out$blen, log.pseudo.prior = qux) save(out, file = "bfst2.rda") } else { .Random.seed <- out$final.seed } time.save <- rbind(time.save, out$time, deparse.level = 0) foo <- apply(time.save, 2, sum) foo.min <- floor(foo[1] / 60) foo.sec <- foo[1] - 60 * foo.min c(foo.min, foo.sec) @ The total time for all runs of the temper function was \Sexpr{foo.min} minutes and \Sexpr{round(foo.sec, 1)} seconds. \subsection{Bayes Factor Calculations} Now we calculate log 10 Bayes factors relative to the model with the highest unnormalized Bayes factor. <>= log.10.unnorm.bayes <- (qux - log(colMeans(out$ibatch))) / log(10) k <- seq(along = log.10.unnorm.bayes)[log.10.unnorm.bayes == min(log.10.unnorm.bayes)] models[k, ] log.10.bayes <- log.10.unnorm.bayes - log.10.unnorm.bayes[k] log.10.bayes @ These are base 10 logarithms of the Bayes factors against the $k$-th model where $k = \Sexpr{k}$. For example, the Bayes factor for the $k$-th model divided by the Bayes factor for the first model is $10^{\Sexpr{round(log.10.bayes[1], 3)}}$. Now we calculate Monte Carlo standard errors two different ways. One is the way the delta method is usually taught. To simplify notation, denote the Bayes factors $$ b_m = b(y \mid m) $$ and their Monte Carlo approximations $\hat{b}_m$. Then the log Bayes factors are $$ g_i(b) = \log_{10} b_i - \log_{10} b_k $$ hence we need to apply the delta method with the function $g_i$, which has derivatives \begin{align*} \frac{\partial g_i(b)}{\partial b_i} & = \frac{1}{b_i \log_e(10)} \\ \frac{\partial g_i(b)}{\partial b_k} & = - \frac{1}{b_k \log_e(10)} \\ \frac{\partial g_i(b)}{\partial b_j} & = 0, \qquad \text{$j \neq i$ and $j \neq k$} \end{align*} <>= fred <- var(out$ibatch) / out$nbatch sally <- colMeans(out$ibatch) mcse.log.10.bayes <- (1 / log(10)) * sqrt(diag(fred) / sally^2 - 2 * fred[ , k] / (sally * sally[k]) + fred[k, k] / sally[k]^2) mcse.log.10.bayes foompter <- cbind(models, log.10.bayes, mcse.log.10.bayes) round(foompter, 5) @ An alternative calculation of the MCSE replaces the actual function of the raw Bayes factors with its best linear approximation $$ \frac{1}{\log_e(10)} \left(\frac{\hat{b}_i - b_i}{b_i} - \frac{\hat{b}_k - b_k}{b_k} \right) $$ and calculates the standard deviation of this quantity by batch means <>= ibar <- colMeans(out$ibatch) herman <- sweep(out$ibatch, 2, ibar, "/") herman <- sweep(herman, 1, herman[ , k], "-") mcse.log.10.bayes.too <- (1 / log(10)) * apply(herman, 2, sd) /sqrt(out$nbatch) all.equal(mcse.log.10.bayes, mcse.log.10.bayes.too) @ \section{Discussion} We hope readers are impressed with the power of this method. The key to the method is pseudopriors adjusted by trial and error. The method could have been invented by any Bayesian who realized that the priors on models, $\prior(m)$ in our notation in Section~\ref{sec:bayes-factors}, do not affect the Bayes factors and hence are irrelevant to calculating Bayes factors. Thus the priors (or pseudopriors in our terminology) should be chosen for reasons of computational convenience, as we have done, rather than to incorporate prior information. The rest of the details of the method are unimportant. The \texttt{temper} function in R is convenient to use for this purpose, but there is no reason to believe that it provides optimal sampling. Samplers carefully designed for each particular application would undoubtedly do better. Our notion of ``padding'' so that the within model parameters have the same dimension for all models follows \citet{carlin-chib} but ``reversible jump'' samplers \citep{green} would undoubtedly do better. Unfortunately, there seems to be no way to code up a function like \texttt{temper} that uses ``reversible jump'' and requires no theoretical work from users that if messed up destroys the algorithm. The \texttt{temper} function is foolproof in the sense that if the log unnormalized density function written by the user (like our \texttt{ludfun}) is correct, then the ST Markov chain has the equilibrium distribution is supposed to have. There is nothing the user can mess up except this user written function. No analog of this for ``reversible jump'' chains is apparent (to your humble author). Two issues remain where the text above said ``see the discussion section for more on this issue.'' The first was about within model priors for the ``padding'' components of within model parameter vectors $g_{\text{pad}}(\theta_{\text{pad}} \mid m)$ in the notation in \eqref{eq:unnormalized-bayes-factors}. Rather than choose these so that they do not depend on the data (as we did), it would be better (if more trouble) to choose them differently for each ``padding'' component, centering $g_{\text{pad}}(\theta_{\text{pad}} \mid m)$ so the distribution of a component of $\theta_{\text{pad}}$ is near to the marginal distribution of the same component in neighboring models (according to the \texttt{neighbors} argument of the \texttt{temper} function). The other remaining issue is adjusting acceptance rates for jumps. There is no way to adjust this other than by changing the number of models and their definitions. But the models we have cannot be changed; if we are to calculate Bayes factors for them, then we must sample them as they are. But we can insert new models between old models. For example, if the acceptance for swaps between model $i$ and model $j$ is too low, then we can insert distribution $k$ between them that has unnormalized density $$ h_k(x) = \sqrt{h_i(x) h_j(x)}. $$ This idea is inherited from simulated tempering; \citep{geyer-thompson} have much discussion of how to insert additional distributions into a tempering network. It is another key issue in using tempering to speed up sampling. It is less obvious in the Bayes factor context, but still an available technique if needed. \begin{thebibliography}{} \bibitem[Carlin and Chib(1995)]{carlin-chib} Carlin, B.~P. and Chib, S. (1995). \newblock Bayesian model choice via Markov chain Monte Carlo methods. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{57}, 473--484. \bibitem[Geyer(1994)]{geyer} Geyer, C.~J. (1994). \newblock On the convergence of Monte Carlo maximum likelihood calculations. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{56} 261--274. \bibitem[Geyer., 2009]{mcmc-R-package} Geyer., C.~J. (2009). \newblock \emph{mcmc: Markov Chain Monte Carlo}. \newblock R package version 0.7-2, available from CRAN. \bibitem[Geyer and Thompson(1995)]{geyer-thompson} Geyer, C.~J., and Thompson, E.~A. (1995). \newblock Annealing Markov chain Monte Carlo with applications to ancestral inference. \newblock \emph{Journal of the American Statistical Association}, \textbf{90}, 909--920. \bibitem[Green(1995)]{green} Green, P.~J. (1995). \newblock Reversible jump {M}arkov chain {M}onte {C}arlo computation and {B}ayesian model determination. \newblock \emph{Biometrika}, \textbf{82}, 711--732. \bibitem[Hoeting et al.(1999)Hoeting, Madigan, Raftery, and Volinsky]{bma} Hoeting, J.~A., Madigan, D., Raftery, A.~E. and Volinsky, C.~T. (1999). \newblock Bayesian model averaging: A tutorial (with discussion). \newblock \emph{Statical Science}, \textbf{19}, 382--417. \newblock The version printed in the journal had the equations messed up in the production process; a corrected version is available at \url{http://www.stat.washington.edu/www/research/online/1999/hoeting.pdf}. \bibitem[Marinari and Parisi(1992)]{marinari-parisi} Marinari, E., and Parisi G. (1992). \newblock Simulated tempering: A new Monte Carlo Scheme. \newblock \emph{Europhysics Letters}, \textbf{19}, 451--458. \bibitem[R Development Core Team(2010)]{rcore} R Development Core Team (2010). \newblock R: A language and environment for statistical computing. \newblock R Foundation for Statistical Computing, Vienna, Austria. \newblock \url{http://www.R-project.org}. \bibitem[Sung and Geyer(2007)]{sung-geyer} Sung, Y.~J. and Geyer, C.~J. (2007). \newblock Monte Carlo likelihood inference for missing data models. \newblock \emph{Annals of Statistics}, \textbf{35}, 990--1011. \bibitem[Thompson and Guo(1991)]{thompson-guo} Thompson, E. A. and Guo, S. W. (1991). \newblock Evaluation of likelihood ratios for complex genetic models. \newblock \emph{IMA J. Math. Appl. Med. Biol.}, \textbf{8}, 149--169. \bibitem[Torrie and Valleau(1977)]{torrie-valleau} Torrie, G.~M., and Valleau, J.~P. (1977). \newblock Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. \newblock \emph{Journal of Computational Physics}, \textbf{23}, 187--199. \end{thebibliography} \end{document} mcmc/vignettes/morph.Rnw0000644000175100001440000006124413045741600015043 0ustar hornikusers\documentclass{article} \usepackage{natbib} \usepackage{graphics} \usepackage{amsmath,amssymb} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage[tableposition=top]{caption} \usepackage{url} \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} \DeclareMathOperator{\E}{E} \newcommand{\inner}[1]{\langle #1 \rangle} % \VignetteIndexEntry{MCMC Morph Example} \begin{document} <>= options(keep.source = TRUE, width = 60) foo <- packageDescription("mcmc") @ \title{Morphometric MCMC (mcmc Package Ver.~\Sexpr{foo$Version})} % $ (Just to make emacs syntax highlighting work properly) \author{Leif T. Johnson \and Charles J. Geyer} \maketitle \section{Overview} This is an example how to use morphometric Markov chains as implemented in the \verb@mcmc@ package in R. Let $X$ be an $\mathbb{R}^k$ valued random variable with probability density function, $f_X$. Let $g$ be a diffeomorphism, and $Y=g(X)$. Then the probability density function of $Y$, $f_Y$ is given by \begin{equation}\label{eq:def-fy} f_Y(y) = f_X\bigl(g^{-1}(y)\bigr) \det\bigl( \nabla g^{-1}(y) \bigr). \end{equation} Since $g$ is a diffeomorphism, we can draw inference about $X$ from information about $Y$ (and vice versa). It is not unusual for $f_X$ to either be known only up to a normalizing constant, or to be analytically intractable in other ways --- such as being high dimensional. A common solution to this problem is to use Markov chain Monte Carlo (MCMC) methods to learn about $f_X$. When using MCMC, a primary concern of the practitioner should be the question ``Does the Markov chain converge fast enough to be useful?'' One very useful convergence rate is called \emph{geometrically ergodic} \citep[Chapter~1]{johnson-thesis}. The \texttt{mcmc} package implements the Metropolis random-walk algorithm for arbitrary log unnormalized probability densities. But the Metropolis random-walk algorithm does not always perform well. As is demonstrated in \citet{johnson-geyer}, for $f_X$ and $f_Y$ related by diffeomorphism as in \eqref{eq:def-fy}, a Metropolis random-walk for $f_Y$ can be geometrically ergodic even though a Metropolis random-walk for $f_X$ is not. Since the transformation is one-to-one, inference about $f_X$ can be drawn from the Markov chain for $f_Y$. The \texttt{morph.metrop} and \texttt{morph} functions in the \texttt{mcmc} package provide this functionality, and this vignette gives a demonstration on how to use them. \section{T Distribution} \label{sec:toy} We start with a univariate example, which is a Student $t$ distribution with three degrees of freedom. Of course, one doesn't need MCMC to simulate this distribution (the R function \texttt{rt} does that), so this is just a toy problem. But it does illustrate some aspects of using variable transformation. A necessary condition for geometric ergodicity of a random-walk Metropolis algorithm is that the target density $\pi$ have a moment generating function \citep{jarner-tweedie}. For a univariate target density, which we have in this section, a sufficient condition for geometric ergodicity of a random-walk Metropolis algorithm is that the target density $\pi$ be exponentially light \citet{mengersen-tweedie}. Thus if we do not use variable transformation, the Markov chain simulated by the \texttt{metrop} function will not be geometrically ergodic. \citet[Example 4.2]{johnson-geyer} show that a $t$ distribution is sub-exponentially light. Hence using the transformations described in their Corollaries~1 and~2 will induce a target density $\pi_\gamma$ for which a Metropolis random-walk will be geometrically ergodic. using the transformation described as $h_2$ in \citet[Corollary~2]{johnson-geyer} will induce a target density for which a Metropolis random-walk will be geometrically ergodic. Passing a positive value for \texttt{b} to \texttt{morph} function will create the aforementioned transformation, $h_2$. It's as simple as <<>>= library(mcmc) h2 <- morph(b=1) @ We can now see the induced density. Note that \texttt{morph} works for log unnormalized densities, so we need exponentiate the induced density to plot it on the usual scale. <<>>= lud <- function(x) dt(x, df=3, log=TRUE) lud.induced <- h2$lud(lud) @ We can plot the two densities, <>= curve(exp(Vectorize(lud.induced)(x)), from = -3, to = 3, lty = 2, xlab = "t", ylab = "density") curve(exp(lud(x)), add = TRUE) legend("topright", c("t density", "induced density"), lty=1:2) @ The \texttt{Vectorize} in this example is necessary because the function \texttt{lud.induced} is not vectorized. Instead, it treats any vector passed as a single input, which is rescaled (using the specified diffeomorphism) and passed to \texttt{lud}. Compare the behavior of \texttt{lud} and \texttt{lud.induced} in the following example. <<>>= lud(1:4) lud(1) foo <- try(lud.induced(1:4)) class(foo) cat(foo, "\n") lud.induced(1) @ Because the function \texttt{dt} is vectorized, the function \texttt{lud} is also vectorized, mapping vectors to vectors, whereas the function \texttt{lud.induced} is not vectorized, mapping vectors to scalars. Before we start using random numbers, we set the seed of the random number generator so this document always produces the same results. <>= set.seed(42) @ To change the results, change the seed or delete the \texttt{set.seed} statement. Running a Markov chain for the induced density is done with \texttt{morph.metrop}. <<>>= out <- morph.metrop(lud, 0, blen=100, nbatch=100, morph=morph(b=1)) @ The content of \texttt{out\$batch} is on the scale of used by \texttt{lud}. Once the transformation has been set, no adjustment is needed (unless you want to change transformations). We start by adjusting the scale. <<>>= # adjust scale to find a roughly 20% acceptance rate out$accept @ An acceptance rate of \Sexpr{round(100 * out$accept, 1)}\% %$ to fix emacs highlighting is probably too high. By increasing the scale of the proposal distribution we can bring it down towards 20\%. <<>>= out <- morph.metrop(out, scale=4) out$accept @ We now use this Markov chain to estimate the expectation of the target distribution. But first we need to check whether our batch length is good. The following code <>= acf(out$batch) @ makes the autocorrelation plot (Figure~\ref{fig:fig0}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot for the sequence of batch means.} \label{fig:fig0} \end{figure} It looks like there is no significant autocorrelation among the batches so the following produces a valid confidence interval for the true unknown mean of the target distribution (since this is a toy problem we actually know the true ``unknown'' mean is zero, but we pretend we don't know that for the purposes of the toy problem) <<>>= t.test(out$batch) @ If we want a point estimate and a Monte Carlo standard error, those are <<>>= colMeans(out$batch) apply(out$batch, 2, sd) / sqrt(out$nbatch) @ If a shorter confidence interval is desired, the Markov chain can be run longer (increase either the number of batches or the batch length, or both). Note that when calculating our estimate and the Monte Carlo standard error we are not concerned with what was happening on the transformed scale. The \texttt{morph.metrop} function seamlessly does this for us. \subsection{Comparison of Morphed and Unmorphed} To show the utility of the transformation, we will study the behavior of the Markov chain with and without the transformation for the same problem as in the preceding section. We will consider two different estimation methods. \begin{enumerate} \item \label{enum:rw} Estimate the mean of the target distribution using a random-walk Metropolis algorithm implemented by the \texttt{metrop} function. \citet{jarner-roberts} demonstrate that a central limit theorem does not hold for these estimates. \item \label{enum:rw-induced} Estimate the mean of the target distribution using a random-walk Metropolis algorithm implemented by the \texttt{morph.metrop} function with argument \texttt{morph = morph(b=1)}. \citet{johnson-geyer} demonstrate that a central limit theorem does hold for these estimates. \end{enumerate} For the former, we need to adjust the scale. <>= out.unmorph <- metrop(lud, 0, blen=1000, nbatch=1) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=4) out.unmorph$accept out.unmorph <- metrop(out.unmorph, scale=6) out.unmorph$accept @ A scale of 6 appears to be about right. Now we do a long run for this sampler. Because this run takes longer than CRAN vingettes are supposed to take, we save the results to a file and load the results from this file if it already exists. <>= lout <- suppressWarnings(try(load("morph1.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.unmorph <- metrop(out.unmorph, blen = 1e5, nbatch = 1e3) save(out.unmorph, file = "morph1.rda") } else { .Random.seed <- out.unmorph$final.seed } out.unmorph$accept @ Let's look at the distribution of batch means. The following code <>= foo <- as.vector(out.unmorph$batch) qqnorm(foo) qqline(foo) @ makes a Q-Q plot of the batch means (Figure~\ref{fig:fig4}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Q-Q plot of batch means (batch length \Sexpr{out.unmorph$blen}) for the unmorphed chain.} \label{fig:fig4} \end{figure} We see bad behavior of the unmorphed chain. These batch means (or at least some batch means for sufficiently long batch length) should look normally distributed, and these don't. Not even close. We do a formal test just to check our interpretation of the plot <>= shapiro.test(foo) @ Now for comparison, we check the morphed chain. <>= lout <- suppressWarnings(try(load("morph2.rda"), silent = TRUE)) if (inherits(lout, "try-error")) { out.morph <- morph.metrop(out, blen = 1e5, nbatch = 1e3) save(out.morph, file = "morph2.rda") } else { .Random.seed <- out.morph$final.seed } out.morph$accept @ The following code <>= foo <- as.vector(out.morph$batch) qqnorm(foo) qqline(foo) @ makes a Q-Q plot of the batch means (Figure~\ref{fig:fig5}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Q-Q plot of batch means (batch length \Sexpr{out.unmorph$blen}) for the morphed chain.} \label{fig:fig5} \end{figure} We see good behavior of the morphed chain. These batch means do look normally distributed. We do a formal test just to check our interpretation of the plot <>= shapiro.test(foo) @ \section{Binomial Distribution with a Conjugate Prior} We demonstrate a morphometric Markov chain using the \texttt{UCBAdmisions} data set included in \texttt{R}, (use \texttt{help(UCBAdmissions)} to see details of this data set). We will model the probability of a student being admitted or rejected, using the sex of the student and the department that the student applied to as predictor variables. For our prior, we naively assume that 30\% of all students are admitted, independent of sex or department. As this is a naive prior, we will only add 5 students to each gender-department combination. This will not give the prior much weight, most of the information in the posterior distribution will be from the data. If we have $L$ observations from a multinomial distribution, then using a multinomial logit-link, with model matrices $M^1,\dots,M^L$, regression parameter $\beta$, observed counts $Y^1,\dots,Y^N$ with observed sample sizes $N^1,\dots,N^L$ and prior probabilities $\xi^1, \dots, \xi^L$ and prior ``sample sizes'' $\nu^1,\dots,\nu^L$ then the posterior distribution of $\beta$ is given by \citep[Sec. 5.1.2]{johnson-thesis} \begin{equation}\label{eq:mult-post-conj-complicated} \pi(\beta|y,n,\xi,\nu) \propto \exp\biggl\{ \sum_{l=1}^L \inner{y^l + \xi^l \nu^l, M^l \beta} - (n^l + \nu^l) \log\bigl( \sum_j e^{M_{j\cdot} \beta} \bigr) \biggr\} \end{equation} where $\inner{a, b}$ denotes the usual inner product between vectors $a$ and $b$. For our application, we can simplify this in two ways. First, we use the posterior counts instead of the sum of the prior and data counts, i.e. use $y^{*l} = y^l + \xi^l \nu^l$ and $n^{*l} = n^l + \nu^l$. Second, to avoid having a direction of recession in $\pi(\beta|\cdot)$, we need to fix the elements of $\beta$ that correspond with one of the response categories. Since we are going to fitting a binomial response, if we set these elements of $\beta$ to be $0$, we may then replace the sequence of model matrices with a single model matrix; $M$ instead of $M^1,\dots,M^L$. The $l$-th row of $M$ will correspond to $M^l$. Label the two response categories $A$ and $B$. Without loss of generality, we will fix the elements of $\beta$ corresponding to category $B$ to 0. Let $x_1,\dots,x_L$ represent the posterior counts of category $A$, and $\beta^*$ represent the corresponding elements of $\beta$ --- these are the elements of $\beta$ we did not fix as 0. The meaning of $n^{*1},\dots,n^{*L}$ is unchanged. Then our simplified unnormalized posterior density is \begin{equation}\label{eq:simplified-posterior} \pi(\beta|x,n^*) \propto \exp\biggl\{ \inner{x, M \beta^*} - \sum_{l=1}^L n^{*l} \log\bigl(1 + e^{(M \beta^*)_l}\bigr) \biggr\}. \end{equation} This can be computed with a very simple \texttt{R} function, we implement it in log form. <>= lud.binom <- function(beta, M, x, n) { MB <- M %*% beta sum(x * MB) - sum(n * log(1 + exp(MB))) } @ Now that we have a function to calculate a log-unnormalized posterior density, we can run the Markov chain. To that, we need the model matrix. First we convert the \texttt{UCAdmissions} data to a \texttt{data.frame}. <>= dat <- as.data.frame(UCBAdmissions) dat.split <- split(dat, dat$Admit) dat.split <- lapply(dat.split, function(d) { val <- as.character(d$Admit[1]) d["Admit"] <- NULL names(d)[names(d) == "Freq"] <- val d }) dat <- merge(dat.split[[1]], dat.split[[2]]) @ Next we build the model matrix. Our model specification allows for an interaction between gender and department, even though our prior assumes that they are independent. <>= formula <- cbind(Admitted, Rejected) ~ (Gender + Dept)^2 mf <- model.frame(formula, dat) M <- model.matrix(formula, mf) @ As stated above, we will take $\nu = 5$ and $\xi=0.30$. That is, we will add 5 students to each gender-department combination, where each combination has a 30\% acceptance rate. <<>>= xi <- 0.30 nu <- 5 @ <>= lud.berkeley <- function(B) lud.binom(B, M, dat$Admitted + xi * nu, dat$Admitted + dat$Rejected + nu) @ This function is suitable for passing to \texttt{metrop} or \texttt{morph.metrop}. We know that using \texttt{morph.metrop} with \texttt{morph=morph(p=3)} will run a geometrically ergodic Markov chain \citep{johnson-geyer}. <<>>= berkeley.out <- morph.metrop(lud.berkeley, rep(0, ncol(M)), blen=1000, nbatch=1, scale=0.1, morph=morph(p=3)) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.05) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, scale=0.02) berkeley.out$accept berkeley.out <- morph.metrop(berkeley.out, blen=10000) berkeley.out$accept @ <<>>= berkeley.out <- morph.metrop(berkeley.out, blen=1, nbatch=100000) @ Estimate the posterior mean acceptance probabilities for each gender-department combination. <<>>= beta <- setNames(colMeans(berkeley.out$batch), colnames(M)) MB <- M %*% beta dat$p <- dat$Admitted / (dat$Admitted + dat$Rejected) dat$p.post <- exp(MB) / (1 + exp(MB)) dat @ The small difference between the data and posterior probabilities is expected, our prior was given very little weight. Using \texttt{morph.metrop} with the setting \texttt{morph=morph(p=3)} in this setting is an efficient way of sampling from the posterior distribution. We can also compare the posterior distribution of admittance probability for each gender-department combination. Table~\ref{tab:post-quant} gives the 5\% and 95\% quantiles for the posterior distribution of the admittance probabilities for each gender-department combination. Figure~\ref{fig:posterior-probs} gives the same quantiles, plus the mean posterior-probability for each gender-department combination. From these we can see that for each department, there is considerable overlap of the distributions of probabilities for males and females. <>= posterior.probabilities <- t(apply(berkeley.out$batch, 1, function(r) { eMB <- exp(M %*% r) eMB / (1 + eMB) })) quants <- apply(posterior.probabilities, 2, quantile, prob=c(0.05, 0.95)) quants.str <- matrix(apply(quants, 2, function(r) sprintf("[%0.2f, %0.2f]", r[1], r[2])), nrow=2, byrow=TRUE) @ \begin{table}[ht] \caption{5\% and 95\% posterior quantiles for admittance probability for each gender-department combination} \begin{center} \begin{tabular}{|l|c|c|c|c|c|c|} \hline Gender & Dept. A & Dept. B & Dept. C & Dept. D & Dept. E. & Dept. F \\ \hline Female & \Sexpr{paste(quants.str[1, 1:6], collapse=" & ")} \\ Male & \Sexpr{paste(quants.str[2, 1:6], collapse=" & ")} \\ \hline \end{tabular} \label{tab:post-quant} \end{center} \end{table} \begin{figure} \begin{center} <>= x <- (0:5) * 2 + 1 plot(x[c(1, 6)] + 0.5 * c(-1, 1), 0:1, xlab="Department", ylab="Probability", xaxt="n", type="n") axis(1, x, LETTERS[1:6]) for(i in 1:6) { lines((x[i]-0.25)*c(1, 1), quants[1:2, i], lwd=2, col="gray") lines((x[i] + 0.25) * c(1, 1), quants[1:2, i + 6], lwd=2, col="gray") points(x[i] + 0.25 * c(-1, 1), dat$p.post[i + c(0, 6)], pch=c("F", "M")) } @ \end{center} \caption{Posterior 5\% and 95\% quantiles and mean, by department and gender.} \label{fig:posterior-probs} \end{figure} \section{Cauchy Location-Scale Model} We are going to do a Cauchy location-scale family objective Bayesianly. \subsection{Data} First we generate some data. <>= n <- 15 mu0 <- 50 sigma0 <- 10 x <- rcauchy(n, mu0, sigma0) round(sort(x), 1) @ \texttt{mu0} and \texttt{sigma0} are the true unknown parameter values (since the data are simulated we actually know these ``unknown'' parameter values, but we must pretend we don't know them and estimate them). \subsection{Prior} The standard objective prior distribution for this situation (insofar as any prior distribution can be said to be an objective standard) is the improper prior $$ g(\mu, \sigma) = \frac{1}{\sigma} $$ which is right Haar measure for the location-scale group, and is the standard prior that comes from the group invariance argument \citep[Section~3.2]{kass-wasserman}. \subsection{Log Unnormalized Posterior} We need a function whose argument is a two-vector <>= lup <- function(theta) { if (any(is.na(theta))) stop("NA or NaN in input to log unnormalized density function") mu <- theta[1] sigma <- theta[2] if (sigma <= 0) return(-Inf) if (any(! is.finite(theta))) return(-Inf) result <- sum(dcauchy(x, mu, sigma, log = TRUE)) - log(sigma) if (! is.finite(result)) { warning(paste("Oops! mu = ", mu, "and sigma =", sigma)) } return(result) } @ \subsection{Laplace Approximation} To have some idea what we are doing, we first maximize the log unnormalized posterior. To do it helps to have good starting points for the optimization. Robust estimators of location and scale are <>= mu.twiddle <- median(x) sigma.twiddle <- IQR(x) c(mu.twiddle, sigma.twiddle) @ The posterior mode is <>= oout <- optim(c(mu.twiddle, sigma.twiddle), lup, control = list(fnscale = -1), hessian = TRUE) stopifnot(oout$convergence == 0) mu.hat <- oout$par[1] sigma.hat <- oout$par[2] c(mu.hat, sigma.hat) @ and the hessian evaluated at the posterior mode (calculated by \texttt{optim} using finite differences) is <>= oout$hessian @ The hessian is nearly diagonal and one can check that theoretically is exactly diagonal. Thus approximate (asymptotic) posterior standard deviations are <>= sqrt(- 1 / diag(oout$hessian)) @ \subsection{Theory} To use the theory in \citet{johnson-geyer} we must verify that the target distribution (the unnormalized posterior) is everywhere positive, and it isn't (it is zero for $\sigma \le 0$). We tried making $\log(\sigma)$ the parameter but this didn't work either because $\log(\sigma)$ goes to infinity so slowly that this stretches out the tails so much that the transformations introduced by \citet{johnson-geyer} can't pull them back in again. We do know \citep[Example~3.4]{johnson-geyer} that if we fix $\sigma$ this is a sub-exponentially light target distribution. Letting $\sigma$ vary can only make this worse. Thus, if we don't do anything and just use the \texttt{metrop} function, then performance will be very bad. So we are going to use the transformations and the \texttt{morph.metrop} function, even though the theory that motivates them does not hold. \subsection{Morph} We want to center the transformation at the posterior mode, and use a radius $r$ that doesn't transform until several approximate standard deviations <>= moo <- morph(b = 0.5, r = 7, center = c(mu.hat, sigma.hat)) mout <- morph.metrop(lup, c(mu.hat, sigma.hat), 1e4, scale = 3, morph = moo) mout$accept mout <- morph.metrop(mout) @ Good enough. An attempt to increase the scale led to error when the transformation functions overflowed. Can't take steps too big with this stuff. The following code <>= acf(mout$batch) @ makes an autocorrelation plot (Figure~\ref{fig:cfig1}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Autocorrelation plot. First component is $\mu$, second is $\sigma$.} \label{fig:cfig1} \end{figure} It looks like lag 10 to 15 is enough to get near independence. Now we want to make marginal density plots. If we just feed our MCMC output to the R function \texttt{density} it undersmooths because it expects independent and identically distributed data rather than autocorrelated data. Thus we feed it subsampled, nearly uncorrelated data to select the bandwidth and then use that bandwidth on the full data. Here's how that works. The following code <>= mu <- mout$batch[ , 1] i <- seq(1, mout$nbatch, by = 15) out.sub <- density(mu[i]) out <- density(mu, bw = out.sub$bw) plot(out) @ makes the density plot (Figure~\ref{fig:cfig2}). \begin{figure} \begin{center} <>= <> @ \end{center} \caption{Density plot for the marginal posterior for $\mu$.} \label{fig:cfig2} \end{figure} And a similar plot for $\sigma$ (Figure~\ref{fig:cfig3}) \begin{figure} \begin{center} <>= sigma <- mout$batch[ , 2] out.sub <- density(sigma[i]) out <- density(sigma, bw = out.sub$bw) plot(out) @ \end{center} \caption{Density plot for the marginal posterior for $\sigma$.} \label{fig:cfig3} \end{figure} \begin{thebibliography}{} \bibitem[Jarner and Roberts(2007)]{jarner-roberts} Jarner, S.F., and G.O. Roberts (2007). \newblock Convergence of heavy-tailed Monte Carlo Markov chain algorithms. \newblock \emph{Scandinavian Journal of Statistics}, 34, 781--815. \bibitem[Jarner and Tweedie(2003)]{jarner-tweedie} Jarner, S.~F., and Tweedie, R.~L. (2003). \newblock Necessary conditions for geometric and polynomial ergodicity of random-walk-type Markov chains. \newblock \emph{Bernoulli}, 9, 559--578. \bibitem[Johnson(2011)]{johnson-thesis} Johnson, L.~T. (2011). \newblock Geometric Ergodicity of a Random-Walk Metropolis Algorithm via Variable Transformation and Computer Aided Reasoning in Statistics. \newblock Ph.~D. thesis. University of Minesota. \url{http://purl.umn.edu/113140} \bibitem[Johnson and Geyer(submitted)]{johnson-geyer} Johnson, L.~T., and C.~J. Geyer (submitted). \newblock Variable Transformation to Obtain Geometric Ergodicity in the Random-walk Metropolis Algorithm. \newblock Revised and resubmitted to \emph{Annals of Statistics}. \bibitem[Kass and Wasserman(1996)]{kass-wasserman} Kass, R.~E., and Wasserman, L. (1996). \newblock Formal rules for selecting prior distributions: A review and annotated bibliography. \newblock \emph{Journal of the American Statistical Association}, 435, 1343--1370. \bibitem[Mengersen and Tweedie(1996)]{mengersen-tweedie} Mengersen, K.L., ad R. L. 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hornikusersmcmc/build/vignette.rds0000644000175100001440000000044513074514644014660 0ustar hornikusersRN0R&#y1$F MFE09\NEyc< .ށ \ rDukK16 9ORXV'nHJԆ and Leif T. Johnson Maintainer: Charles J. Geyer Depends: R (>= 3.0.2) Imports: stats, compiler Suggests: xtable, Iso ByteCompile: TRUE Description: Simulates continuous distributions of random vectors using Markov chain Monte Carlo (MCMC). Users specify the distribution by an R function that evaluates the log unnormalized density. Algorithms are random walk Metropolis algorithm (function metrop), simulated tempering (function temper), and morphometric random walk Metropolis (Johnson and Geyer, 2012, , function morph.metrop), which achieves geometric ergodicity by change of variable. License: MIT + file LICENSE URL: http://www.stat.umn.edu/geyer/mcmc/, https://github.com/cjgeyer/mcmc NeedsCompilation: yes Packaged: 2017-04-15 22:07:33 UTC; geyer Repository: CRAN Date/Publication: 2017-04-16 10:23:50 UTC mcmc/ChangeLog0000644000175100001440000000630013074505364012767 0ustar hornikusers 0.4.1 fixed documentation for metrop 0.5 changed licence to X11 (was AFL, which apparently sucks) changed to NAMESPACE cleaned up latex in inst/doc, added Makefile removed src/Makevars had to get rid of R_IsNaNorNA in src/metrop.c generally, lots of changes to get ready for CRAN all of the tests/*.Rout.save redone because the digits in numerical printing seem to have changed -- cannot get bit for bit identical with earlier R (even before code changes). changed first arg of metrop from "o" to "obj" 0.5-1 changed package vignette to fix bogus MCSE calculation (bogosity still admitted and explained in appendix). 0.6 added initseq function 0.7 added temper function -- not tested yet 0.7-1 fix bug && to || much confusion about whether my_i and my_j were 0-origin or 1-origin forgot to allocate debug_acceptd confusion about dimension of state (nx + 1 vector if serial, ncomp x nx matrix if parallel) and <= nx versus < nx in loop bounds bug where my_swapped_proposal_log_dens was used twice when one should have been my_swapped_coproposal_log_dens bug in dimension of debug_state when parallel (last dim nx not nx + 1) accepted swap didn't actually swap, fixed all the accept decisions when log_hastings_ratio < 0 were backwards batch means accumulation has = instead of += batch means divided by nbatch rather than blen accepti entries stored in wrong places in output structure ripped out old system for caching lud values, put in new that caches all (more efficient, but real reason was old was broken somehow) of course, its first implementation had 0-origin vs 1-origin confuse yet another instance 0-origin vs 1-origin confusion (my_i should be my_i - 1 length of output structure (len_result) determined incorrectly, fixed caching of current lud value broken in serial case (stored in i not j) after all that, everything now checked even with --use-valgrind and --use-gct 0.7-3 add bugs section to initseq.Rd fix example in temper.Rd to actually be for temper rather than metrop fix comment in temper.c add inst/doc/debug.Rnw a vignette about debugging and debug output 0.7-4 some changes to /inst/doc/demo.Rnw 0.7-5 add inst/doc/bfst.Rnw a vignette about Bayes factors and serial tempering 0.8 earlier versions calculated the Hastings ratio wrong in doing serial tempering, ignoring the number of neighbors each component had, hence computing wrong answers when components differed in number of neighbors 0.9 morph, morph.metrop functions and morph.pdf vignette bumped required R version to 2.10.0 (required by xtable) 0.9-1 added PACKAGE = "mcmc" where missing to .C and .Call 0.9-3 move vignettes to vignettes directory changed LICENCE file yet again as per CRAN requirements 0.9-4 cleaned up some tests import from stats, as required by R-3.3.0 0.9-5 use registration of native routines described in Sections 5.4 and 6.15 of Writing R Extensions add calls to cmpfun inside metrop and temper mcmc/man/0000755000175100001440000000000013063023671011763 5ustar hornikusersmcmc/man/metrop.Rd0000644000175100001440000002757113063023671013574 0ustar hornikusers\name{metrop} \alias{metrop} \alias{metrop.function} \alias{metrop.metropolis} \title{Metropolis Algorithm} \description{ Markov chain Monte Carlo for continuous random vector using a Metropolis algorithm. } \usage{ metrop(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) \method{metrop}{function}(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) \method{metrop}{metropolis}(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, ...) } \arguments{ \item{obj}{Either an \R function or an object of class \code{"metropolis"} from a previous invocation of this function. If a function, it evaluates the log unnormalized probability density of the desired equilibrium distribution of the Markov chain. Its first argument is the state vector of the Markov chain. Other arguments arbitrary and taken from the \code{...} arguments of this function. It should return \code{-Inf} for points of the state space having probability zero under the desired equilibrium distribution. See also Details and Warning. If an object of class \code{"metropolis"}, any missing arguments (including the log unnormalized density function) are taken from this object. Also \code{initial} is ignored and the initial state of the Markov chain is the final state from the run recorded in \code{obj}. } \item{initial}{a real vector, the initial state of the Markov chain. Must be feasible, see Details. Ignored if \code{obj} is of class \code{"metropolis"}.} \item{nbatch}{the number of batches.} \item{blen}{the length of batches.} \item{nspac}{the spacing of iterations that contribute to batches.} \item{scale}{controls the proposal step size. If scalar or vector, the proposal is \code{x + scale * z} where \code{x} is the current state and \code{z} is a standard normal random vector. If matrix, the proposal is \code{x + scale \%*\% z}.} \item{outfun}{controls the output. If a function, then the batch means of \code{outfun(state, ...)} are returned. If a numeric or logical vector, then the batch means of \code{state[outfun]} (if this makes sense). If missing, the the batch means of \code{state} are returned.} \item{debug}{if \code{TRUE} extra output useful for testing.} \item{...}{additional arguments for \code{obj} or \code{outfun}.} } \details{ Runs a \dQuote{random-walk} Metropolis algorithm, terminology introduced by Tierney (1994), with multivariate normal proposal producing a Markov chain with equilibrium distribution having a specified unnormalized density. Distribution must be continuous. Support of the distribution is the support of the density specified by argument \code{obj}. The initial state must satisfy \code{obj(state, ...) > -Inf}. Description of a complete MCMC analysis (Bayesian logistic regression) using this function can be found in the vignette \code{vignette("demo", "mcmc")}. Suppose the function coded by the log unnormalized function (either \code{obj} or \code{obj$lud}) is actually a log unnormalized density, that is, if \eqn{w} denotes that function, then \eqn{e^w}{exp(w)} integrates to some value strictly between zero and infinity. Then the \code{metrop} function always simulates a reversible, Harris ergodic Markov chain having the equilibrium distribution with this log unnormalized density. The chain is not guaranteed to be geometrically ergodic. In fact it cannot be geometrically ergodic if the tails of the log unnormalized density are sufficiently heavy. The \code{\link{morph.metrop}} function deals with this situation. } \value{ an object of class \code{"mcmc"}, subclass \code{"metropolis"}, which is a list containing at least the following components: \item{accept}{fraction of Metropolis proposals accepted.} \item{batch}{\code{nbatch} by \code{p} matrix, the batch means, where \code{p} is the dimension of the result of \code{outfun} if \code{outfun} is a function, otherwise the dimension of \code{state[outfun]} if that makes sense, and the dimension of \code{state} when \code{outfun} is missing.} \item{accept.batch}{a vector of length \code{nbatch}, the batch means of the acceptances.} \item{initial}{value of argument \code{initial}.} \item{final}{final state of Markov chain.} \item{initial.seed}{value of \code{.Random.seed} before the run.} \item{final.seed}{value of \code{.Random.seed} after the run.} \item{time}{running time of Markov chain from \code{system.time()}.} \item{lud}{the function used to calculate log unnormalized density, either \code{obj} or \code{obj$lud} from a previous run.} \item{nbatch}{the argument \code{nbatch} or \code{obj$nbatch}.} \item{blen}{the argument \code{blen} or \code{obj$blen}.} \item{nspac}{the argument \code{nspac} or \code{obj$nspac}.} \item{outfun}{the argument \code{outfun} or \code{obj$outfun}.} Description of additional output when \code{debug = TRUE} can be found in the vignette \code{debug} (\url{../doc/debug.pdf}). } \section{Warning}{ If \code{outfun} is missing or not a function, then the log unnormalized density can be defined without a \ldots argument and that works fine. One can define it starting \code{ludfun <- function(state)} and that works or \code{ludfun <- function(state, foo, bar)}, where \code{foo} and \code{bar} are supplied as additional arguments to \code{metrop}. If \code{outfun} is a function, then both it and the log unnormalized density function can be defined without \ldots arguments \emph{if they have exactly the same arguments list} and that works fine. Otherwise it doesn't work. Define these functions by \preformatted{ ludfun <- function(state, foo) outfun <- function(state, bar) } and you get an error about unused arguments. Instead define these functions by \preformatted{ ludfun <- function(state, foo, \ldots) outfun <- function(state, bar, \ldots) } and supply \code{foo} and \code{bar} as additional arguments to \code{metrop}, and that works fine. In short, the log unnormalized density function and \code{outfun} need to have \ldots in their arguments list to be safe. Sometimes it works when \ldots is left out and sometimes it doesn't. Of course, one can avoid this whole issue by always defining the log unnormalized density function and \code{outfun} to have only one argument \code{state} and use global variables (objects in the \R global environment) to specify any other information these functions need to use. That too follows the \R way. But some people consider that bad programming practice. } \section{Philosophy of MCMC}{ This function follows the philosophy of MCMC explained the introductory chapter of the \emph{Handbook of Markov Chain Monte Carlo} (Geyer, 2011). This function automatically does batch means in order to reduce the size of output and to enable easy calculation of Monte Carlo standard errors (MCSE), which measure error due to the Monte Carlo sampling (not error due to statistical sampling --- MCSE gets smaller when you run the computer longer, but statistical sampling variability only gets smaller when you get a larger data set). All of this is explained in the package vignette \code{vignette("demo", "mcmc")} and in Section 1.10 of Geyer (2011). This function does not apparently do \dQuote{burn-in} because this concept does not actually help with MCMC (Geyer, 2011, Section 1.11.4) but the re-entrant property of this function does allow one to do \dQuote{burn-in} if one wants. Assuming \code{ludfun}, \code{start.value}, \code{scale} have been already defined and are, respectively, an \R function coding the log unnormalized density of the target distribution, a valid state of the Markov chain, and a useful scale factor, \preformatted{ out <- metrop(ludfun, start.value, nbatch = 1, blen = 1e5, scale = scale) out <- metrop(out, nbatch = 100, blen = 1000) } throws away a run of 100 thousand iterations before doing another run of 100 thousand iterations that is actually useful for analysis, for example, \preformatted{ apply(out$batch, 2, mean) apply(out$batch, 2, sd) / sqrt(out$nbatch) } give estimates of posterior means and their MCSE assuming the batch length (here 1000) was long enough to contain almost all of the signifcant autocorrelation (see Geyer, 2011, Section 1.10, for more on MCSE). The re-entrant property of this function (the second run starts where the first one stops) assures that this is really \dQuote{burn-in}. The re-entrant property allows one to do very long runs without having to do them in one invocation of this function. \preformatted{ out2 <- metrop(out) out3 <- metrop(out2) batch <- rbind(out$batch, out2$batch, out3$batch) } produces a result as if the first run had been three times as long. } \section{Tuning}{ The \code{scale} argument must be adjusted so that the acceptance rate is not too low or too high to get reasonable performance. The rule of thumb is that the acceptance rate should be about 25\%. But this recommendation (Gelman, et al., 1996) is justified by analysis of a toy problem (simulating a spherical multivariate normal distribution) for which MCMC is unnecessary. There is no reason to believe this is optimal for all problems (if it were optimal, a stronger theorem could be proved). Nevertheless, it is clear that at very low acceptance rates the chain makes little progress (because in most iterations it does not move) and that at very high acceptance rates the chain also makes little progress (because unless the log unnormalized density is nearly constant, very high acceptance rates can only be achieved by very small values of \code{scale} so the steps the chain takes are also very small. Even in the Gelman, et al. (1996) result, the optimal rate for spherical multivariate normal depends on dimension. It is 44\% for \eqn{d = 1} and 23\% for \eqn{d = \infty}{d = infinity}. Geyer and Thompson (1995) have an example, admittedly for simulated tempering (see \code{\link{temper}}) rather than random-walk Metropolis, in which no acceptance rate less than 70\% produces an ergodic Markov chain. Thus 25\% is merely a rule of thumb. We only know we don't want too high or too low. Probably 1\% or 99\% is very inefficient. } \references{ Gelman, A., Roberts, G. O., and Gilks, W. R. (1996) Efficient Metropolis jumping rules. In \emph{Bayesian Statistics 5: Proceedings of the Fifth Valencia International Meeting}. Edited by J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith. Oxford University Press, Oxford, pp. 599--607. Geyer, C. J. (2011) Introduction to MCMC. In \emph{Handbook of Markov Chain Monte Carlo}. Edited by S. P. Brooks, A. E. Gelman, G. L. Jones, and X. L. Meng. Chapman & Hall/CRC, Boca Raton, FL, pp. 3--48. Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. \emph{Journal of the American Statistical Association} \bold{90} 909--920. Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion). \emph{Annals of Statistics} \bold{22} 1701--1762. } \seealso{ \code{\link{morph.metrop}} and \code{\link{temper}} } \examples{ h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) out <- metrop(h, rep(0, 5), 1000) out$accept # acceptance rate too low out <- metrop(out, scale = 0.1) out$accept t.test(out$accept.batch)$conf.int # acceptance rate o. k. (about 25 percent) plot(out$batch[ , 1]) # but run length too short (few excursions from end to end of range) out <- metrop(out, nbatch = 1e4) out$accept plot(out$batch[ , 1]) hist(out$batch[ , 1]) acf(out$batch[ , 1], lag.max = 250) # looks like batch length of 250 is perhaps OK out <- metrop(out, blen = 250, nbatch = 100) apply(out$batch, 2, mean) # Monte Carlo estimates of means apply(out$batch, 2, sd) / sqrt(out$nbatch) # Monte Carlo standard errors t.test(out$accept.batch)$conf.int acf(out$batch[ , 1]) # appears that blen is long enough } \keyword{misc} mcmc/man/olbm.Rd0000644000175100001440000000233213063023671013203 0ustar hornikusers\name{olbm} \alias{olbm} \title{Overlapping Batch Means} \description{ Variance of sample mean of time series calculated using overlapping batch means. } \usage{ olbm(x, batch.length, demean = TRUE) } \arguments{ \item{x}{a matrix or time series object. Each column of \code{x} is treated as a scalar time series.} \item{batch.length}{length of batches.} \item{demean}{when \code{demean = TRUE} (the default) the sample mean is subtracted from each batch mean when estimating the variance. Using \code{demean = FALSE} would essentially assume the true mean is known to be zero, which might be useful in a toy problem where the answer is known.} } \value{ The estimated variance of the sample mean. } \seealso{ \code{\link{ts}} } \examples{ h <- function(x) if (all(x >= 0) && sum(x) <= 1) return(1) else return(-Inf) out <- metrop(h, rep(0, 5), 1000) out <- metrop(out, scale = 0.1) out <- metrop(out, nbatch = 1e4) foo <- olbm(out$batch, 150) # monte carlo estimates (true means are same by symmetry) apply(out$batch, 2, mean) # monte carlo standard errors (true s. d. are same by symmetry) sqrt(diag(foo)) # check that batch length is reasonable acf(out$batch, lag.max = 200) } \keyword{ts} mcmc/man/logit.Rd0000644000175100001440000000101413045741240013363 0ustar hornikusers\name{logit} \docType{data} \alias{logit} \title{Simulated logistic regression data.} \description{ Like it says } \usage{data(logit)} \format{ A data frame with variables \describe{ \item{x1}{quantitative predictor.} \item{x2}{quantitative predictor.} \item{x3}{quantitative predictor.} \item{x4}{quantitative predictor.} \item{y}{Bernoulli response.} } } \examples{ library(mcmc) data(logit) out <- glm(y ~ x1 + x2 + x3 + x4, family = binomial, data = logit) summary(out) } \keyword{datasets} mcmc/man/morph.Rd0000644000175100001440000001255113045741600013402 0ustar hornikusers\name{morph} \encoding{UTF-8} \alias{morph} \alias{morph.identity} \title{Variable Transformation} \description{ Utility functions for variable transformation. } \usage{ morph(b, r, p, center) morph.identity() } \arguments{ \item{b}{Positive real number. May be missing.} \item{r}{Non-negative real number. May be missing. If \code{p} is specified, \code{r} defaults to 0.} \item{p}{Real number strictly greater than 2. May be missing. If \code{r} is specified, \code{p} defaults to 3.} \item{center}{Real scalar or vector. May be missing. If \code{center} is a vector it should be the same length of the state of the Markov chain, \code{center} defaults to 0} } \section{Warning}{ The equations for the returned \code{transform} function (see below) do not have a general analytical solution when \code{p} is not equal to 3. This implementation uses numerical approximation to calculate \code{transform} when \code{p} is not equal to 3. If computation speed is a factor, it is advisable to use \code{p=3}. This is not a factor when using \code{\link{morph.metrop}}, as \code{transform} is only called once during setup, and not at all while running the Markov chain. } \details{ The \code{morph} function facilitates using variable transformations by providing functions to (using \eqn{X} for the original random variable with the pdf \eqn{f_X}{f.X}, and \eqn{Y} for the transformed random variable with the pdf \eqn{f_Y}{f.Y}): \itemize{ \item Calculate the log unnormalized probability density for \eqn{Y} induced by the transformation. \item Transform an arbitrary function of \eqn{X} to a function of \eqn{Y}. \item Transform values of \eqn{X} to values of \eqn{Y}. \item Transform values of \eqn{Y} to values of \eqn{X} (the inverse transformation). } for a select few transformations. \code{morph.identity} implements the identity transformation, \eqn{Y=X}. The parameters \code{r}, \code{p}, \code{b} and \code{center} specify the transformation function. In all cases, \code{center} gives the center of the transformation, which is the value \eqn{c} in the equation \deqn{Y = f(X - c).} If no parameters are specified, the identity transformation, \eqn{Y=X}, is used. The parameters \code{r}, \code{p} and \code{b} specify a function \eqn{g}, which is a monotonically increasing bijection from the non-negative reals to the non-negative reals. Then \deqn{f(X) = g\bigl(|X|\bigr) \frac{X}{|X|}}{f(X) = g(|X|) * X / |X|} where \eqn{|X|} represents the Euclidean norm of the vector \eqn{X}. The inverse function is given by \deqn{f^{-1}(Y) = g^{-1}\bigl(|Y|\bigr) \frac{Y}{|Y|}.}{f^{-1}(Y) = g^{-1}(|Y|) * Y / |Y|.} The parameters \code{r} and \code{p} are used to define the function \deqn{g_1(x) = x + (x-r)^p I(x > r)}{g1(x) = x + (x-r)^p * I(x > r)} where \eqn{I( \cdot )}{I(•)} is the indicator function. We require that \code{r} is non-negative and \code{p} is strictly greater than 2. The parameter \code{b} is used to define the function \deqn{g_2(x) = \bigl(e^{bx} - e / 3\bigr) I(x > \frac{1}{b}) + \bigl(x^3 b^3 e / 6 + x b e / 2\bigr) I(x \leq \frac{1}{b})}{ g2(x) = (exp(b * x) - exp(1) / 3) * I(x > 1 / b) + (x^3 * b^3 exp(1) / 6 + x * b * exp(1) / 2) * I(x <= 1 / b).} We require that \eqn{b} is positive. The parameters \code{r}, \code{p} and \code{b} specify \eqn{f^{-1}} in the following manner: \itemize{ \item If one or both of \code{r} and \code{p} is specified, and \code{b} is not specified, then \deqn{f^{-1}(X) = g_1(|X|) \frac{X}{|X|}.}{f^{-1}(X) = g1(|X|) * X / |X|.} If only \code{r} is specified, \code{p = 3} is used. If only \code{p} is specified, \code{r = 0} is used. \item If only \code{b} is specified, then \deqn{f^{-1}(X) = g_2(|X|) \frac{X}{|X|}.}{f^{-1}(X) = g2(|X|) * X / |X|.} \item If one or both of \code{r} and \code{p} is specified, and \code{b} is also specified, then \deqn{f^{-1}(X) = g_2(g_1(|X|)) \frac{X}{|X|}.}{f^{-1}(X) = g2(g1(|X|)) * X / |X|.} } } \value{ a list containing the functions \itemize{ \item \code{outfun(f)}, a function that operates on functions. \code{outfun(f)} returns the function \code{function(state, ...) f(inverse(state), ...)}. \item \code{inverse}, the inverse transformation function. \item \code{transform}, the transformation function. \item \code{lud}, a function that operates on functions. As input, \code{lud} takes a function that calculates a log unnormalized probability density, and returns a function that calculates the log unnormalized density by transforming a random variable using the \code{transform} function. \code{lud(f) = function(state, ...) f(inverse(state), ...) + log.jacobian(state)}, where \code{log.jacobian} represents the function that calculate the log Jacobian of the transformation. \code{log.jacobian} is not returned. } } \examples{ # use an exponential transformation, centered at 100. b1 <- morph(b=1, center=100) # original log unnormalized density is from a t distribution with 3 # degrees of freedom, centered at 100. lud.transformed <- b1$lud(function(x) dt(x - 100, df=3, log=TRUE)) d.transformed <- Vectorize(function(x) exp(lud.transformed(x))) \dontrun{ curve(d.transformed, from=-3, to=3, ylab="Induced Density") } } \seealso{ \code{\link{morph.metrop}} } \keyword{misc} mcmc/man/morph.metrop.Rd0000644000175100001440000001527413045741600014714 0ustar hornikusers\name{morph.metrop} \alias{morph.metrop} \alias{morph.metrop.function} \alias{morph.metrop.morph.metropolis} \title{Morphometric Metropolis Algorithm} \description{ Markov chain Monte Carlo for continuous random vector using a Metropolis algorithm for an induced density. } \usage{ morph.metrop(obj, initial, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, morph, ...) } \arguments{ \item{obj}{see \code{\link{metrop}}.} \item{initial}{see \code{\link{metrop}}.} \item{nbatch}{see \code{\link{metrop}}.} \item{blen}{see \code{\link{metrop}}.} \item{nspac}{see \code{\link{metrop}}.} \item{scale}{see \code{\link{metrop}}.} \item{outfun}{unlike for \code{\link{metrop}} must be a function or missing; if missing the identity function, \code{function(x) x}, is used.} \item{debug}{see \code{\link{metrop}}.} \item{morph}{morph object used for transformations. See \code{\link{morph}}.} \item{...}{see \code{\link{metrop}}.} } \details{ \code{morph.metrop} implements morphometric methods for Markov chains. The caller specifies a log unnormalized probability density and a transformation. The transformation specified by the \code{morph} parameter is used to induce a new log unnormalized probability density, a Metropolis algorithm is run for the induced density. The Markov chain is transformed back to the original scale. Running the Metropolis algorithm for the induced density, instead of the original density, can result in a Markov chain with better convergence properties. For more details see Johnson and Geyer (submitted). Except for \code{morph}, all parameters are passed to \code{\link{metrop}}, transformed when necessary. The \code{scale} parameter is \emph{not} transformed. If \eqn{X} is a real vector valued continuous random variable, and \eqn{Y = f(X)} where \eqn{f} is a diffeomorphism, then the pdf of \eqn{Y} is given by \deqn{f_Y(y) = f_X(f^{-1}(y)) | \nabla f^{-1}(y) |}{ fY(y) = fX(f^{-1}(y)) | del f^{-1}(y) |} where \eqn{f_X}{fX} is the pdf of \eqn{X} and \eqn{\nabla f^{-1}}{del f^{-1}} is the Jacobian of \eqn{f^{-1}}. Because \eqn{f} is a diffeomorphism, a Markov chain for \eqn{f_Y}{fY} may be transformed into a Markov chain for \eqn{f_X}{fX}. Furthermore, these Markov chains are isomorphic (Johnson and Geyer, submitted) and have the same convergence rate. The \code{\link{morph}} variable provides a diffeomorphism, \code{morph.metrop} uses this diffeomorphism to induce the log unnormalized density, \eqn{\log f_Y}{log fY} based on the user supplied log unnormalized density, \eqn{\log f_X}{log fX}. \code{morph.metrop} runs a Metropolis algorithm for \eqn{\log f_Y}{log fY} and transforms the resulting Markov chain into a Markov chain for \eqn{f_X}{fX}. The user accessible output components are the same as those that come from \code{\link{metrop}}, see the documentation for \code{\link{metrop}} for details. Subsequent calls of \code{morph.metrop} may change to the transformation by specifying a new value for \code{morph}. Any of the other parameters to \code{morph.metrop} may also be modified in subsequent calls. See \code{\link{metrop}} for more details. The general idea is that a random-walk Metropolis sampler (what \code{\link{metrop}} does) will not be geometrically ergodic unless the tails of the unnormalized density decrease superexponentially fast (so the tails of the log unnormalized density decrease faster than linearly). It may not be geometrically ergodic even then (see Johnson and Geyer, submitted, for the complete theory). The transformations used by this function (provided by \code{\link{morph}}) can produce geometrically ergodic chains when the tails of the log unnormalized density are too light for \code{\link{metrop}} to do so. When the tails of the unnormalized density are exponentially light but not superexponentially light (so the tails of the log unnormalized density are asymptotically linear, as in the case of exponential family models when conjugate priors are used, for example logistic regression, Poisson regression with log link, or log-linear models for categorical data), one should use \code{\link{morph}} with \code{b = 0} (the default), which produces a transformation of the form \eqn{g_1}{g1} in the notation used in the details section of the help for \code{\link{morph}}. This will produce a geometrically ergodic sampler if other features of the log unnormalized density are well behaved. For example it will do so for the exponential family examples mentioned above. (See Johnson and Geyer, submitted, for the complete theory.) The transformation \eqn{g_1}{g1} behaves like a shift transformation on a ball of radius \code{r} centered at \code{center}, so these arguments to \code{\link{morph}} should be chosen so that a sizable proportion of the probability under the original (untransformed) unnormalized density is contained in this ball. This function will work when \code{r = 0} and \code{center = 0} (the defaults) are used, but may not work as well as when \code{r} and \code{center} are well chosen. When the tails of the unnormalized density are not exponentially light (so the tails of the log unnormalized density decrease sublinearly, as in the case of univariate and multivariate \eqn{t} distributions), one should use \code{\link{morph}} with \code{r > 0} and \code{p = 3}, which produces a transformation of the form \eqn{g_2}{g2} composed with \eqn{g_1}{g1} in the notation used in the details section of the help for \code{\link{morph}}. This will produce a geometrically ergodic sampler if other features of the log unnormalized density are well behaved. For example it will do so for the \eqn{t} examples mentioned above. (See Johnson and Geyer, submitted, for the complete theory.) } \value{ an object of class \code{mcmc}, subclass \code{morph.metropolis}. This object is a list containing all of the elements from an object returned by \code{\link{metrop}}, plus at least the following components: \item{morph}{the morph object used for the transformations.} \item{morph.final}{the final state of the Markov chain on the transformed scale.} } \examples{ out <- morph.metrop(function(x) dt(x, df=3, log=TRUE), 0, blen=100, nbatch=100, morph=morph(b=1)) # change the transformation. out <- morph.metrop(out, morph=morph(b=2)) out$accept # accept rate is high, increase the scale. out <- morph.metrop(out, scale=4) # close to 0.20 is about right. out$accept } \references{ Johnson, L. T. and Geyer, C. J. (submitted) Variable Transformation to Obtain Geometric Ergodicity in the Random-walk Metropolis Algorithm. } \seealso{ \code{\link{metrop}}, \code{\link{morph}}. } \keyword{misc} mcmc/man/initseq.Rd0000644000175100001440000001060113045741600013723 0ustar hornikusers\name{initseq} \alias{initseq} \title{Initial Sequence Estimators} \description{ Variance of sample mean of functional of reversible Markov chain using methods of Geyer (1992). } \usage{ initseq(x) } \arguments{ \item{x}{a numeric vector that is a scalar-valued functional of a reversible Markov chain.} } \details{ Let \deqn{\gamma_k = \textrm{cov}(X_i, X_{i + k})}{gamma[k] = cov(x[i], x[i + k])} considered as a function of the lag \eqn{k} be the autocovariance function of the input time series. Define \deqn{\Gamma_k = \gamma_{2 k} + \gamma_{2 k + 1}}{Gamma[k] = gamma[2 k] + gamma[2 k + 1]} the sum of consecutive pairs of autocovariances. Then Theorem 3.1 in Geyer (1992) says that \eqn{\Gamma_k}{Gamma[k]} considered as a function of \eqn{k} is strictly positive, strictly decreasing, and strictly convex, assuming the input time series is a scalar-valued functional of a reversible Markov chain. All of the MCMC done by this package is reversible. This \R function estimates the \dQuote{big gamma} function, \eqn{\Gamma_k}{Gamma[k]} considered as a function of \eqn{k}, subject to three different constraints, (1) nonnegative, (2) nonnegative and nonincreasing, and (3) nonnegative, nonincreasing, and convex. It also estimates the variance in the Markov chain central limit theorem (CLT) \deqn{\gamma_0 + 2 \sum_{k = 1}^\infty \gamma_k = - \gamma_0 + 2 \sum_{k = 0}^\infty \Gamma_k}{- gamma0 + 2 * sum(gamma) = - gamma0 + 2 * sum(Gamma)} \strong{Note:} The batch means provided by \code{\link{metrop}} are also scalar functionals of a reversible Markov chain. Thus these initial sequence estimators applied to the batch means give valid standard errors for the mean of the match means even when the batch length is too short to provide a valid estimate of asymptotic variance. One does, of course, have to multiply the asymptotic variance of the batch means by the batch length to get the asymptotic variance for the unbatched chain. } \value{ a list containing the following components: \item{gamma0}{the scalar \eqn{\gamma_0}{gamma[0]}, the marginal variance of \code{x}.} \item{Gamma.pos}{the vector \eqn{\Gamma}{Gamma}, estimated so as to be nonnegative, where, as always, \R uses one-origin indexing so \code{Gamma.pos[1]} is \eqn{\Gamma_0}{Gamma[0]}.} \item{Gamma.dec}{the vector \eqn{\Gamma}{Gamma}, estimated so as to be nonnegative and nonincreasing, where, as always, \R uses one-origin indexing so \code{Gamma.dec[1]} is \eqn{\Gamma_0}{Gamma[0]}.} \item{Gamma.con}{the vector \eqn{\Gamma}{Gamma}, estimated so as to be nonnegative and nonincreasing and convex, where, as always, \R uses one-origin indexing so \code{Gamma.con[1]} is \eqn{\Gamma_0}{Gamma[0]}.} \item{var.pos}{the scalar \code{- gamma0 + 2 * sum(Gamma.pos)}, which is the asymptotic variance in the Markov chain CLT. Divide by \code{length(x)} to get the approximate variance of the sample mean of \code{x}.} \item{var.dec}{the scalar \code{- gamma0 + 2 * sum(Gamma.dec)}, which is the asymptotic variance in the Markov chain CLT. Divide by \code{length(x)} to get the approximate variance of the sample mean of \code{x}.} \item{var.con}{the scalar \code{- gamma0 + 2 * sum(Gamma.con)}, which is the asymptotic variance in the Markov chain CLT. Divide by \code{length(x)} to get the approximate variance of the sample mean of \code{x}.} } \section{Bugs}{ Not precisely a bug, but \code{var.pos}, \code{var.dec}, and \code{var.con} can be negative. This happens only when the chain is way too short to estimate the variance, and even then rarely. But it does happen. } \references{ Geyer, C. J. (1992) Practical Markov Chain Monte Carlo. \emph{Statistical Science} \bold{7} 473--483. } \seealso{ \code{\link{metrop}} } \examples{ n <- 2e4 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) out <- initseq(x) \dontrun{ plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l") lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, col = "red") lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, col = "blue") } # asymptotic 95\% confidence interval for mean of x mean(x) + c(-1, 1) * qnorm(0.975) * sqrt(out$var.con / length(x)) # estimated asymptotic variance out$var.con # theoretical asymptotic variance (1 + rho) / (1 - rho) * 1 / (1 - rho^2) # illustrating use with batch means bm <- apply(matrix(x, nrow = 5), 2, mean) initseq(bm)$var.con * 5 } \keyword{ts} mcmc/man/foo.Rd0000644000175100001440000000072613045741240013041 0ustar hornikusers\name{foo} \docType{data} \alias{foo} \title{Simulated logistic regression data.} \description{ Like it says } \usage{data(foo)} \format{ A data frame with variables \describe{ \item{x1}{quantitative predictor.} \item{x2}{quantitative predictor.} \item{x3}{quantitative predictor.} \item{y}{Bernoulli response.} } } \examples{ library(mcmc) data(foo) out <- glm(y ~ x1 + x2 + x3, family = binomial, data = foo) summary(out) } \keyword{datasets} mcmc/man/temper.Rd0000644000175100001440000003052013063023671013546 0ustar hornikusers\name{temper} \alias{temper} \alias{temper.function} \alias{temper.tempering} \title{Simulated Tempering and Umbrella Sampling} \description{ Markov chain Monte Carlo (MCMC) for continuous random vectors using parallel or serial tempering, the latter also called umbrella sampling and simulated tempering. The chain simulates \eqn{k} different distributions on the same state space. In parallel tempering, all the distributions are simulated in each iteration. In serial tempering, only one of the distributions is simulated (a random one). In parallel tempering, the state is a \eqn{k \times p}{k by p} matrix, each row of which is the state for one of the distributions. In serial tempering the state of the Markov chain is a pair \eqn{(i, x)}, where \eqn{i} is an integer between 1 and \eqn{k} and \eqn{x} is a vector of length \eqn{p}. This pair is represented as a single real vector \code{c(i, x)}. The variable \eqn{i} says which distribution \eqn{x} is a simulation for. } \usage{ temper(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, \dots) \method{temper}{function}(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, \dots) \method{temper}{tempering}(obj, initial, neighbors, nbatch, blen = 1, nspac = 1, scale = 1, outfun, debug = FALSE, parallel = FALSE, \dots) } \arguments{ \item{obj}{either an \R function or an object of class \code{"tempering"} from a previous run. If a function, it should evaluate the log unnormalized density \eqn{\log h(i, x)}{log h(i, x)} of the desired equilibrium distribution of the Markov chain for serial tempering (the same function is used for both serial and parallel tempering, see Details below for further explanation). If an object of class \code{"tempering"}, the log unnormalized density function is \code{obj$lud}, and missing arguments of \code{temper} are obtained from the corresponding elements of \code{obj}. The first argument of the log unnormalized density function is the is an \R vector \code{c(i, x)}, where \code{i} says which distribution \code{x} is supposed to be a simulation for. Other arguments are arbitrary and taken from the \code{\dots} arguments of \code{temper}. The log unnormalized density function should return \code{-Inf} in regions of the state space having probability zero.} \item{initial}{for serial tempering, a real vector \code{c(i, x)} as described above. For parallel tempering, a real \eqn{k \times p}{k by p} matrix as described above. In either case, the initial state of the Markov chain. Ignored if \code{obj} has class \code{"tempering"}.} \item{neighbors}{a logical symmetric matrix of dimension \code{k} by \code{k}. Elements that are \code{TRUE} indicate jumps or swaps attempted by the Markov chain. Ignored if \code{obj} has class \code{"tempering"}.} \item{nbatch}{the number of batches.} \item{blen}{the length of batches.} \item{nspac}{the spacing of iterations that contribute to batches.} \item{scale}{controls the proposal step size for real elements of the state vector. For serial tempering, proposing a new value for the \eqn{x} part of the state \eqn{(i, x)}. For parallel tempering, proposing a new value for the \eqn{x_i}{x[i]} part of the state \eqn{(x_1, \ldots, x_k)}{(x[1], \ldots, x[k])}. In either case, the proposal is a real vector of length \eqn{p}. If scalar or vector, the proposal is \code{x + scale * z} where \code{x} is the part \eqn{x} or \eqn{x_i}{x[i]} of the state the proposal may replace. If matrix, the proposal is \code{x + scale \%*\% z}. If list, the length must be \code{k}, and each element must be scalar, vector, or matrix, and operate as described above. The \eqn{i}-th component of the list is used to update \eqn{x} when the state is \eqn{(i, x)} or \eqn{x_i}{x[i]} otherwise.} \item{outfun}{controls the output. If a function, then the batch means of \code{outfun(state, \dots)} are returned. The argument \code{state} is like the argument \code{initial} of this function. If missing, the batch means of the real part of the state vector or matrix are returned, and for serial tempering the batch means of a multivariate Bernoulli indicating the current component are returned.} \item{debug}{if \code{TRUE} extra output useful for testing.} \item{parallel}{if \code{TRUE} does parallel tempering, if \code{FALSE} does serial tempering. Ignored if \code{obj} has class \code{"tempering"}.} \item{...}{additional arguments for \code{obj} or \code{outfun}.} } \details{ Serial tempering simulates a mixture of distributions of a continuous random vector. The number of components of the mixture is \code{k}, and the dimension of the random vector is \code{p}. Denote the state \eqn{(i, x)}, where \eqn{i} is a positive integer between 1 and \eqn{k}, and let \eqn{h(i, x)} denote the unnormalized joint density of their equilibrium distribution. The logarithm of this function is what \code{obj} or \code{obj$lud} calculates. The mixture distribution is the marginal for \eqn{x} derived from the equilibrium distribution \eqn{h(i, x)}, that is, \deqn{h(x) = \sum_{i = 1}^k h(i, x)}{h(x) = sum[i = 1 to k] h(i, x)} Parallel tempering simulates a product of distributions of a continuous random vector. Denote the state \eqn{(x_1, \ldots, x_k)}{(x[1], \ldots, x[k])}, then the unnormalized joint density of the equilibrium distribution is \deqn{h(x_1, \ldots, x_k) = \prod_{i = 1}^k h(i, x_i)}{h(x[1], \dots, x[k]) = prod[i = 1 to k] h(i, x[i])} The update mechanism of the Markov chain combines two kinds of elementary updates: jump/swap updates (jump for serial tempering, swap for parallel tempering) and within-component updates. Each iteration of the Markov chain one of these elementary updates is done. With probability 1/2 a jump/swap update is done, and with probability 1/2 a with-component update is done. Within-component updates are the same for both serial and parallel tempering. They are \dQuote{random-walk} Metropolis updates with multivariate normal proposal, the proposal distribution being determined by the argument \code{scale}. In serial tempering, the \eqn{x} part of the current state \eqn{(i, x)} is updated preserving \eqn{h(i, x)}. In parallel tempering, an index \eqn{i} is chosen at random and the part of the state \eqn{x_i}{x[i]} representing that component is updated, again preserving \eqn{h(i, x)}. Jump updates choose uniformly at random a neighbor of the current component: if \eqn{i} indexes the current component, then it chooses uniformly at random a \eqn{j} such that \code{neighbors[i, j] == TRUE}. It then does does a Metropolis-Hastings update for changing the current state from \eqn{(i, x)} to \eqn{(j, x)}. Swap updates choose a component uniformly at random and a neighbor of that component uniformly at random: first an index \eqn{i} is chosen uniformly at random between 1 and \eqn{k}, then an index \eqn{j} is chosen uniformly at random such that \code{neighbors[i, j] == TRUE}. It then does does a Metropolis-Hastings update for swapping the states of the two components: interchanging \eqn{x_i}{x[i, ]} and \eqn{x_j}{x[j, ]} while preserving \eqn{h(x_1, \ldots, x_k)}{h(x[1], \dots, x[k])}. The initial state must satisfy \code{lud(initial, ...) > - Inf} for serial tempering or must satisfy \code{lud(initial[i, ], ...) > - Inf} for each \code{i} for parallel tempering, where \code{lud} is either \code{obj} or \code{obj$lud}. That is, the initial state must have positive probability. } \value{ an object of class \code{"mcmc"}, subclass \code{"tempering"}, which is a list containing at least the following components: \item{batch}{the batch means of the continuous part of the state. If \code{outfun} is missing, an \code{nbatch} by \code{k} by \code{p} array. Otherwise, an \code{nbatch} by \code{m} matrix, where \code{m} is the length of the result of \code{outfun}.} \item{ibatch}{(returned for serial tempering only) an \code{nbatch} by \code{k} matrix giving batch means for the multivariate Bernoulli random vector that is all zeros except for a 1 in the \code{i}-th place when the current state is \eqn{(i, x)}.} \item{acceptx}{fraction of Metropolis within-component proposals accepted. A vector of length \code{k} giving the acceptance rate for each component.} \item{accepti}{fraction of Metropolis jump/swap proposals accepted. A \code{k} by \code{k} matrix giving the acceptance rate for each allowed jump or swap component. \code{NA} for elements such that the corresponding elements of \code{neighbors} is \code{FALSE}.} \item{initial}{value of argument \code{initial}.} \item{final}{final state of Markov chain.} \item{initial.seed}{value of \code{.Random.seed} before the run.} \item{final.seed}{value of \code{.Random.seed} after the run.} \item{time}{running time of Markov chain from \code{system.time()}.} \item{lud}{the function used to calculate log unnormalized density, either \code{obj} or \code{obj$lud} from a previous run.} \item{nbatch}{the argument \code{nbatch} or \code{obj$nbatch}.} \item{blen}{the argument \code{blen} or \code{obj$blen}.} \item{nspac}{the argument \code{nspac} or \code{obj$nspac}.} \item{outfun}{the argument \code{outfun} or \code{obj$outfun}.} Description of additional output when \code{debug = TRUE} can be found in the vignette \code{debug} (\url{../doc/debug.pdf}). } \section{Warning}{ If \code{outfun} is missing, then the log unnormalized density function can be defined without a \ldots argument and that works fine. One can define it starting \code{ludfun <- function(state)} and that works or \code{ludfun <- function(state, foo, bar)}, where \code{foo} and \code{bar} are supplied as additional arguments to \code{temper} and that works too. If \code{outfun} is a function, then both it and the log unnormalized density function can be defined without \ldots arguments \emph{if they have exactly the same arguments list} and that works fine. Otherwise it doesn't work. Define these functions by \preformatted{ ludfun <- function(state, foo) outfun <- function(state, bar) } and you get an error about unused arguments. Instead define these functions by \preformatted{ ludfun <- function(state, foo, \ldots) outfun <- function(state, bar, \ldots) } and supply \code{foo} and \code{bar} as additional arguments to \code{metrop}, and that works fine. In short, the log unnormalized density function and \code{outfun} need to have \ldots in their arguments list to be safe. Sometimes it works when \ldots is left out and sometimes it doesn't. Of course, one can avoid this whole issue by always defining the log unnormalized density function and \code{outfun} to have only one argument \code{state} and use global variables (objects in the \R global environment) to specify any other information these functions need to use. That too follows the \R way. But some people consider that bad programming practice. } \examples{ d <- 9 witch.which <- c(0.1, 0.3, 0.5, 0.7, 1.0) ncomp <- length(witch.which) neighbors <- matrix(FALSE, ncomp, ncomp) neighbors[row(neighbors) == col(neighbors) + 1] <- TRUE neighbors[row(neighbors) == col(neighbors) - 1] <- TRUE ludfun <- function(state, log.pseudo.prior = rep(0, ncomp)) { stopifnot(is.numeric(state)) stopifnot(length(state) == d + 1) icomp <- state[1] stopifnot(icomp == as.integer(icomp)) stopifnot(1 <= icomp && icomp <= ncomp) stopifnot(is.numeric(log.pseudo.prior)) stopifnot(length(log.pseudo.prior) == ncomp) theta <- state[-1] if (any(theta > 1.0)) return(-Inf) bnd <- witch.which[icomp] lpp <- log.pseudo.prior[icomp] if (any(theta > bnd)) return(lpp) return(- d * log(bnd) + lpp) } # parallel tempering thetas <- matrix(0.5, ncomp, d) out <- temper(ludfun, initial = thetas, neighbors = neighbors, nbatch = 20, blen = 10, nspac = 5, scale = 0.56789, parallel = TRUE, debug = TRUE) # serial tempering theta.initial <- c(1, rep(0.5, d)) # log pseudo prior found by trial and error qux <- c(0, 9.179, 13.73, 16.71, 20.56) out <- temper(ludfun, initial = theta.initial, neighbors = neighbors, nbatch = 50, blen = 30, nspac = 2, scale = 0.56789, parallel = FALSE, debug = FALSE, log.pseudo.prior = qux) } \keyword{misc} mcmc/.Rinstignore0000644000175100001440000000002313063023671013507 0ustar hornikusersdesignDoc/Makefile mcmc/LICENSE0000644000175100001440000000012413045741600012211 0ustar hornikusersYEAR: 2005, 2009, 2010, 2012 COPYRIGHT HOLDER: Charles J. Geyer and Leif T. Johnson