LearnBayes/0000755000176200001440000000000012341651465012306 5ustar liggesusersLearnBayes/inst/0000755000176200001440000000000012341620471013254 5ustar liggesusersLearnBayes/inst/doc/0000755000176200001440000000000012341620470014020 5ustar liggesusersLearnBayes/inst/doc/BinomialInference.R0000644000176200001440000000234712341620470017522 0ustar liggesusers### R code from vignette source 'BinomialInference.Rnw' ################################################### ### code chunk number 1: BinomialInference.Rnw:17-20 ################################################### library(LearnBayes) beta.par <- beta.select(list(p=0.5, x=0.2), list(p=0.75, x=.28)) beta.par ################################################### ### code chunk number 2: BinomialInference.Rnw:32-33 ################################################### triplot(beta.par, c(6, 4)) ################################################### ### code chunk number 3: BinomialInference.Rnw:40-43 ################################################### beta.post.par <- beta.par + c(6, 4) post.sample <- rbeta(1000, beta.post.par[1], beta.post.par[2]) quantile(post.sample, c(0.05, 0.95)) ################################################### ### code chunk number 4: BinomialInference.Rnw:50-51 ################################################### predplot(beta.par, 10, 6) ################################################### ### code chunk number 5: BinomialInference.Rnw:60-65 ################################################### n <- 20 s <- 0:n pred.probs <- pbetap(beta.par, n, s) plot(s, pred.probs, type="h") discint(cbind(s, pred.probs), 0.90) LearnBayes/inst/doc/DiscreteBayes.pdf0000644000176200001440000032542112341503203017242 0ustar liggesusers%PDF-1.5 % 1 0 obj << /Length 269 >> stream concordance:DiscreteBayes.tex:DiscreteBayes.Rnw:1 5 1 1 0 14 1 1 2 1 0 1 1 3 0 1 5 7 0 1 2 4 1 1 2 1 0 1 1 3 0 1 5 7 0 1 2 2 1 1 2 12 0 1 2 6 1 1 2 1 0 2 1 3 0 1 2 1 4 7 0 1 2 3 1 1 2 1 0 1 1 3 0 1 2 1 1 1 2 4 0 1 2 1 1 1 2 4 0 1 2 2 1 1 2 10 0 2 2 5 0 2 2 17 0 1 2 1 1 endstream endobj 5 0 obj << /Length 927 /Filter /FlateDecode >> stream x}Vn8}W}E h Xvh"y-Es$FdRÙ-?AmĐ-6ʺPgD925NYslQ4F0piINM;. 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IT\;-EK\$ezDq!"r+""bH4-D&EB$E<{-6֏Ȉ9bܸxr v{]{b( İZHű8bt]x;ϿN5btF&>dw0pj>Z? endstream endobj startxref 108848 %%EOF LearnBayes/inst/doc/MultilevelModeling.R0000644000176200001440000000507012341620470017746 0ustar liggesusers### R code from vignette source 'MultilevelModeling.Rnw' ################################################### ### code chunk number 1: MultilevelModeling.Rnw:17-24 ################################################### d <- data.frame(Name=c("Clemente", "Robinson", "Howard", "Johnstone", "Berry", "Spencer", "Kessinger", "Alvarado", "Santo", "Swaboda", "Petrocelli", "Rodriguez", "Scott", "Unser", "Williams", "Campaneris", "Munson", "Alvis"), Hits=c(18, 17, 16, 15, 14, 14, 13, 12, 11, 11, 10, 10, 10, 10, 10, 9, 8, 7), At.Bats=45) ################################################### ### code chunk number 2: MultilevelModeling.Rnw:59-64 ################################################### library(LearnBayes) laplace.fit <- laplace(betabinexch, c(0, 0), d[, c("Hits", "At.Bats")]) laplace.fit ################################################### ### code chunk number 3: MultilevelModeling.Rnw:68-73 ################################################### mcmc.fit <- rwmetrop(betabinexch, list(var=laplace.fit$var, scale=2), c(0, 0), 5000, d[, c("Hits", "At.Bats")]) ################################################### ### code chunk number 4: MultilevelModeling.Rnw:77-81 ################################################### mycontour(betabinexch, c(-1.5, -0.5, 2, 12), d[, c("Hits", "At.Bats")], xlab="Logit ETA", ylab="Log K") with(mcmc.fit, points(par)) ################################################### ### code chunk number 5: MultilevelModeling.Rnw:88-99 ################################################### eta <- with(mcmc.fit, exp(par[, 1]) / (1 + exp(par[, 1]))) K <- exp(mcmc.fit$par[, 2]) p.estimate <- function(j, eta, K){ yj <- d[j, "Hits"] nj <- d[j, "At.Bats"] p.sim <- rbeta(5000, yj + K * eta, nj - yj + K * (1 - eta)) quantile(p.sim, c(0.05, 0.50, 0.95)) } E <- t(sapply(1:18, p.estimate, eta, K)) rownames(E) <- d[, "Name"] round(E, 3) ################################################### ### code chunk number 6: MultilevelModeling.Rnw:105-115 ################################################### plot(d$Hits / 45, E[, 2], pch=19, ylim=c(.15, .40), xlab="Observed AVG", ylab="True Probability", main="90 Percent Probability Intervals") for (j in 1:18) lines(d$Hits[j] / 45 * c(1, 1), E[j, c(1, 3)]) abline(a=0, b=1, col="blue") abline(h=mean(d$Hits) / 45, col="red") legend("topleft", legend=c("Individual", "Combined"), lty=1, col=c("blue", "red")) LearnBayes/inst/doc/BinomialInference.Rnw0000644000176200001440000000531612341620470020066 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Learning About a Binomial Proportion} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Learning About a Binomial Proportion} \author{Jim Albert} \maketitle \section*{Constructing a Beta Prior} Suppose we are interested in the proportion $p$ on sunny days in my town. The function {\tt bayes.select} is a convenient tool for specifying a beta prior based on knowledge of two prior quantiles. Suppose my prior median for the proportion of sunny days is $.2$ and my 75th percentile is $.28$. <<>>= library(LearnBayes) beta.par <- beta.select(list(p=0.5, x=0.2), list(p=0.75, x=.28)) beta.par @ A beta(2.95, 10.82) prior matches this prior information \section*{Updating with Data} Next, I observe the weather for 10 days and observe 6 sunny days. (There are 6 ``successes" and 4 ``failures".) The posterior distribution is beta with shape parameters 2.95 + 6 and 10.82 + 4. \section*{Triplot} The {\tt triplot} function shows the prior, likelihood, and posterior on the same display; the inputs are the vector of prior parameters and the data vector. <>= triplot(beta.par, c(6, 4)) @ \section*{Simulating from Posterior to Perform Inference} One can perform inference about the proportion $p$ by simulating a large number of draws from the posterior and summarizing the simulated sample. Here the {\tt rbeta} function is used to simulate from the beta posterior and the {\tt quantile} function is used to construct a 90 percent probability interval for $p$. <<>>= beta.post.par <- beta.par + c(6, 4) post.sample <- rbeta(1000, beta.post.par[1], beta.post.par[2]) quantile(post.sample, c(0.05, 0.95)) @ \section*{Predictive Checking} One can check the suitability of this model by means of a predictive check. The function {\tt predplot} displays the prior predictive density for the number of successes and overlays the observed number of successes. <>= predplot(beta.par, 10, 6) @ The observed data is in the tail of the predictive distribution suggesting some incompability of the prior information and the sample. \section*{Prediction of a Future Sample} Suppose we want to predict the number of sunny days in the future 20 days. The function {\tt pbetap} computes the posterior predictive distribution with a beta prior. The inputs are the vector of beta prior parameters, the future sample size, and the vector of number of successes in the future experiment. <>= n <- 20 s <- 0:n pred.probs <- pbetap(beta.par, n, s) plot(s, pred.probs, type="h") discint(cbind(s, pred.probs), 0.90) @ The probability that we will observe between 0 and 8 successes in the future sample is .92. \end{document}LearnBayes/inst/doc/DiscreteBayes.Rnw0000644000176200001440000000736712341620470017253 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Bayes using Discrete Priors} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Bayes using Discrete Priors} \author{Jim Albert} \maketitle \section*{Learning About a Proportion} \subsection*{A Discrete Prior} Consider a population of ``successes" and ``failures" where the proportion of successes is $p$. Suppose $p$ takes on the discrete set of values 0, .01, ..., .99, 1 and one assigns a uniform prior on these values. We enter the values of $p$ and the associated probabilities into the vectors {\tt p} and {\tt prior}, respectively. <<>>= p <- seq(0, 1, by = 0.01) prior <- 1 / 101 + 0 * p @ <>= plot(p, prior, type="h", main="Prior Distribution") @ \subsection*{Posterior Distribution} Suppose one takes a random sample from the population without replacement and observes 20 successes and 12 failiures. The function {\tt pdisc} in the {\tt LearnBayes} package computes the associated posterior probabilities for $p$. The inputs to {\tt pdisc} are the prior (vector of values of $p$ and vector of prior probabilities) and a vector containing the number of successes and failures. <<>>= library(LearnBayes) post <- pdisc(p, prior, c(20, 12)) @ <>= plot(p, post, type="h", main="Posterior Distribution") @ A highest probability interval for a discrete distribution is obtained using the {\tt discint} function. This function has two inputs: the probability distribution matrix where the first column contains the values and the second column contains the probabilities, and the desired probability content. To illustrate, we compute a 90 percent probability interval for $p$ from the posterior distribution. <<>>= discint(cbind(p, post), 0.90) @ The probability that $p$ falls in the interval (0.49, 0.75) is approximately 0.90. \subsection*{Prediction} Suppose a new sample of size 20 is to be taken and we're interested in predicting the number of successes. The current opinion about the proportion is reflected in the posterior distribution stored in the vectors {\tt p} and {\tt post}. We store the possible number of successes in the future sample in {\tt s} and the function {\tt pdiscp} computes the corresponding predictive probabilities. <<>>= n <- 20 s <- 0:20 pred.probs <- pdiscp(p, post, n, s) @ <>= plot(s, pred.probs, type="h", main="Predictive Distribution") @ \section*{Learning About a Poisson Mean} Discrete models can be used for other sampling distributions using the {\tt discrete.bayes} function. To illustrate, suppose the number of accidents in a particular year is Poisson with mean $\lambda$. A priori one believes that $\lambda$ is equally likely to take on the values 20, 21, ..., 30. We put the prior probabilities 1/11, ..., 1/11 in the vector {\tt prior} and use the {\tt names} function to name the components of this vector with the values of $\lambda$. <<>>= prior <- rep(1/11, 11) names(prior) <- 20:30 @ One observes the number of accidents for ten weeks -- these values are placed in the vector {\tt y}: <<>>= y <- c(24, 25, 31, 31, 22, 21, 26, 20, 16, 22) @ To compute the posterior probabilities, we use the function {\tt discrete.bayes}; the inputs are the Poisson sampling density {\tt dpois}, the vector of prior probabilities {\tt prior}, and the vector of observations {\tt y}. <<>>= post <- discrete.bayes(dpois, prior, y) @ One can display the posterior probabilities by use of the {\tt print} method, one displays the posterior probabilites by the {\tt plot} method, and one summarizes the posterior distribution by the {\tt summary} method. <<>>= print(post) @ <>= plot(post) @ <<>>= summary(post) @ \end{document}LearnBayes/inst/doc/MultilevelModeling.Rnw0000644000176200001440000001321112341620470020307 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Multilevel Modeling} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Multilevel Modeling} \author{Jim Albert} \maketitle \section*{Efron and Morris Baseball Data} Efron and Morris, in a famous 1975 JASA paper, introduced the problem of estimating the true batting averages for 18 players during the 1971 baseball season. In the table, we observe the number of hits for each player in the first 35 batting opportunities in the season. <<>>= d <- data.frame(Name=c("Clemente", "Robinson", "Howard", "Johnstone", "Berry", "Spencer", "Kessinger", "Alvarado", "Santo", "Swaboda", "Petrocelli", "Rodriguez", "Scott", "Unser", "Williams", "Campaneris", "Munson", "Alvis"), Hits=c(18, 17, 16, 15, 14, 14, 13, 12, 11, 11, 10, 10, 10, 10, 10, 9, 8, 7), At.Bats=45) @ \section*{The Multilevel Model} One can simultaneously estimate the true batting averages by the following multilevel model. We assume the hits for the $j$th player $y_j$ has a binomial distribution with sample size $n_j$ and probability of success $p_j$, $j = 1, ..., 18$. The true batting averages $p_1, .., p_{18}$ are assumed to be a random sample from a beta($a, b$) distribution. It is convenient to reparameterize $a$ and $b$ into the mean $\eta = a / (a + b)$ and precision $K = a + b$. We assign $(\eta, K)$ the noninformative prior $$ g(\eta, K) \propto \frac{1}{\eta (1 - \eta)}\frac{1}{(1 + K)^2} $$ After data $y$ is observed, the posterior distribution of the parameters $(\{p_j\}, \eta, K)$ has the convenient representation $$ g(\{p_j\}, \eta, K | y) = g(\eta, K | y) \times g(\{p_j\} | \eta, K, y). $$ Conditional on $\eta$ and $K$, the posterior distributions of $p_1, ..., p_{18}$ are independent, where $$ p_j \sim Beta(y_j + K \eta, n_j - y_j + K ( 1 - \eta)). $$ The posterior density of $(\eta, K)$ is given by $$ g(\eta, K| y) \propto \prod_{j=1}^{18} \left(\frac{B(y_j + K \eta, n_j - y_j + K (1 - \eta))} {B(K \eta, n_j - y_j + K (1 - \eta))}\right) \frac{1}{\eta (1 - \eta)}\frac{1}{(1 + K)^2}. $$ \section*{Simulation of the Posterior of $(\eta, K)$} For computational purposes, it is convenient to reparameterize $\eta$ and $K$ to the real-valued parameters $$ \theta_1 = \log \frac{\eta}{1 - \eta}, \theta_2 = \log K. $$ The log posterior of the vector $\theta = (\theta_1, \theta_2)$ is programmed in the function {\tt betaabinexch}. We initially use the {\tt laplace} function to find the posterior mode and associated variance-covariance matrix. The inputs are the log posterior function, an initial guess at the mode, and the data. <<>>= library(LearnBayes) laplace.fit <- laplace(betabinexch, c(0, 0), d[, c("Hits", "At.Bats")]) laplace.fit @ The outputs from {\tt laplace} are used to inform the inputs of a random walk Metropolis algorithm in the function {\tt rwmetrop}. The inputs are the function defining the log posterior, the estimate of the variance-covarance matrix and scale for the proposal density, the starting value in the Markov Chain, and the data. <<>>= mcmc.fit <- rwmetrop(betabinexch, list(var=laplace.fit$var, scale=2), c(0, 0), 5000, d[, c("Hits", "At.Bats")]) @ To demonstrate that this MCMC algorithm produces a reasonable sample from the posterior, the {\tt mycontour} function displays a contour graph of the exact posterior density and the {\tt points} function is used to overlay 5000 draws from the MCMC algorithm. <>= mycontour(betabinexch, c(-1.5, -0.5, 2, 12), d[, c("Hits", "At.Bats")], xlab="Logit ETA", ylab="Log K") with(mcmc.fit, points(par)) @ \section*{Simulation of the Posterior of the Probabilities} One can simulate from the joint posterior of $(\{p_j\}, \eta, K)$, by (1) simulating $(\eta, K)$ from its marginal posterior, and (2) simulating $p_1, ..., p_{18}$ from the conditional distribution $[\{p_j\} | \eta, K]$. In the R script, I store the simulated draws from the posterior of $K$ and $\eta$ in the vectors {\tt K} and {\tt eta}. Then the function {\tt p.estimate} simulates draws from the posterior of the $j$th probability and computes a 90\% probability interval by extracting the 5th and 95th percentiles. I repeat this process for all 18 players by the {\tt sapply} function and display the 90\% intervals for all players. <<>>= eta <- with(mcmc.fit, exp(par[, 1]) / (1 + exp(par[, 1]))) K <- exp(mcmc.fit$par[, 2]) p.estimate <- function(j, eta, K){ yj <- d[j, "Hits"] nj <- d[j, "At.Bats"] p.sim <- rbeta(5000, yj + K * eta, nj - yj + K * (1 - eta)) quantile(p.sim, c(0.05, 0.50, 0.95)) } E <- t(sapply(1:18, p.estimate, eta, K)) rownames(E) <- d[, "Name"] round(E, 3) @ The following graph displays the 90 percent probability intervals for the players' true batting averages. The blue line represents {\it individual estimates} where each batting probability is estimated by the observed batting average. The red line represents the {\it combined estimate} where one combines all of the data. The multilevel estimate represented by the dot is a compromise between the individual estimate and the combined estimate. <>= plot(d$Hits / 45, E[, 2], pch=19, ylim=c(.15, .40), xlab="Observed AVG", ylab="True Probability", main="90 Percent Probability Intervals") for (j in 1:18) lines(d$Hits[j] / 45 * c(1, 1), E[j, c(1, 3)]) abline(a=0, b=1, col="blue") abline(h=mean(d$Hits) / 45, col="red") legend("topleft", legend=c("Individual", "Combined"), lty=1, col=c("blue", "red")) @ \end{document}LearnBayes/inst/doc/MCMCintro.R0000644000176200001440000000412112341620470015734 0ustar liggesusers### R code from vignette source 'MCMCintro.Rnw' ################################################### ### code chunk number 1: MCMCintro.Rnw:34-42 ################################################### minmaxpost <- function(theta, data){ mu <- theta[1] sigma <- exp(theta[2]) dnorm(data$min, mu, sigma, log=TRUE) + dnorm(data$max, mu, sigma, log=TRUE) + (data$n - 2) * log(pnorm(data$max, mu, sigma) - pnorm(data$min, mu, sigma)) } ################################################### ### code chunk number 2: MCMCintro.Rnw:51-55 ################################################### data <- list(n=10, min=52, max=84) library(LearnBayes) fit <- laplace(minmaxpost, c(70, 2), data) fit ################################################### ### code chunk number 3: MCMCintro.Rnw:60-64 ################################################### mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) mycontour(lbinorm, c(45, 95, 1.5, 4), list(m=fit$mode, v=fit$var), add=TRUE, col="red") ################################################### ### code chunk number 4: MCMCintro.Rnw:73-78 ################################################### mcmc.fit <- rwmetrop(minmaxpost, list(var=fit$v, scale=3), c(70, 2), 10000, data) ################################################### ### code chunk number 5: MCMCintro.Rnw:82-83 ################################################### mcmc.fit$accept ################################################### ### code chunk number 6: MCMCintro.Rnw:88-92 ################################################### mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) points(mcmc.fit$par) ################################################### ### code chunk number 7: MCMCintro.Rnw:105-110 ################################################### mu <- mcmc.fit$par[, 1] sigma <- exp(mcmc.fit$par[, 2]) P.75 <- mu + 0.674 * sigma plot(density(P.75), main="Posterior Density of Upper Quartile") LearnBayes/inst/doc/BayesFactors.Rnw0000644000176200001440000001232112341620470017074 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Bayes Factors} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Bayes Factors} \author{Jim Albert} \maketitle \section*{Models for Fire Calls} To motivate the discussion of plausible models, the website \newline {\tt http://www.franklinvillefire.org/callstatistics.htm} gives the number of fire calls for each month in Franklinville, NC for the last several years. Suppose we observe the fire call counts $y_1, ..., y_N$ for $N$ consecutive months. Here is a general model for these data. \begin{itemize} \item $y_1, ..., y_N$ are independent $f(y | \theta)$ \item $\theta$ has a prior $g(\theta)$ \end{itemize} Also suppose we have some prior beliefs about the mean fire count $E(y)$. We believe that this mean is about 70 and the standard deviation of this guess is 10. Given this general model structure, we have to think of possible choices for $f$, the sampling density. We think of the popular distributions, say Poisson, normal, exponential, etc. Also we should think about different choices for the prior density. For the prior, there are many possible choices -- we typically choose one that can represent my prior information. Once we decide on several plausible choices of sampling density and prior, then we'll compare the models by Bayes factors. To do this, we compute the prior predictive density of the actual data for each possible model. The Laplace method provides a convenient and accurate approximation to the logarithm of the predictive density and we'll use the function {\tt laplace} from the {\tt LearnBayes} package. Continuing our example, suppose our prior beliefs about the mean count of fire calls $\theta$ is Gamma(280, 4). (Essentially this says that our prior guess at $\theta$ is 70 and the prior standard deviation is about 4.2.) But we're unsure about the sampling model -- it could be (model $M_1$) Poisson($\theta$), (model $M_2$) normal with mean $\theta$ and standard deviation 12, or (model $M_3$) normal with mean $\theta$ and standard deviation 6. To get some sense about the best sampling model, a histogram of the fire call counts are graphed below. I have overlaid fitted Poisson and normal distributions where I estimate $\theta$ by the sample mean. The Poisson model appears to be the best fit, followed by the Normal model with standard deviation 6, and the Normal model with standard deviation 12. We want to formalize this comparison by computation of Bayes factors. <>= fire.counts <- c(75, 88, 84, 99, 79, 68, 86, 109, 73, 85, 101, 85, 75, 81, 64, 77, 83, 83, 88, 83, 78, 83, 78, 80, 82, 90, 74, 72, 69, 72, 76, 76, 104, 86, 92, 88) hist(fire.counts, probability=TRUE, ylim=c(0, .08)) x <- 60:110 lines(x, dpois(x, lambda=mean(fire.counts)), col="red") lines(x, dnorm(x, mean=mean(fire.counts), sd=12), col="blue") lines(x, dnorm(x, mean=mean(fire.counts), sd=6), col="green") legend("topright", legend=c("M1: Poisson(theta)", "M2: N(theta, 12)", "M3: N(theta, 6)"), col=c("red", "blue", "green"), lty=1) @ \section*{Bayesian Model Comparison} Under the general model, the predictive density of $y$ is given by the integral $$ f(y) = \int \prod_{j=1}^N f(y_j | \theta) g(\theta) d\theta. $$ This density can be approximated by the Laplace method implemented in the {\tt laplace} function. One compares the suitability of two Bayesian models by comparing the corresponding values of the predictive density. The Bayes factor in support of model $M_1$ over model $M_2$ is given by the ratio $$ BF_{12} = \frac{f_1(y)}{f_2(y)}. $$ Computationally, it is convenient to compute the predictive densities on the log scale, so the Bayes factor can be expressed as $$ BF_{12} = \exp \left(\log f_1(y) - \log f_2(y)\right). $$ To compute the predictive density for a model, say model $M_1$, we initially define a function {\tt model.1} which gives the log posterior. <<>>= model.1 <- function(theta, y){ sum(log(dpois(y, theta))) + dgamma(theta, shape=280, rate=4) } @ Then the log predictive density at $y$ is computed by using the {\tt laplace} function with inputs the function name, a guess at the posterior mode, and the data (vector of fire call counts). The component {\tt int} gives the log of $f(y)$ <<>>= library(LearnBayes) log.pred.1 <- laplace(model.1, 80, fire.counts)$int log.pred.1 @ We similarly find the predictive densities of the models $M_2$ and $M_3$ by defining functions for the corresponding posteriors and using {\tt laplace}: <<>>= model.2 <- function(theta, y){ sum(log(dnorm(y, theta, 6))) + dgamma(theta, shape=280, rate=4) } model.3 <- function(theta, y){ sum(log(dnorm(y, theta, 12))) + dgamma(theta, shape=280, rate=4) } log.pred.2 <- laplace(model.2, 80, fire.counts)$int log.pred.3 <- laplace(model.3, 80, fire.counts)$int @ Displaying the three models and predictive densities, we see that model $M_1$ is preferred to $M_3$ which is preferred to model $M_2$. <<>>= data.frame(Model=1:3, log.pred=c(log.pred.1, log.pred.2, log.pred.3)) @ The Bayes factor in support of model $M_1$ over model $M_3$ is given by <<>>= exp(log.pred.1 - log.pred.3) @ \end{document}LearnBayes/inst/doc/BayesFactors.pdf0000644000176200001440000041651712341613726017124 0ustar liggesusers%PDF-1.5 % 1 0 obj << /Length 178 >> stream concordance:BayesFactors.tex:BayesFactors.Rnw:1 5 1 1 0 27 1 1 4 3 0 5 1 1 4 7 0 1 2 18 1 1 5 7 0 2 2 1 0 2 1 6 0 1 2 1 1 1 5 4 0 1 4 3 0 2 1 3 0 1 2 1 1 1 2 10 0 2 2 7 0 1 2 5 1 endstream endobj 4 0 obj << /Length 2706 /Filter /FlateDecode >> stream xڝYK6Wʂ {Z;l4_?ͻ.tvvq]Ԩp\;UfqY7ai|+'o s*KN>POyfY0 θ/w]-Vs9(*]LJ3?|DKќ|$Xʸve*iA 6SڷNgڪJgrUXYEY 1Yivx&[ߝ}#]';dTX.*U&G)pULvYf ϲdߝam^S|YׯQ}/MrDjZҒ؀Y}˻z$v Ųk Kw{cSCqDr*MڻpʾFtҿs^녟ZhJ4y+t{ٺ6>eb KF멌u54h.By~K^T6|tZeiY҇aDb_>`S$CcAR KĜ'qW$xyXF|9ՠvcRo >ay,DYU.؆uf]j~%"WUD V¼c-њ-W:/S+u6F֌hZN^F4s4N<)LuLJE 3@Ӡkc Z]!jl6e rtFLzM㎵b_x^-E߭r+7Mx@: 3JӠPj_S7'B0p;Y Fy1b̔P$*#zF+mK+v߮3Ke䁑d-Puj{w.U6 bNZd_bUA, _A!T1G" ka*`-8+Ž~X}< '8`{{8  t ( ?O##작W*c}qT#9E"WUq6, K}o34e9( Bi{$qTМ3MT*zlxP0E'~!Zr!ZM G&f(l1 ;I"W96TQ*4#WpKi |i#~4ĢD׋Qm>-㥹jA+'#ȋ}xU=r&O5O~^{,..!l&){&V{쒍.|̙ﺰ $IA"u*wW0!H:^-EUJyV!a^69WJ&WC6+WX XNd(cʪ 0/,g{#Ep.(YU$ͯB70}ox \cURo\2*:+0swOߞ)өt kzF'kT"<ySwR$f! $EqE>έ=4þc<rL" B*#{em* f$7tY|y+1*&rPNĦE:<iM9wɶwOB,S_"d#7Y9 lݾ9, endstream endobj 20 0 obj << /Length 1100 /Filter /FlateDecode >> stream xWYo6~$>Z`v"S{Crd|b C֌87xt8u1Lb-2 UdYrZBr,ҷsdA}iod&k4T{YtjW'U$/l} Fd^œ{r  00zS5Eq #_^&k w6mzx(W(w3JqG=f` kExHh"o!v.BH:5YaDM:^O( \ XX;I"WP}6$1PI1AɅQ\5JOEx}[܋q&.}őS̩K޹=n:9xI^v%8da/::ݔ;}qx7-n(!tq3"I$2,z+_~_T6荟%DHyCő&$m%)ZU0-a0gRXi3bކ=0 m} ;P!b؎OX'އq9 _#88c r9ѐ6oYkcbyx> /ExtGState << >>/ColorSpace << /sRGB 26 0 R >>>> /Length 1583 /Filter /FlateDecode >> stream xn\7 Z@Y&mZH$H\u qQFv3@#?!ʐbڼYY?כ/ן ;G)Tp6Dsre>r+ssn/'ޙ#{z9yDNp󶐡bo[8G[B4ws~E*V3E F-Qr#F^CW.}6ۮ,f6w&w\3F[ ʕi'%k"^An.ݜ_v БLI._ޗGSuD#6eݲ^+hV3 'FK b4z'4ݺڞdnnΆ}U?Sq̲2ro[+2Mj *hH)r@>ÒkA-AmL]t+EYlJuE=81EĹ_ D!YVZGtz~SB%z;9iTrTm=Y<],1XJ֩]YI#' z}6~3'(q58VzDؘj7xI&V`*(@]`cKo"RZ>}*6ȱag5BPXl%˃Sd)ex$p~N"p>ܟ1F_-$[E/ZO\kT=ՁIׇ s0,k!V# U)"kQAY5%Qh_ymO%5]w& Ui4; rܑ#= PEh*:5 tTtA'KtXD'UQDGt@ȊIZ扎BQutDOtH: Mtnk舝x#a4đDGbHkV4), 蠦Z:oe.Uej:]UWt|?@)jqAWcnCh}&Uk"?3ozp}xͅhi{R0 endstream endobj 28 0 obj << /Alternate /DeviceRGB /N 3 /Length 2596 /Filter /FlateDecode >> stream xwTSϽ7PkhRH H.*1 J"6DTpDQ2(C"QDqpId߼y͛~kg}ֺLX Xňg` lpBF|،l *?Y"1P\8=W%Oɘ4M0J"Y2Vs,[|e92<se'9`2&ctI@o|N6(.sSdl-c(2-yH_/XZ.$&\SM07#1ؙYrfYym";8980m-m(]v^DW~ emi]P`/u}q|^R,g+\Kk)/C_|Rax8t1C^7nfzDp 柇u$/ED˦L L[B@ٹЖX!@~(* {d+} G͋љς}WL$cGD2QZ4 E@@A(q`1D `'u46ptc48.`R0) @Rt CXCP%CBH@Rf[(t CQhz#0 Zl`O828.p|O×X ?:0FBx$ !i@ڐH[EE1PL ⢖V6QP>U(j MFkt,:.FW8c1L&ӎ9ƌaX: rbl1 {{{;}#tp8_\8"Ey.,X%%Gщ1-9ҀKl.oo/O$&'=JvMޞxǥ{=Vs\x ‰N柜>ucKz=s/ol|ϝ?y ^d]ps~:;/;]7|WpQoH!ɻVsnYs}ҽ~4] =>=:`;cܱ'?e~!ańD#G&}'/?^xI֓?+\wx20;5\ӯ_etWf^Qs-mw3+?~O~ endstream endobj 31 0 obj << /Length 2072 /Filter /FlateDecode >> stream xYK6С! =lEQ=$9h͢ 9(KhzEQpftI,Zd4K&y̓$y> x_5Jƫ 'waqoCө)7@z rIx#)rsfj9 4kXH JՐ~tBn!EdhTl,rlNp^jh ܋3 گzG&\Pԯ(ɵ[b4*S2?vjO|q]~l dYx(k?qSNO݆=y -o=#=TkNCt堯Rm}8qqTӏEؓ T9lcf QGFȿdJ#IqS3s3N8D{u;EmaHG$, <鍧(l! 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)>:#)q2Ad&u MN $ ~f/YxZ5QU ُp\W-".UKeWa[ endstream endobj startxref 270997 %%EOF LearnBayes/inst/doc/BayesFactors.R0000644000176200001440000000423612341620470016535 0ustar liggesusers### R code from vignette source 'BayesFactors.Rnw' ################################################### ### code chunk number 1: BayesFactors.Rnw:34-46 ################################################### fire.counts <- c(75, 88, 84, 99, 79, 68, 86, 109, 73, 85, 101, 85, 75, 81, 64, 77, 83, 83, 88, 83, 78, 83, 78, 80, 82, 90, 74, 72, 69, 72, 76, 76, 104, 86, 92, 88) hist(fire.counts, probability=TRUE, ylim=c(0, .08)) x <- 60:110 lines(x, dpois(x, lambda=mean(fire.counts)), col="red") lines(x, dnorm(x, mean=mean(fire.counts), sd=12), col="blue") lines(x, dnorm(x, mean=mean(fire.counts), sd=6), col="green") legend("topright", legend=c("M1: Poisson(theta)", "M2: N(theta, 12)", "M3: N(theta, 6)"), col=c("red", "blue", "green"), lty=1) ################################################### ### code chunk number 2: BayesFactors.Rnw:67-71 ################################################### model.1 <- function(theta, y){ sum(log(dpois(y, theta))) + dgamma(theta, shape=280, rate=4) } ################################################### ### code chunk number 3: BayesFactors.Rnw:74-77 ################################################### library(LearnBayes) log.pred.1 <- laplace(model.1, 80, fire.counts)$int log.pred.1 ################################################### ### code chunk number 4: BayesFactors.Rnw:81-91 ################################################### model.2 <- function(theta, y){ sum(log(dnorm(y, theta, 6))) + dgamma(theta, shape=280, rate=4) } model.3 <- function(theta, y){ sum(log(dnorm(y, theta, 12))) + dgamma(theta, shape=280, rate=4) } log.pred.2 <- laplace(model.2, 80, fire.counts)$int log.pred.3 <- laplace(model.3, 80, fire.counts)$int ################################################### ### code chunk number 5: BayesFactors.Rnw:95-96 ################################################### data.frame(Model=1:3, log.pred=c(log.pred.1, log.pred.2, log.pred.3)) ################################################### ### code chunk number 6: BayesFactors.Rnw:99-100 ################################################### exp(log.pred.1 - log.pred.3) LearnBayes/inst/doc/MCMCintro.Rnw0000644000176200001440000001201512341620470016302 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Markov Chain Monte Carlo} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Markov Chain Monte Carlo} \author{Jim Albert} \maketitle \section*{A Selected Data Problem} Here is an interesting problem with ``selected data". Suppose you are measuring the speeds of cars driving on an interstate. You assume the speeds are normally distributed with mean $\mu$ and standard deviation $\sigma$. You see 10 cars pass by and you only record the minimum and maximum speeds. What have you learned about the normal parameters? First we focus on the construction of the likelihood. Given values of the normal parameters, what is the probability of observing minimum = $x$ and the maximum = $y$ in a sample of size n? Essentially we're looking for the joint density of two order statistics which is a standard result. Let $f$ and $F $denote the density and cdf of a normal density with mean $\mu$ and standard deviation $\sigma$. Then the joint density of $(x, y)$ is given by $$f(x, y | \mu, \sigma) \propto f(x) f(y) [F(y) - F(x)]^{n-2}, x < y$$ After we observe data, the likelihood is this sampling density viewed as function of the parameters. Suppose we take a sample of size 10 and we observe $x = 52, y = 84$. Then the likelihood is given by $$ L(\mu, \sigma) \propto f(52) f(84) [F(84) - F(52)]^{8} $$ \section*{Defining the log posterior} First I write a short function {\tt minmaxpost} that computes the logarithm of the posterior density. The arguments to this function are $\theta = (\mu, \log \sigma)$ and data which is a list with components {\tt n}, {\tt min}, and {\tt max}. I'd recommend using the R functions {\tt pnorm} and {\tt dnorm} in computing the density -- it saves typing errors. <<>>= minmaxpost <- function(theta, data){ mu <- theta[1] sigma <- exp(theta[2]) dnorm(data$min, mu, sigma, log=TRUE) + dnorm(data$max, mu, sigma, log=TRUE) + (data$n - 2) * log(pnorm(data$max, mu, sigma) - pnorm(data$min, mu, sigma)) } @ \section*{Normal approximation to posterior} We work with the parameterization $(\mu, \log \sigma)$ which will give us a better normal approximation. A standard noninformative prior is uniform on $(\mu, \log \sigma)$. The function {\tt laplace} is used to summarize this posterior. The arguments to {\tt laplace} are the name of the log posterior function, an initial estimate at $\theta$, and the data that is used in the log posterior function. The output of laplace includes mode, the posterior mode, and var, the corresponding estimate at the variance-covariance matrix. <<>>= data <- list(n=10, min=52, max=84) library(LearnBayes) fit <- laplace(minmaxpost, c(70, 2), data) fit @ In this example, this gives a pretty good approximation in this situation. The {\tt mycontour} function is used to display contours of the exact posterior and overlay the matching normal approximation using a second application of {\tt mycontour}. <>= mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) mycontour(lbinorm, c(45, 95, 1.5, 4), list(m=fit$mode, v=fit$var), add=TRUE, col="red") @ \section*{Random Walk Metropolis Sampling} The {\tt rwmetrop} function implements the M-H random walk algorithm. There are four inputs: (1) the function defining the log posterior, (2) a list containing var, the estimated var-cov matrix, and scale, the M-H random walk scale constant, (3) the starting value in the Markov Chain simulation, (4) the number of iterations of the algorithm, and (5) any data and prior parameters used in the log posterior density. Here we use {\tt fit\$v} as our estimated var-cov matrix, use a scale value of 3, start the simulation at $(\mu, \log \sigma) = (70, 2)$ and try 10,000 iterations. <<>>= mcmc.fit <- rwmetrop(minmaxpost, list(var=fit$v, scale=3), c(70, 2), 10000, data) @ I display the acceptance rate -- here it is 19\% which is a reasonable value. <<>>= mcmc.fit$accept @ We display the contours of the exact posterior and overlay the simulated draws. <>= mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) points(mcmc.fit$par) @ It appears like we have been successful in getting a good sample from this posterior distribution. \section*{Random Walk Metropolis Sampling} To illustrate simulation-based inference, suppose one is interested in learning about the upper quartile $$ P.75 = \mu + 0.674 \times \sigma $$ of the car speed distribution. For each simulated draw of $(\mu, \sigma)$ from the posterior, we compute the upper quartile $P.75$. 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Gy'S^OGXf?}/d3~% ?B=dJtpď~Pe{~7㓼*!7o:%zG'o^^n3jрV%$v$/=>HNO'%@Yצ[~>O3DE[U 5+Q9nO|y>f[e#Ip]|D2*m>FVȋq1Α% +ܠ.2)kX5ͦ0˧b7]^¶CbRoi IXVy?K>O+ʥ]Bp'a8ؚ/3yB,|yE>aw.=;;|%EN5<قݨ.l&ϫ0`/ܩ?yutopn)ݼICbo(·gz|9mIjC8HB,A$)JuMQqSWT3q̦MII/֩XhUmZT<7y-SHojSSWwV3o#嗗/)m_ [⤠Ki~񨓭w? b2'U9)+*h NkPQ't;U.S| lO<Ψʻ\ɶ{2-Q'y"ߒ^ K_/jZ;-:py1oc^W5~Ihݕ\Wt5da\tݕtW_juu\Ėu;>mtMv׽rZѽ6|[bw˗k\3pz\y}mjU Wkl["zҁnw׽r*-K;wWz~h^jˆJƙqfPxPVl<" endstream endobj 86 0 obj << /Type /XRef /Index [0 87] /Size 87 /W [1 3 1] /Root 84 0 R /Info 85 0 R /ID [<123C219F96B36FD4736E020D9CD22D92> <123C219F96B36FD4736E020D9CD22D92>] /Length 254 /Filter /FlateDecode >> stream x%йJA{[ĸ|~ƥ 鬴AGKBJsl~/33cf Yh!dT #)':{KNT8,Ԙ#VFI$]{vڹI7z<)G?kh釟h d$d~Q2F$Sd̐Y2G;]?=ͺj ުJ72EKU{j[UF̧ rXϾ$^^r#6 endstream endobj startxref 284213 %%EOF LearnBayes/inst/doc/DiscreteBayes.R0000644000176200001440000000511112341620470016667 0ustar liggesusers### R code from vignette source 'DiscreteBayes.Rnw' ################################################### ### code chunk number 1: DiscreteBayes.Rnw:21-23 ################################################### p <- seq(0, 1, by = 0.01) prior <- 1 / 101 + 0 * p ################################################### ### code chunk number 2: DiscreteBayes.Rnw:25-28 ################################################### plot(p, prior, type="h", main="Prior Distribution") ################################################### ### code chunk number 3: DiscreteBayes.Rnw:35-37 ################################################### library(LearnBayes) post <- pdisc(p, prior, c(20, 12)) ################################################### ### code chunk number 4: DiscreteBayes.Rnw:39-42 ################################################### plot(p, post, type="h", main="Posterior Distribution") ################################################### ### code chunk number 5: DiscreteBayes.Rnw:47-48 ################################################### discint(cbind(p, post), 0.90) ################################################### ### code chunk number 6: DiscreteBayes.Rnw:57-60 ################################################### n <- 20 s <- 0:20 pred.probs <- pdiscp(p, post, n, s) ################################################### ### code chunk number 7: DiscreteBayes.Rnw:63-66 ################################################### plot(s, pred.probs, type="h", main="Predictive Distribution") ################################################### ### code chunk number 8: DiscreteBayes.Rnw:72-74 ################################################### prior <- rep(1/11, 11) names(prior) <- 20:30 ################################################### ### code chunk number 9: DiscreteBayes.Rnw:78-79 ################################################### y <- c(24, 25, 31, 31, 22, 21, 26, 20, 16, 22) ################################################### ### code chunk number 10: DiscreteBayes.Rnw:83-84 ################################################### post <- discrete.bayes(dpois, prior, y) ################################################### ### code chunk number 11: DiscreteBayes.Rnw:89-90 ################################################### print(post) ################################################### ### code chunk number 12: DiscreteBayes.Rnw:93-94 ################################################### plot(post) ################################################### ### code chunk number 13: DiscreteBayes.Rnw:97-98 ################################################### summary(post) LearnBayes/NAMESPACE0000644000176200001440000000003712341075160013515 0ustar liggesusersexportPattern("^[[:alpha:]]+") LearnBayes/demo/0000755000176200001440000000000012255064641013230 5ustar liggesusersLearnBayes/demo/Chapter.6.10.R0000644000176200001440000000203111127266510015314 0ustar liggesusers############################################################# # Section 6.10 Analysis of the Stanford Heart Transplant Data ############################################################# library(LearnBayes) data(stanfordheart) start=c(0,3,-1) laplacefit=laplace(transplantpost,start,stanfordheart) laplacefit proposal=list(var=laplacefit$var,scale=2) s=rwmetrop(transplantpost,proposal,start,10000,stanfordheart) s$accept par(mfrow=c(2,2)) tau=exp(s$par[,1]) plot(density(tau),main="TAU") lambda=exp(s$par[,2]) plot(density(lambda),main="LAMBDA") p=exp(s$par[,3]) plot(density(p),main="P") apply(exp(s$par),2,quantile,c(.05,.5,.95)) S=readline(prompt="Type to continue : ") par(mfrow=c(1,1)) t=seq(1,240) p5=0*t; p50=0*t; p95=0*t for (j in 1:240) { S=(lambda/(lambda+t[j]))^p q=quantile(S,c(.05,.5,.95)) p5[j]=q[1]; p50[j]=q[2]; p95[j]=q[3]} windows() plot(t,p50,type="l",ylim=c(0,1),ylab="Prob(Survival)", xlab="time") lines(t,p5,lty=2) lines(t,p95,lty=2) LearnBayes/demo/Chapter.8.8.R0000644000176200001440000000104411106370646015253 0ustar liggesusers################################################################### # Section 8.8 A Test of Independence in a Two-Way Contingency Table ################################################################### library(LearnBayes) data=matrix(c(11,9,68,23,3,5),c(2,3)) data chisq.test(data) a=matrix(rep(1,6),c(2,3)) a ctable(data,a) log.K=seq(2,7) compute.log.BF=function(log.K) log(bfindep(data,exp(log.K),100000)$bf) log.BF=sapply(log.K,compute.log.BF) BF=exp(log.BF) round(data.frame(log.K,log.BF,BF),2)LearnBayes/demo/Chapter.4.4.R0000644000176200001440000000341211127273754015251 0ustar liggesusers################################################### # Section 4.4 A Bioassay Experiment ################################################### library(LearnBayes) x = c(-0.86, -0.3, -0.05, 0.73) n = c(5, 5, 5, 5) y = c(0, 1, 3, 5) data = cbind(x, n, y) glmdata = cbind(y, n - y) results = glm(glmdata ~ x, family = binomial) summary(results) # when x = -.7, median and 90th percentile of p are (.2,.4) # when x = +.6, median and 90th percentile of p are (.8, .95) a1.b1=beta.select(list(p=.5,x=.2),list(p=.9,x=.5)) a2.b2=beta.select(list(p=.5,x=.8),list(p=.9,x=.98)) prior=rbind(c(-0.7, 4.68, 1.12), c(0.6, 2.10, 0.74)) data.new=rbind(data, prior) # plot prior ####################################### plot(c(-1,1),c(0,1),type="n",xlab="Dose",ylab="Prob(death)") lines(-0.7*c(1,1),qbeta(c(.25,.75),a1.b1[1],a1.b1[2]),lwd=4) lines(0.6*c(1,1),qbeta(c(.25,.75),a2.b2[1],a2.b2[2]),lwd=4) points(c(-0.7,0.6),qbeta(.5,c(a1.b1[1],a2.b2[1]),c(a1.b1[2],a2.b2[2])), pch=19,cex=2) text(-0.3,.2,"Beta(1.12, 3.56)") text(.2,.8,"Beta(2.10, 0.74)") response=rbind(a1.b1,a2.b2) x=c(-0.7,0.6) fit = glm(response ~ x, family = binomial) curve(exp(fit$coef[1]+fit$coef[2]*x)/ (1+exp(fit$coef[1]+fit$coef[2]*x)),add=T) ####################################################### S=readline(prompt="Type to continue : ") windows() mycontour(logisticpost,c(-3,3,-1,9),data.new, xlab="beta0", ylab="beta1") s=simcontour(logisticpost,c(-2,3,-1,11),data.new,1000) points(s) S=readline(prompt="Type to continue : ") windows() plot(density(s$y),xlab="beta1") S=readline(prompt="Type to continue : ") theta=-s$x/s$y windows() hist(theta,xlab="LD-50",breaks=20) quantile(theta,c(.025,.975)) LearnBayes/demo/Chapter.6.2.R0000644000176200001440000000107211106347520015237 0ustar liggesusers#################################################### # Section 6.2 Introduction to Discrete Markov Chains #################################################### P=matrix(c(.5,.5,0,0,0,0,.25,.5,.25,0,0,0,0,.25,.5,.25,0,0, 0,0,.25,.5,.25,0,0,0,0,.25,.5,.25,0,0,0,0,.5,.5), nrow=6,ncol=6,byrow=TRUE) P s=array(0,c(50000,1)) s[1]=3 for (j in 2:50000) s[j]=sample(1:6,size=1,prob=P[s[j-1],]) m=c(500,2000,8000,50000) for (i in 1:4) print(table(s[1:m[i]])/m[i]) w=matrix(c(.1,.2,.2,.2,.2,.1),nrow=1,ncol=6) w%*%P LearnBayes/demo/Chapter.6.7.R0000644000176200001440000000164111106351254015245 0ustar liggesusers################################################################## # Section 6.7 Learning about a Normal Population from Grouped Data ################################################################## library(LearnBayes) d=list(int.lo=c(-Inf,seq(66,74,by=2)), int.hi=c(seq(66,74,by=2), Inf), f=c(14,30,49,70,33,15)) y=c(rep(65,14),rep(67,30),rep(69,49),rep(71,70),rep(73,33), rep(75,15)) mean(y) log(sd(y)) start=c(70,1) fit=laplace(groupeddatapost,start,d) fit modal.sds=sqrt(diag(fit$var)) proposal=list(var=fit$var,scale=2) fit2=rwmetrop(groupeddatapost,proposal,start,10000,d) fit2$accept post.means=apply(fit2$par,2,mean) post.sds=apply(fit2$par,2,sd) cbind(c(fit$mode),modal.sds) cbind(post.means,post.sds) mycontour(groupeddatapost,c(69,71,.6,1.3),d, xlab="mu",ylab="log sigma") points(fit2$par[5001:10000,1],fit2$par[5001:10000,2]) LearnBayes/demo/Chapter.5.6.R0000644000176200001440000000065511106346072015251 0ustar liggesusers###################################################### # Section 5.6 The Example ###################################################### library(LearnBayes) data(cancermortality) fit=laplace(betabinexch,c(-7,6),cancermortality) fit npar=list(m=fit$mode,v=fit$var) mycontour(lbinorm,c(-8,-4.5,3,16.5),npar, xlab="logit eta", ylab="log K") se=sqrt(diag(fit$var)) fit$mode-1.645*se fit$mode+1.645*se LearnBayes/demo/Chapter.3.6.R0000644000176200001440000000163611127272322015246 0ustar liggesusers####################################################### # Section 3.6 A Bayesian Test of the Fairness of a Coin ####################################################### library(LearnBayes) pbinom(5, 20, 0.5) n = 20 y = 5 a = 10 p = 0.5 m1 = dbinom(y, n, p) * dbeta(p, a, a)/dbeta(p, a + y, a + n - y) lambda = dbinom(y, n, p)/(dbinom(y, n, p) + m1) lambda pbetat(p,.5,c(a,a),c(y,n-y)) prob.fair=function(log.a) { a = exp(log.a) m2 = dbinom(y, n, p) * dbeta(p, a, a)/ dbeta(p, a + y, a + n - y) dbinom(y, n, p)/(dbinom(y, n, p) + m2) } n = 20; y = 5; p = 0.5 curve(prob.fair(x), from=-4, to=5, xlab="log a", ylab="Prob(coin is fair)", lwd=2) S=readline(prompt="Type to continue : ") n=20 y=5 a=10 p=.5 m2=0 for (k in 0:y) m2=m2+dbinom(k,n,p)*dbeta(p,a,a)/dbeta(p,a+k,a+n-k) lambda=pbinom(y,n,p)/(pbinom(y,n,p)+m2) lambda LearnBayes/demo/Chapter.3.3.R0000644000176200001440000000212411106333772015240 0ustar liggesusers########################################################## # Section 3.3 Estimating a Heart Transplant Mortality Rate ########################################################## alpha=16;beta=15174 yobs=1; ex=66 y=0:10 lam=alpha/beta py=dpois(y, lam*ex)*dgamma(lam, shape = alpha, rate = beta)/dgamma(lam, shape= alpha + y, rate = beta + ex) cbind(y, round(py, 3)) lambdaA = rgamma(1000, shape = alpha + yobs, rate = beta + ex) ex = 1767; yobs=4 y = 0:10 py = dpois(y, lam * ex) * dgamma(lam, shape = alpha, rate = beta)/dgamma(lam, shape = alpha + y, rate = beta + ex) cbind(y, round(py, 3)) lambdaB = rgamma(1000, shape = alpha + yobs, rate = beta + ex) par(mfrow = c(2, 1)) plot(density(lambdaA), main="HOSPITAL A", xlab="lambdaA", lwd=3) curve(dgamma(x, shape = alpha, rate = beta), add=TRUE) legend("topright",legend=c("prior","posterior"),lwd=c(1,3)) plot(density(lambdaB), main="HOSPITAL B", xlab="lambdaB", lwd=3) curve(dgamma(x, shape = alpha, rate = beta), add=TRUE) legend("topright",legend=c("prior","posterior"),lwd=c(1,3)) LearnBayes/demo/Chapter.9.4.R0000644000176200001440000000134311127237534015253 0ustar liggesusers############################################## # Section 9.4 Survival Modeling ############################################## library(LearnBayes) data(chemotherapy) attach(chemotherapy) library(survival) survreg(Surv(time,status)~factor(treat)+age,dist="weibull") start=c(-.5,9,.5,-.05) d=cbind(time,status,treat-1,age) fit=laplace(weibullregpost,start,d) fit proposal=list(var=fit$var,scale=1.5) bayesfit=rwmetrop(weibullregpost,proposal,fit$mode,10000,d) bayesfit$accept par(mfrow=c(2,2)) sigma=exp(bayesfit$par[,1]) mu=bayesfit$par[,2] beta1=bayesfit$par[,3] beta2=bayesfit$par[,4] hist(beta1,xlab="treatment",main="") hist(beta2,xlab="age",main="") hist(sigma,xlab="sigma",main="") LearnBayes/demo/Chapter.6.8.R0000644000176200001440000000260111127272506015250 0ustar liggesusers################################################## # Section 6.8 Example of Output Analysis ################################################## library(LearnBayes) d=list(int.lo=c(-Inf,seq(66,74,by=2)), int.hi=c(seq(66,74,by=2), Inf), f=c(14,30,49,70,33,15)) library(coda) library(lattice) start=c(70,1) fit=laplace(groupeddatapost,start,d) start=c(65,1) proposal=list(var=fit$var,scale=0.2) bayesfit=rwmetrop(groupeddatapost,proposal,start,10000,d) dimnames(bayesfit$par)[[2]]=c("mu","log sigma") xyplot(mcmc(bayesfit$par[-c(1:2000),]),col="black") S=readline(prompt="Type to continue : ") windows() par(mfrow=c(2,1)) autocorr.plot(mcmc(bayesfit$par[-c(1:2000),]),auto.layout=FALSE) summary(mcmc(bayesfit$par[-c(1:2000),])) batchSE(mcmc(bayesfit$par[-c(1:2000),]), batchSize=50) S=readline(prompt="Type to continue : ") start=c(70,1) proposal=list(var=fit$var,scale=2.0) bayesfit=rwmetrop(groupeddatapost,proposal,start,10000,d) dimnames(bayesfit$par)[[2]]=c("mu","log sigma") sim.parameters=mcmc(bayesfit$par[-c(1:2000),]) windows() xyplot(mcmc(bayesfit$par[-c(1:2000),]),col="black") s=readline(prompt="Type to continue : ") windows() par(mfrow=c(2,1)) autocorr.plot(sim.parameters,auto.layout=FALSE) summary(sim.parameters) batchSE(sim.parameters, batchSize=50) LearnBayes/demo/Chapter.4.3.R0000644000176200001440000000156011127272336015245 0ustar liggesusers################################################### # Section 4.3 A Multinomial Model ################################################### library(LearnBayes) alpha = c(728, 584, 138) theta = rdirichlet(1000, alpha) hist(theta[, 1] - theta[, 2], main="") S=readline(prompt="Type to continue : ") ########################################### data(election.2008) attach(election.2008) prob.Obama=function(j) { p=rdirichlet(5000, 500*c(M.pct[j],O.pct[j],100-M.pct[j]-O.pct[j])/100+1) mean(p[,2]>p[,1]) } Obama.win.probs=sapply(1:51,prob.Obama) sim.election=function() { winner=rbinom(51,1,Obama.win.probs) sum(EV*winner) } sim.EV=replicate(1000,sim.election()) windows() hist(sim.EV,min(sim.EV):max(sim.EV),col="blue") abline(v=365,lwd=3) # Obama received 365 votes text(375,30,"Actual \n Obama \n total") LearnBayes/demo/Chapter.8.3.R0000644000176200001440000000170011106370000015225 0ustar liggesusers############################################### # Section 8.3 A One-Sided Test of a Normal Mean ############################################### library(LearnBayes) pmean=170; pvar=25 probH=pnorm(175,pmean,sqrt(pvar)) probA=1-probH prior.odds=probH/probA prior.odds weights=c(182, 172, 173, 176, 176, 180, 173, 174, 179, 175) xbar=mean(weights) sigma2=3^2/length(weights) post.precision=1/sigma2+1/pvar post.var=1/post.precision post.mean=(xbar/sigma2+pmean/pvar)/post.precision c(post.mean,sqrt(post.var)) post.odds=pnorm(175,post.mean,sqrt(post.var))/ (1-pnorm(175,post.mean,sqrt(post.var))) post.odds BF = post.odds/prior.odds BF postH=probH*BF/(probH*BF+probA) postH z=sqrt(length(weights))*(mean(weights)-175)/3 1-pnorm(z) weights=c(182, 172, 173, 176, 176, 180, 173, 174, 179, 175) data=c(mean(weights),length(weights),3) prior.par=c(170,1000) mnormt.onesided(175,prior.par,data) LearnBayes/demo/Chapter.4.5.R0000644000176200001440000000143511127272374015252 0ustar liggesusers########################################### # Section 4.5 Comparing Two Proportions ########################################### library(LearnBayes) sigma=c(2,1,.5,.25) plo=.0001;phi=.9999 par(mfrow=c(2,2)) for (i in 1:4) mycontour(howardprior,c(plo,phi,plo,phi),c(1,1,1,1,sigma[i]), main=paste("sigma=",as.character(sigma[i])), xlab="p1",ylab="p2") S=readline(prompt="Type to continue : ") sigma=c(2,1,.5,.25) windows() par(mfrow=c(2,2)) for (i in 1:4) { mycontour(howardprior,c(plo,phi,plo,phi), c(1+3,1+15,1+7,1+5,sigma[i]), main=paste("sigma=",as.character(sigma[i])), xlab="p1",ylab="p2") lines(c(0,1),c(0,1)) } s=simcontour(howardprior,c(plo,phi,plo,phi), c(1+3,1+15,1+7,1+5,2),1000) sum(s$x>s$y)/1000 LearnBayes/demo/Chapter.8.4.R0000644000176200001440000000051411106370070015237 0ustar liggesusers################################################# # Section 8.4 A Two-Sided Test of a Normal Mean ################################################# library(LearnBayes) weights=c(182, 172, 173, 176, 176, 180, 173, 174, 179, 175) data=c(mean(weights),length(weights),3) t=c(.5,1,2,4,8) mnormt.twosided(170,.5,t,data) LearnBayes/demo/Chapter.1.3.R0000644000176200001440000000270711127272142015241 0ustar liggesusers# Chapter 1.3 R commands # Section 1.3.2 x=rnorm(10,mean=50,sd=10) y=rnorm(10,mean=50,sd=10) m=length(x) n=length(y) sp=sqrt(((m-1)*sd(x)^2+(n-1)*sd(y)^2)/(m+n-2)) t.stat=(mean(x)-mean(y))/(sp*sqrt(1/m+1/n)) tstatistic=function(x,y) { m=length(x) n=length(y) sp=sqrt(((m-1)*sd(x)^2+(n-1)*sd(y)^2)/(m+n-2)) t.stat=(mean(x)-mean(y))/(sp*sqrt(1/m+1/n)) return(t.stat) } data.x=c(1,4,3,6,5) data.y=c(5,4,7,6,10) tstatistic(data.x, data.y) S=readline(prompt="Type to continue : ") # Section 1.3.3 # simulation algorithm for normal populations alpha=.1; m=10; n=10 # sets alpha, m, n N=10000 # sets the number of simulations n.reject=0 # counter of num. of rejections for (i in 1:N) { x=rnorm(m,mean=0,sd=1) # simulates xs from population 1 y=rnorm(n,mean=0,sd=1) # simulates ys from population 2 t.stat=tstatistic(x,y) # computes the t statistic if (abs(t.stat)>qt(1-alpha/2,n+m-2)) n.reject=n.reject+1 # reject if |t| exceeds critical pt } true.sig.level=n.reject/N # est. is proportion of rejections s=readline(prompt="Type to continue : ") # simulation algorithm for normal and exponential populations # storing the values of the t statistic in vector tstat m=10; n=10 my.tsimulation=function() tstatistic(rnorm(m,mean=10,sd=2), rexp(n,rate=1/10)) tstat.vector=replicate(10000, my.tsimulation()) plot(density(tstat.vector),xlim=c(-5,8),ylim=c(0,.4),lwd=3) curve(dt(x,df=18),add=TRUE) LearnBayes/demo/Chapter.8.7.R0000644000176200001440000000073211106370356015253 0ustar liggesusers################################################### # Section 8.7 Is a Baseball Hitter Really Streaky? ################################################### library(LearnBayes) data(jeter2004) attach(jeter2004) data=cbind(H,AB) data1=regroup(data,5) log.marg=function(logK) laplace(bfexch,0,list(data=data1,K=exp(logK)))$int log.K=seq(2,6) K=exp(log.K) log.BF=sapply(log.K,log.marg) BF=exp(log.BF) round(data.frame(log.K,K,log.BF,BF),2)LearnBayes/demo/Chapter.3.4.R0000644000176200001440000000322011127272272015237 0ustar liggesusers#################################################### # Section 3.4 An Illustration of Bayesian Robustness #################################################### library(LearnBayes) quantile1=list(p=.5,x=100); quantile2=list(p=.95,x=120) normal.select(quantile1, quantile2) mu = 100 tau = 12.16 sigma = 15 n = 4 se = sigma/sqrt(4) ybar = c(110, 125, 140) tau1 = 1/sqrt(1/se^2 + 1/tau^2) mu1 = (ybar/se^2 + mu/tau^2) * tau1^2 summ1=cbind(ybar, mu1, tau1) summ1 tscale = 20/qt(0.95, 2) tscale par(mfrow=c(1,1)) curve(1/tscale*dt((x-mu)/tscale,2), from=60, to=140, xlab="theta", ylab="Prior Density") curve(dnorm(x,mean=mu,sd=tau), add=TRUE, lwd=3) legend("topright",legend=c("t density","normal density"), lwd=c(1,3)) S=readline(prompt="Type to continue : ") norm.t.compute=function(ybar) { theta = seq(60, 180, length = 500) like = dnorm(theta,mean=ybar,sd=sigma/sqrt(n)) prior = dt((theta - mu)/tscale, 2) post = prior * like post = post/sum(post) m = sum(theta * post) s = sqrt(sum(theta^2 * post) - m^2) c(ybar, m, s) } summ2=t(sapply(c(110, 125, 140),norm.t.compute)) dimnames(summ2)[[2]]=c("ybar","mu1 t","tau1 t") summ2 cbind(summ1,summ2) theta=seq(60, 180, length=500) normpost = dnorm(theta, mu1[3], tau1) normpost = normpost/sum(normpost) windows() plot(theta,normpost,type="l",lwd=3,ylab="Posterior Density") like = dnorm(theta,mean=140,sd=sigma/sqrt(n)) prior = dt((theta - mu)/tscale, 2) tpost = prior * like / sum(prior * like) lines(theta,tpost) legend("topright",legend=c("t prior","normal prior"),lwd=c(1,3)) LearnBayes/demo/Chapter.2.3.R0000644000176200001440000000135611127272164015245 0ustar liggesusers#################################### # Section 2.3 Using a Discrete Prior #################################### library(LearnBayes) p = seq(0.05, 0.95, by = 0.1) prior = c(1, 5.2, 8, 7.2, 4.6, 2.1, 0.7, 0.1, 0, 0) prior = prior/sum(prior) plot(p, prior, type = "h", ylab="Prior Probability") S=readline(prompt="Type to continue : ") data = c(11, 16) post = pdisc(p, prior, data) round(cbind(p, prior, post),2) library(lattice) PRIOR=data.frame("prior",p,prior) POST=data.frame("posterior",p,post) names(PRIOR)=c("Type","P","Probability") names(POST)=c("Type","P","Probability") data=rbind(PRIOR,POST) windows() xyplot(Probability~P|Type,data=data,layout=c(1,2),type="h",lwd=3,col="black") LearnBayes/demo/Chapter.9.3.R0000644000176200001440000000207511127267500015251 0ustar liggesusers############################################## # Section 9.3 Modeling Using Zellner's g Prior ############################################## library(LearnBayes) # illustrating the role of the parameter c data(puffin) X=cbind(1, puffin$Distance - mean(puffin$Distance)) c.prior=c(0.1,0.5,5,2) fit=vector("list",4) for (j in 1:4) { prior=list(b0=c(8,0), c0=c.prior[j]) fit[[j]]=blinreg(puffin$Nest, X, 1000, prior) } BETA=NULL for (j in 1:4) { s=data.frame(Prior=paste("c =",as.character(c.prior[j])), beta0=fit[[j]]$beta[,1],beta1=fit[[j]]$beta[,2]) BETA=rbind(BETA,s) } library(lattice) with(BETA,xyplot(beta1~beta0|Prior,type=c("p","g"),col="black")) S=readline(prompt="Type to continue : ") # model selection data=list(y=puffin$Nest, X=cbind(1,puffin$Grass,puffin$Soil)) prior=list(b0=c(0,0,0), c0=100) beta.start=with(puffin,lm(Nest~Grass+Soil)$coef) laplace(reg.gprior.post,c(beta.start,0),list(data=data,prior=prior))$int X=puffin[,-1]; y=puffin$Nest; c=100 bayes.model.selection(y,X,c,constant=FALSE) LearnBayes/demo/Chapter.7.3.R0000644000176200001440000000051411106543744015247 0ustar liggesusers############################################## # Section 7.3 Individual or Combined Estimates ############################################## library(LearnBayes) data(hearttransplants) attach(hearttransplants) plot(log(e), y/e, xlim=c(6,9.7), xlab="log(e)", ylab="y/e") text(log(e),y/e,labels=as.character(y),pos=4) LearnBayes/demo/Chapter.2.6.R0000644000176200001440000000115511106332434015237 0ustar liggesusers######################## # Section 2.6 Prediction ######################## library(LearnBayes) p=seq(0.05, 0.95, by=.1) prior = c(1, 5.2, 8, 7.2, 4.6, 2.1, 0.7, 0.1, 0, 0) prior=prior/sum(prior) m=20; ys=0:20 pred=pdiscp(p, prior, m, ys) cbind(0:20,pred) ab=c(3.26, 7.19) m=20; ys=0:20 pred=pbetap(ab, m, ys) p=rbeta(1000,3.26, 7.19) y = rbinom(1000, 20, p) table(y) freq=table(y) ys=as.integer(names(freq)) predprob=freq/sum(freq) plot(ys,predprob,type="h",xlab="y", ylab="Predictive Probability") dist=cbind(ys,predprob) covprob=.9 discint(dist,covprob) LearnBayes/demo/Chapter.3.2.R0000644000176200001440000000070711106333562015241 0ustar liggesusers###################################################################### # Section 3.2 Normal Distribution with Known Mean but Unknown Variance ###################################################################### library(LearnBayes) data(footballscores) attach(footballscores) d = favorite - underdog - spread n = length(d) v = sum(d^2) P = rchisq(1000, n)/v s = sqrt(1/P) hist(s) quantile(s, probs = c(0.025, 0.5, 0.975)) LearnBayes/demo/Chapter.5.4.R0000644000176200001440000000072611127272406015250 0ustar liggesusers##################################################### # Section 5.4 A Beta-Binomial Model for Overdispersion ##################################################### library(LearnBayes) data(cancermortality) mycontour(betabinexch0,c(.0001,.003,1,20000),cancermortality, xlab="eta",ylab="K") S=readline(prompt="Type to continue : ") windows() mycontour(betabinexch,c(-8,-4.5,3,16.5),cancermortality, xlab="logit eta",ylab="log K") LearnBayes/demo/Chapter.7.7.R0000644000176200001440000000254311127266772015265 0ustar liggesusers######################################################### # Section 7.7 Simulating from the Posterior ######################################################### library(LearnBayes) data(hearttransplants) attach(hearttransplants) datapar = list(data = hearttransplants, z0 = 0.53) start=c(2, -7) fit = laplace(poissgamexch, start, datapar) fit par(mfrow = c(1, 1)) mycontour(poissgamexch, c(0, 8, -7.3, -6.6), datapar, xlab="log alpha",ylab="log mu") S=readline(prompt="Type to continue : ") start = c(4, -7) fitgibbs = gibbs(poissgamexch, start, 1000, c(1,.15), datapar) fitgibbs$accept windows() mycontour(poissgamexch, c(0, 8, -7.3, -6.6), datapar, xlab="log alpha",ylab="log mu") points(fitgibbs$par[, 1], fitgibbs$par[, 2]) S=readline(prompt="Type to continue : ") windows() plot(density(fitgibbs$par[, 1], bw = 0.2)) alpha = exp(fitgibbs$par[, 1]) mu = exp(fitgibbs$par[, 2]) lam1 = rgamma(1000, y[1] + alpha, e[1] + alpha/mu) alpha = exp(fitgibbs$par[, 1]) mu = exp(fitgibbs$par[, 2]) S=readline(prompt="Type to continue : ") windows() plot(log(e), y/e, pch = as.character(y)) for (i in 1:94) { lami = rgamma(1000, y[i] + alpha, e[i] + alpha/mu) probint = quantile(lami, c(0.05, 0.95)) lines(log(e[i]) * c(1, 1), probint) } LearnBayes/demo/Chapter.7.10.R0000644000176200001440000000166111127267332015330 0ustar liggesusers################################################# # Section 7.10 Posterior Predictive Model Checking ################################################# library(LearnBayes) data(hearttransplants) attach(hearttransplants) datapar = list(data = hearttransplants, z0 = 0.53) start = c(4, -7) fitgibbs = gibbs(poissgamexch, start, 1000, c(1,.15), datapar) lam94=rgamma(1000,y[94]+alpha,e[94]+alpha/mu) ys94=rpois(1000,e[94]*lam94) hist(ys94,breaks=seq(-0.5,max(ys94)+0.5)) lines(y[94]*c(1,1),c(0,100),lwd=3) S=readline(prompt="Type to continue : ") prob.out=function(i) { lami=rgamma(1000,y[i]+alpha,e[i]+alpha/mu) ysi=rpois(1000,e[i]*lami) pleft=sum(ysi<=y[i])/1000 pright=sum(ysi>=y[i])/1000 min(pleft,pright) } pout.exchange=sapply(1:94,prob.out) windows() plot(pout,pout.exchange,xlab="P(extreme), equal means", ylab="P(extreme), exchangeable") abline(0,1) LearnBayes/demo/Chapter.10.4.R0000644000176200001440000000516111127267664015334 0ustar liggesusers################################################### # Section 10.4 Estimating a Table of Means ################################################### library(LearnBayes) data(iowagpa) rlabels = c("91-99", "81-90", "71-80", "61-70", "51-60", "41-50", "31-40", "21-30") clabels = c("16-18", "19-21", "22-24", "25-27", "28-30") gpa = matrix(iowagpa[, 1], nrow = 8, ncol = 5, byrow = T) dimnames(gpa) = list(HSR = rlabels, ACTC = clabels) gpa samplesizes = matrix(iowagpa[, 2], nrow = 8, ncol = 5, byrow = T) dimnames(samplesizes) = list(HSR = rlabels, ACTC = clabels) samplesizes act = seq(17, 29, by = 3) matplot(act, t(gpa), type = "l", lwd = 3, xlim = c(17, 34), col=1:8, lty=1:8) legend(30, 3, lty = 1:8, lwd = 3, legend = c("HSR=9", "HSR=8", "HSR=7", "HSR=6", "HSR=5", "HSR=4", "HSR=3", "HSR=2"), col=1:8) S=readline(prompt="Type to continue : ") MU = ordergibbs(iowagpa, 5000) postmeans = apply(MU, 2, mean) postmeans = matrix(postmeans, nrow = 8, ncol = 5) postmeans=postmeans[seq(8,1,-1),] dimnames(postmeans)=list(HSR=rlabels,ACTC=clabels) round(postmeans,2) windows() matplot(act, t(postmeans), type = "l", lty=1:8, lwd = 3, col = 1, xlim = c(17, 34)) legend(30, 3, lty = 1:8, lwd = 2, legend = c("HSR=9", "HSR=8", "HSR=7", "HSR=6", "HSR=5", "HSR=4", "HSR=3", "HSR=2")) postsds = apply(MU, 2, sd) postsds = matrix(postsds, nrow = 8, ncol = 5) postsds=postsds[seq(8,1,-1),] dimnames(postsds)=list(HSR=rlabels,ACTC=clabels) round(postsds,3) s=.65 se=s/sqrt(samplesizes) round(postsds/se,2) S=readline(prompt="Type to continue : ") FIT=hiergibbs(iowagpa,5000) windows() par(mfrow=c(2,1)) plot(density(FIT$beta[,2]),xlab=expression(beta[2]), main="HIGH SCHOOL RANK") plot(density(FIT$beta[,3]),xlab=expression(beta[3]), main="ACT SCORE") quantile(FIT$beta[,2],c(.025,.25,.5,.75,.975)) quantile(FIT$beta[,3],c(.025,.25,.5,.75,.975)) quantile(FIT$var,c(.025,.25,.5,.75,.975)) posterior.means = apply(FIT$mu, 2, mean) posterior.means = matrix(posterior.means, nrow = 8, ncol = 5, byrow = T) S=readline(prompt="Type to continue : ") windows() par(mfrow=c(1,1)) matplot(act, t(posterior.means), type = "l", lwd = 3, lty=1:8, col=1, xlim = c(17, 34)) legend(30, 3, lty = 1:8, lwd = 2, legend = c("HSR=9", "HSR=8", "HSR=7", "HSR=6", "HSR=5", "HSR=4", "HSR=3", "HSR=2")) p=1-pnorm((2.5-FIT$mu)/.65) prob.success=apply(p,2,mean) prob.success=matrix(prob.success,nrow=8,ncol=5,byrow=T) dimnames(prob.success)=list(HSR=rlabels,ACTC=clabels) round(prob.success,3) LearnBayes/demo/Chapter.3.5.R0000644000176200001440000000133511127272306015243 0ustar liggesusers####################################################### # Section 3.5 Mixtures of Conjugate Priors ####################################################### library(LearnBayes) curve(.5*dbeta(x, 6, 14) + .5*dbeta(x, 14, 6), from=0, to=1, xlab="P", ylab="Density") S=readline(prompt="Type to continue : ") probs=c(.5,.5) beta.par1=c(6, 14) beta.par2=c(14, 6) betapar=rbind(beta.par1, beta.par2) data=c(7,3) post=binomial.beta.mix(probs,betapar,data) post windows() curve(post$probs[1]*dbeta(x,13,17)+post$probs[2]*dbeta(x,21,9), from=0, to=1, lwd=3, xlab="P", ylab="DENSITY") curve(.5*dbeta(x,6,12)+.5*dbeta(x,12,6),0,1,add=TRUE) legend("topleft",legend=c("Prior","Posterior"),lwd=c(1,3)) LearnBayes/demo/Chapter.7.2.R0000644000176200001440000000142311127242034015235 0ustar liggesusers##################################################### # Section 7.2 Introduction to Hierarchical Modeling ##################################################### library(LearnBayes) library(lattice) data(sluggerdata) # fit logistic model for home run data for a particular player logistic.fit=function(player) { d=subset(sluggerdata,Player==player) x=d$Age; x2=d$Age^2 response=cbind(d$HR, d$AB-d$HR) list(Age=x, p=glm(response~x+x2,family=binomial)$fitted) } names=unique(sluggerdata$Player); newdata=NULL for (j in 1:9) { fit=logistic.fit(as.character(names[j])) newdata=rbind(newdata,data.frame(as.character(names[j]),fit$Age,fit$p)) } names(newdata)=c("Player","Age","Fitted") xyplot(Fitted~Age|Player, data=newdata, type="l",lwd=3,col="black") LearnBayes/demo/Chapter.7.5.R0000644000176200001440000000103511106353124015237 0ustar liggesusers######################################################## # Section 7.5 Modeling a Prior Belief of Exchangeability ######################################################## library(LearnBayes) pgexchprior=function(lambda,pars) { alpha=pars[1]; a=pars[2]; b=pars[3] (alpha-1)*log(prod(lambda))-(2*alpha+a)*log(alpha*sum(lambda)+b) } alpha=c(5,20,80,400); par(mfrow=c(2,2)) for (j in 1:4) mycontour(pgexchprior,c(.001,5,.001,5),c(alpha[j],10,10), main=paste("ALPHA = ",alpha[j]),xlab="LAMBDA 1",ylab="LAMBDA 2") LearnBayes/demo/00Index0000644000176200001440000000453211127271034014357 0ustar liggesusersChapter.1.2 Exploring a Student Dataset Chapter.1.3 Exploring the Robustness of the t Statistic Chapter.2.3 Learning About a Proportion - Using a Discrete Prior Chapter.2.4 Learning About a Proportion - Using a Beta Prior Chapter.2.5 Learning About a Proportion - Using a Histogram Prior Chapter.2.6 Learning About a Proportion - Prediction Chapter.3.2 Normal Distribution with Known Mean, Unknown Variance Chapter.3.3 Estimating a Heart Transplant Mortality Rate Chapter.3.4 Learning about a Normal Mean with Known Variance Chapter.3.5 Mixtures of Conjugate Priors Chapter.3.6 A Bayesian Test of the Fairness of a Coin Chapter.4.2 Normal Data with Both Parameters Unknown Chapter.4.3 A Multinomial Model Chapter.4.4 A Bioassay Experiment Chapter.4.5 Comparing Two Proportions Chapter.5.4 A Beta-Binomial Model for Overdispersion Chapter.5.6 Approximations Based on Posterior Modes for Beta-Binomial Model Chapter.5.7 Monte Carlo Method for Computing Integrals Chapter.5.8 Rejection Sampling Chapter.5.9 Importance Sampling Chapter.5.10 Sampling Importance Resampling Chapter.6.2 Discrete Markov Chains Chapter.6.7 MCMC - Learning About a Normal Population Based on Grouped Data Chapter.6.8 MCMC Output Analysis Chapter.6.9 Modeling Data with Cauchy Errors Chapter.6.10 Analysis of the Stanford Heart Transplant Data Chapter.7.2 Introduction to Career Trajectory Example Chapter.7.3 Introduction to Heart Transplant Mortality Data Chapter.7.4 Checking Assumption of Equal Mortality Rates Chapter.7.5 Exchangeable Model for Mortality Rates Chapter.7.7 Simulating from Posterior from Exchangeable Model Chapter.7.8 Illustration of Posterior Inferences Chapter.7.9 Bayesian Sensitivity Analysis Chapter.7.10 Posterior Predictive Model Checking Chapter.8.3 One-Sided Test of a Normal Mean Chapter.8.4 Two-Sided Test of a Normal Mean Chapter.8.6 Models for Soccer Goal Scoring Chapter.8.7 Test if Baseball Player is Streaky Chapter.8.8 Test of Independence in a Two-Way Contingency Table Chapter.9.2 Normal Linear Regression Chapter.9.3 Model Selection Using Zellner's g Prior Chapter.9.4 Survival Modeling Chapter.10.2 Robust Modeling Chapter.10.3 Binary Response Regression with Probit Link Chapter.10.4 Estimating Table of Means with Belief of Order Restriction LearnBayes/demo/Chapter.10.2.R0000644000176200001440000000114011127267540015314 0ustar liggesusers############################### # Section 10.2 Robust Modeling ############################### library(LearnBayes) data(darwin) attach(darwin) fit=robustt(difference,4,10000) plot(density(fit$mu),xlab="mu") mean.lambda=apply(fit$lam,2,mean) lam5=apply(fit$lam,2,quantile,.05) lam95=apply(fit$lam,2,quantile,.95) S=readline(prompt="Type to continue : ") windows() plot(difference,mean.lambda,lwd=2,ylim=c(0,3),ylab="Lambda") for (i in 1:length(difference)) lines(c(1,1)*difference[i],c(lam5[i],lam95[i])) points(difference,0*difference-.05,pch=19,cex=2) LearnBayes/demo/Chapter.6.9.R0000644000176200001440000000305411127267702015256 0ustar liggesusers################################################### # Section 6.9 Modeling Data with Cauchy Errors ################################################### library(LearnBayes) data(darwin) attach(darwin) mean(difference) log(sd(difference)) laplace(cauchyerrorpost,c(21.6,3.6),difference) laplace(cauchyerrorpost,.1*c(21.6,3.6),difference)$mode c(24.7-4*sqrt(34.96),24.7+4*sqrt(34.96)) c(2.77-4*sqrt(.138),2.77+4*sqrt(.138)) mycontour(cauchyerrorpost,c(-10,60,1,4.5),difference, xlab="mu",ylab="log sigma") S=readline(prompt="Type to continue : ") fitlaplace=laplace(cauchyerrorpost,c(21.6,3.6), difference) windows() mycontour(lbinorm,c(-10,60,1,4.5),list(m=fitlaplace$mode, v=fitlaplace$var), xlab="mu",ylab="log sigma") proposal=list(var=fitlaplace$var,scale=2.5) start=c(20,3) m=1000 s=rwmetrop(cauchyerrorpost,proposal,start,m,difference) S=readline(prompt="Type to continue : ") windows() mycontour(cauchyerrorpost,c(-10,60,1,4.5),difference, xlab="mu",ylab="log sigma") points(s$par[,1],s$par[,2]) fitgrid=simcontour(cauchyerrorpost,c(-10,60,1,4.5),difference, 50000) proposal=list(var=fitlaplace$var,scale=2.5) start=c(20,3) fitrw=rwmetrop(cauchyerrorpost,proposal,start,50000, difference) proposal2=list(var=fitlaplace$var,mu=t(fitlaplace$mode)) fitindep=indepmetrop(cauchyerrorpost,proposal2,start,50000, difference) fitgibbs=gibbs(cauchyerrorpost,start,50000,c(12,.75), difference) apply(fitrw$par,2,mean) apply(fitrw$par,2,sd) LearnBayes/demo/Chapter.5.7.R0000644000176200001440000000040411117604764015251 0ustar liggesusers######################################################### # Section 5.7 Monte Carlo Method for Computing Integrals ######################################################### p=rbeta(1000, 14.26, 23.19) est=mean(p^2) se=sd(p^2)/sqrt(1000) c(est,se) LearnBayes/demo/Chapter.5.10.R0000644000176200001440000000106511106347330015316 0ustar liggesusers############################################## # Section 5.10 Sampling Importance Resampling ############################################## library(LearnBayes) data(cancermortality) fit=laplace(betabinexch,c(-7,6),cancermortality) tpar=list(m=fit$mode,var=2*fit$var,df=4) theta.s=sir(betabinexch,tpar,10000,cancermortality) S=bayes.influence(theta.s,cancermortality) plot(c(0,0,0),S$summary,type="b",lwd=3,xlim=c(-1,21), ylim=c(5,11), xlab="Observation removed",ylab="log K") for (i in 1:20) lines(c(i,i,i),S$summary.obs[i,],type="b") LearnBayes/demo/Chapter.1.2.R0000644000176200001440000000150211127272130015225 0ustar liggesusers# Section 1.2 R commands # Section 1.2.2 library(LearnBayes) data(studentdata) studentdata[1,] attach(studentdata) # Section 1.2.3 table(Drink) barplot(table(Drink),xlab="Drink",ylab="Count") S=readline(prompt="Type to continue : ") windows() hours.of.sleep = WakeUp - ToSleep summary(hours.of.sleep) hist(hours.of.sleep,main="") S=readline(prompt="Type to continue : ") # Section 1.2.4 windows() boxplot(hours.of.sleep~Gender, ylab="Hours of Sleep") female.Haircut=Haircut[Gender=="female"] male.Haircut=Haircut[Gender=="male"] summary(female.Haircut) summary(male.Haircut) S=readline(prompt="Type to continue : ") # Section 1.2.5 windows() plot(jitter(ToSleep),jitter(hours.of.sleep)) fit=lm(hours.of.sleep~ToSleep) fit abline(fit) LearnBayes/demo/Chapter.7.8.R0000644000176200001440000000260411127267064015257 0ustar liggesusers########################################################## # Section 7.8 Posterior Inferences ########################################################## library(LearnBayes) data(hearttransplants) attach(hearttransplants) datapar = list(data = hearttransplants, z0 = 0.53) start=c(2, -7) fit = laplace(poissgamexch, start, datapar) fit par(mfrow = c(1, 1)) mycontour(poissgamexch, c(0, 8, -7.3, -6.6), datapar, xlab="log alpha",ylab="log mu") S=readline(prompt="Type to continue : ") start = c(4, -7) fitgibbs = gibbs(poissgamexch, start, 1000, c(1,.15), datapar) alpha = exp(fitgibbs$par[, 1]) mu = exp(fitgibbs$par[, 2]) shrink=function(i) mean(alpha/(alpha + e[i] * mu)) shrinkage=sapply(1:94, shrink) S=readline(prompt="Type to continue : ") windows() plot(log(e), shrinkage) mrate=function(i) mean(rgamma(1000, y[i] + alpha, e[i] + alpha/mu)) hospital=1:94 meanrate=sapply(hospital,mrate) hospital[meanrate==min(meanrate)] ########################################################### sim.lambda=function(i) rgamma(1000,y[i]+alpha,e[i]+alpha/mu) LAM=sapply(1:94,sim.lambda) compare.rates <- function(x) { nc <- NCOL(x) ij <- as.matrix(expand.grid(1:nc, 1:nc)) m <- as.matrix(x[,ij[,1]] > x[,ij[,2]]) matrix(colMeans(m), nc, nc, byrow = TRUE) } better=compare.rates(LAM) better[1:24,85] LearnBayes/demo/Chapter.7.4.R0000644000176200001440000000130711127266602015247 0ustar liggesusers############################################## # Section 7.4 Equal Mortality Rates? ############################################## library(LearnBayes) data(hearttransplants) attach(hearttransplants) sum(y) sum(e) lambda=rgamma(1000,shape=277,rate=294681) ys94=rpois(1000,e[94]*lambda) hist(ys94,breaks=seq(0.5,max(ys94)+0.5)) lines(c(y[94],y[94]),c(0,120),lwd=3) S=readline(prompt="Type to continue : ") lambda=rgamma(1000,shape=277,rate=294681) prob.out=function(i) { ysi=rpois(1000,e[i]*lambda) pleft=sum(ysi<=y[i])/1000 pright=sum(ysi>=y[i])/1000 min(pleft,pright) } pout=sapply(1:94,prob.out) windows() plot(log(e),pout,ylab="Prob(extreme)") LearnBayes/demo/Chapter.8.6.R0000644000176200001440000000133111106370214015237 0ustar liggesusers################################################# # Section 8.6 Models for Soccer Goals ################################################# library(LearnBayes) data(soccergoals) attach(soccergoals) datapar=list(data=goals,par=c(4.57,1.43)) fit1=laplace(logpoissgamma,.5,datapar) datapar=list(data=goals,par=c(1,.5)) fit2=laplace(logpoissnormal,.5,datapar) datapar=list(data=goals,par=c(2,.5)) fit3=laplace(logpoissnormal,.5,datapar) datapar=list(data=goals,par=c(1,2)) fit4=laplace(logpoissnormal,.5,datapar) postmode=c(fit1$mode,fit2$mode,fit3$mode,fit4$mode) postsd=sqrt(c(fit1$var,fit2$var,fit3$var,fit4$var)) logmarg=c(fit1$int,fit2$int,fit3$int,fit4$int) cbind(postmode,postsd,logmarg) LearnBayes/demo/Chapter.9.2.R0000644000176200001440000000537311127267464015265 0ustar liggesusers################################ # Section 9.2.6 An Example ################################ library(LearnBayes) data(birdextinct) attach(birdextinct) logtime=log(time) plot(nesting,logtime) out = (logtime > 3) text(nesting[out], logtime[out], label=species[out], pos = 2) S=readline(prompt="Type to continue : ") windows() plot(jitter(size),logtime,xaxp=c(0,1,1)) S=readline(prompt="Type to continue : ") windows() plot(jitter(status),logtime,xaxp=c(0,1,1)) ##### Least-squares fit fit=lm(logtime~nesting+size+status,data=birdextinct,x=TRUE,y=TRUE) summary(fit) ##### Sampling from posterior theta.sample=blinreg(fit$y,fit$x,5000) S=readline(prompt="Type to continue : ") windows() par(mfrow=c(2,2)) hist(theta.sample$beta[,2],main="NESTING", xlab=expression(beta[1])) hist(theta.sample$beta[,3],main="SIZE", xlab=expression(beta[2])) hist(theta.sample$beta[,4],main="STATUS", xlab=expression(beta[3])) hist(theta.sample$sigma,main="ERROR SD", xlab=expression(sigma)) apply(theta.sample$beta,2,quantile,c(.05,.5,.95)) quantile(theta.sample$sigma,c(.05,.5,.95)) S=readline(prompt="Type to continue : ") ###### Estimating mean extinction times cov1=c(1,4,0,0) cov2=c(1,4,1,0) cov3=c(1,4,0,1) cov4=c(1,4,1,1) X1=rbind(cov1,cov2,cov3,cov4) mean.draws=blinregexpected(X1,theta.sample) c.labels=c("A","B","C","D") windows() par(mfrow=c(2,2)) for (j in 1:4) hist(mean.draws[,j], main=paste("Covariate set",c.labels[j]),xlab="log TIME") S=readline(prompt="Type to continue : ") ######## Predicting extinction times cov1=c(1,4,0,0) cov2=c(1,4,1,0) cov3=c(1,4,0,1) cov4=c(1,4,1,1) X1=rbind(cov1,cov2,cov3,cov4) pred.draws=blinregpred(X1,theta.sample) c.labels=c("A","B","C","D") windows() par(mfrow=c(2,2)) for (j in 1:4) hist(pred.draws[,j], main=paste("Covariate set",c.labels[j]),xlab="log TIME") S=readline(prompt="Type to continue : ") ######### Model checking via posterior predictive distribution pred.draws=blinregpred(fit$x,theta.sample) pred.sum=apply(pred.draws,2,quantile,c(.05,.95)) par(mfrow=c(1,1)) ind=1:length(logtime) windows() matplot(rbind(ind,ind),pred.sum,type="l",lty=1,col=1, xlab="INDEX",ylab="log TIME") points(ind,logtime,pch=19) out=(logtime>pred.sum[2,]) text(ind[out], logtime[out], label=species[out], pos = 4) S=readline(prompt="Type to continue : ") ######### Model checking via bayes residuals prob.out=bayesresiduals(fit,theta.sample,2) windows() par(mfrow=c(1,1)) plot(nesting,prob.out) out = (prob.out > 0.35) text(nesting[out], prob.out[out], label=species[out], pos = 4) LearnBayes/demo/Chapter.4.2.R0000644000176200001440000000111311106344376015237 0ustar liggesusers###################################################### # Section 4.2 Normal Data with Both Parameters Unknown ###################################################### library(LearnBayes) data(marathontimes) attach(marathontimes) d = mycontour(normchi2post, c(220, 330, 500, 9000), time, xlab="mean",ylab="variance") S = sum((time - mean(time))^2) n = length(time) sigma2 = S/rchisq(1000, n - 1) mu = rnorm(1000, mean = mean(time), sd = sqrt(sigma2)/sqrt(n)) points(mu, sigma2) quantile(mu, c(0.025, 0.975)) quantile(sqrt(sigma2), c(0.025, 0.975)) LearnBayes/demo/Chapter.7.9.R0000644000176200001440000000300211127236512015243 0ustar liggesusers################################################# # Section 7.9 Bayesian Sensitivity Analysis ################################################# library(LearnBayes) data(hearttransplants) attach(hearttransplants) datapar = list(data = hearttransplants, z0 = 0.53) start = c(4, -7) fitgibbs = gibbs(poissgamexch, start, 1000, c(1,.15), datapar) sir.old.new=function(theta, prior, prior.new) { log.g=log(prior(theta)) log.g.new=log(prior.new(theta)) wt=exp(log.g.new-log.g-max(log.g.new-log.g)) probs=wt/sum(wt) n=length(probs) indices=sample(1:n,size=n,prob=probs,replace=TRUE) theta[indices] } prior=function(theta) 0.53*exp(theta)/(exp(theta)+0.53)^2 prior.new=function(theta) 5*exp(theta)/(exp(theta)+5)^2 log.alpha=fitgibbs$par[, 1] log.alpha.new=sir.old.new(log.alpha, prior, prior.new) ############ drawing figure library(lattice) draw.graph=function() { LOG.ALPHA=data.frame("prior",log.alpha) names(LOG.ALPHA)=c("Prior","log.alpha") LOG.ALPHA.NEW=data.frame("new.prior",log.alpha.new) names(LOG.ALPHA.NEW)=c("Prior","log.alpha") D=densityplot(~log.alpha,group=Prior,data=rbind(LOG.ALPHA,LOG.ALPHA.NEW), plot.points=FALSE,main="Original Prior and Posterior (solid), New Prior and Posterior (dashed)", lwd=4,adjust=2,lty=c(1,2),xlab="log alpha",xlim=c(-3,5),col="black") update(D, panel=function(...){ panel.curve(prior(x),lty=1,lwd=2,col="black") panel.curve(prior.new(x),lty=2, lwd=2,col="black") panel.densityplot(...) })} draw.graph() LearnBayes/demo/Chapter.10.3.R0000644000176200001440000000276211127267554015335 0ustar liggesusers############################################################# # Section 10.3 Binary Response Regression with a Probit Link ############################################################# ################################################# # Section 10.3.1. Missing data and Gibbs sampling ################################################# library(LearnBayes) data(donner) attach(donner) X=cbind(1,age,male) fit=glm(survival~X-1,family=binomial(link=probit)) summary(fit) m=10000 fit=bayes.probit(survival,X,m) apply(fit$beta,2,mean) apply(fit$beta,2,sd) a=seq(15,65) X1=cbind(1,a,1) p.male=bprobit.probs(X1,fit$beta) plot(a,apply(p.male,2,quantile,.5),type="l",ylim=c(0,1), xlab="age",ylab="Probability of Survival") lines(a,apply(p.male,2,quantile,.05),lty=2) lines(a,apply(p.male,2,quantile,.95),lty=2) S=readline(prompt="Type to continue : ") ################################################### # Section 10.3.2 Proper priors and model selection ################################################### library(LearnBayes) data(donner) y=donner$survival X=cbind(1,donner$age,donner$male) beta0=c(0,0,0); c0=100 P0=t(X)%*%X/c0 bayes.probit(y,X,1000,list(beta=beta0,P=P0))$log.marg bayes.probit(y,X[,-2],1000, list(beta=beta0[-2],P=P0[-2,-2]))$log.marg bayes.probit(y,X[,-3],1000, list(beta=beta0[-3],P=P0[-3,-3]))$log.marg bayes.probit(y,X[,-c(2,3)],1000, list(beta=beta0[-c(2,3)],P=P0[-c(2,3),-c(2,3)]))$log.marg LearnBayes/demo/Chapter.2.5.R0000644000176200001440000000144611127272226015246 0ustar liggesusers##################################### # Section 2.5 Using a Histogram Prior ##################################### library(LearnBayes) midpt = seq(0.05, 0.95, by = 0.1) prior = c(1, 5.2, 8, 7.2, 4.6, 2.1, 0.7, 0.1, 0, 0) prior = prior/sum(prior) curve(histprior(x,midpt,prior), from=0, to=1, ylab="Prior density",ylim=c(0,.3)) s = 11 f = 16 S=readline(prompt="Type to continue : ") windows() curve(histprior(x,midpt,prior) * dbeta(x,s+1,f+1), from=0, to=1, ylab="Posterior density") S=readline(prompt="Type to continue : ") p = seq(0, 1, length=500) post = histprior(p, midpt, prior) * dbeta(p, s+1, f+1) post = post/sum(post) ps = sample(p, replace = TRUE, prob = post) windows() hist(ps, xlab="p", main="") LearnBayes/demo/Chapter.2.4.R0000644000176200001440000000131411127217336015240 0ustar liggesusers################################ # Section 2.4 Using a Beta Prior ############################# library(LearnBayes) quantile2=list(p=.9,x=.5) quantile1=list(p=.5,x=.3) ab=beta.select(quantile1,quantile2) a = ab[1] b = ab[2] s = 11 f = 16 curve(dbeta(x,a+s,b+f), from=0, to=1, xlab="p",ylab="Density",lty=1,lwd=4) curve(dbeta(x,s+1,f+1),add=TRUE,lty=2,lwd=4) curve(dbeta(x,a,b),add=TRUE,lty=3,lwd=4) legend(.7,4,c("Prior","Likelihood","Posterior"), lty=c(3,2,1),lwd=c(3,3,3)) 1 - pbeta(0.5, a + s, b + f) qbeta(c(0.05, 0.95), a + s, b + f) ps = rbeta(1000, a + s, b + f) windows() hist(ps,xlab="p") sum(ps >= 0.5)/1000 quantile(ps, c(0.05, 0.95)) LearnBayes/demo/Chapter.5.9.R0000644000176200001440000000300611106346666015256 0ustar liggesusers############################################# # Section 5.9 Importance Sampling ############################################# library(LearnBayes) data(cancermortality) fit=laplace(betabinexch,c(-7,6),cancermortality) betabinexch.cond=function (log.K, data) { eta = exp(-6.818793)/(1 + exp(-6.818793)) K = exp(log.K) y = data[, 1]; n = data[, 2]; N = length(y) logf=0*log.K for (j in 1:length(y)) logf = logf + lbeta(K * eta + y[j], K * (1 - eta) + n[j] - y[j]) - lbeta(K * eta, K * (1 - eta)) val = logf + log.K - 2 * log(1 + K) return(exp(val-max(val))) } I=integrate(betabinexch.cond,2,16,cancermortality) par(mfrow=c(2,2)) curve(betabinexch.cond(x,cancermortality)/I$value,from=3,to=16, ylab="Density", xlab="log K",lwd=3, main="Densities") curve(dnorm(x,8,2),add=TRUE) legend("topright",legend=c("Exact","Normal"),lwd=c(3,1)) curve(betabinexch.cond(x,cancermortality)/I$value/ dnorm(x,8,2),from=3,to=16, ylab="Weight",xlab="log K", main="Weight = g/p") curve(betabinexch.cond(x,cancermortality)/I$value,from=3,to=16, ylab="Density", xlab="log K",lwd=3, main="Densities") curve(1/2*dt(x-8,df=2),add=TRUE) legend("topright",legend=c("Exact","T(2)"),lwd=c(3,1)) curve(betabinexch.cond(x,cancermortality)/I$value/ (1/2*dt(x-8,df=2)),from=3,to=16, ylab="Weight",xlab="log K", main="Weight = g/p") tpar=list(m=fit$mode,var=2*fit$var,df=4) myfunc=function(theta) return(theta[2]) s=impsampling(betabinexch,tpar,myfunc,10000,cancermortality) cbind(s$est,s$se) LearnBayes/demo/Chapter.5.8.R0000644000176200001440000000150511106346626015253 0ustar liggesusers######################################################### # Section 5.8 Rejection Sampling ######################################################### library(LearnBayes) data(cancermortality) fit=laplace(betabinexch,c(-7,6),cancermortality) betabinT=function(theta,datapar) { data=datapar$data tpar=datapar$par d=betabinexch(theta,data)-dmt(theta,mean=c(tpar$m), S=tpar$var,df=tpar$df,log=TRUE) return(d) } tpar=list(m=fit$mode,var=2*fit$var,df=4) datapar=list(data=cancermortality,par=tpar) start=c(-6.9,12.4) fit1=laplace(betabinT,start,datapar) fit1$mode betabinT(fit1$mode,datapar) theta=rejectsampling(betabinexch,tpar,-569.2813,10000,cancermortality) dim(theta) mycontour(betabinexch,c(-8,-4.5,3,16.5),cancermortality, xlab="logit eta",ylab="log K") points(theta[,1],theta[,2]) LearnBayes/data/0000755000176200001440000000000012341620471013210 5ustar liggesusersLearnBayes/data/marathontimes.txt.gz0000644000176200001440000000011412341620471017237 0ustar liggesusers=10 CݧI10q0<ɶϥ,C6Ĕ\}~aъJmCxULearnBayes/data/achievement.txt.gz0000644000176200001440000000160012341620471016655 0ustar liggesusersMUKnTA \;iKDX /!d<\ǻ}jwxsp;bﷆOÞnmIoXoK~-<6Bc!#mtNOLDӄ3-6CD &DFՏakD1lƲ4lݕ`C(Ad(,b+?{Rq i'ՙ.jC@.t/%7 j?_d⹫z85a}˂ oFLearnBayes/data/bermuda.grass.txt.gz0000644000176200001440000000043112341620471017123 0ustar liggesusersE;n1 EzaH껉p*q݇ F0́pAwu)^- #OCzS4NT snv(QF-9=a$_$ɾ%ۄj Zm hiP+P vui(f~yFL{ * HCR'PB^ށk^%iPm Ȯqdl`MvTێmA:mtv@mlA+3Ym6)LearnBayes/data/iowagpa.txt.gz0000644000176200001440000000041112341620471016013 0ustar liggesusers51R@! 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RV5L` ΰ']v$т Z? /[i;X% F#D}5"E;hR9X3#V ;^= e@ja+/V,BL޿ٴ &x2Ţi{(8 ө1($[%1ꃃ“s|\| ]{t랱!89fa^w'sS`^NxJ/A %pSdosB4/܋kE(>❇ bZ"Jw3Be83w*K>6t"yT 6ȴ>텅ϙ,Pm Ct^i?19K;ϔ< 7z=׫ҐBg }E뽾dmJyY T;Q!zYw)6i6gq1zCAYNN6;G0);(rPT>ɱbGڦ,5<@2rZv &>j96 [KŴr#Q\{LłeN2Au>MIQ3]$¢Q1LearnBayes/data/birthweight.txt.gz0000644000176200001440000000022512341620471016707 0ustar liggesusers=;1 D4kA" 4&t7e׽˷?CzHx#T5b`NW%d3Y'*jeT #pfDDHppܣ;MwS}1;.K it>ʃHn&IM^0P"ګ %D i f=wp6u&7(a <5:]?'aM% +#⛚&GZuXD -{~vt'lՁ^Le*J\KKlǛBN @v7(?B{U2fk.馜Y0 /LearnBayes/data/chemotherapy.txt.gz0000644000176200001440000000033212341620471017056 0ustar liggesusers59nD1 Ck4n shL#/(_qm6쁪 uI _ \ɡĔɢ׽ &2Gҋ[ǵls#.g/V)Xֻ+,aNFh>/UNs|rKto$shϽew8iI<ʚP[lSQ'8CͅLearnBayes/data/puffin.txt.gz0000644000176200001440000000051612341620471015661 0ustar liggesusers5QKjC1 \K1d,'%@H/#=w9l cRkpHgJ  bpWcp=,xC6(z lFԂpan;{V+Œ%ЏþCͳ 238Ď9.d[)-h1qhz/ wQk\n"{yȦ`N"xͳ8sw!oLearnBayes/data/donner.txt.gz0000644000176200001440000000022712341620471015656 0ustar liggesusers]90 k50) { log.f=sum(y*log(fit$fitted)+(1-y)*log(1-fit$fitted)) log.g=dmnorm(beta.s,beta0,solve(BI),log=TRUE) log.marg=log.f+log.g-log(post.ord/m) } return(list(beta=Mb,log.marg=log.marg)) } LearnBayes/R/blinreg.R0000644000176200001440000000160411054056652014252 0ustar liggesusersblinreg=function (y, X, m, prior=NULL) { if(length(prior)>0) { c0=prior$c0; beta0=matrix(prior$b0,c(1,length(prior$b)))} fit = lm(y ~ 0 + X) bhat = matrix(fit$coef, c(1, fit$rank)) s2 = sum(fit$residuals^2)/fit$df.residual if(length(prior)==0) { shape = fit$df.residual/2 rate = fit$df.residual/2 * s2 beta.m = bhat vbeta = vcov(fit)/s2 } else { shape = length(y)/2 rate = fit$df.residual/2 * s2 + (beta0 - bhat) %*% t(X) %*% X %*% t(beta0 - bhat)/2/(c0+1) beta.m = c0/(c0+1)*(beta0/c0 + bhat) vbeta = vcov(fit)/s2*c0/(c0+1) } sigma = sqrt(1/rgamma(m, shape = shape, rate = rate)) beta = rmnorm(m, mean=rep(0, fit$rank), varcov=vbeta) beta = array(1, c(m, 1)) %*% beta.m + array(sigma, c(m, fit$rank))*beta return(list(beta = beta, sigma = sigma)) } LearnBayes/R/histprior.R0000644000176200001440000000030110537542316014646 0ustar liggesusershistprior=function(p,midpts,prob) { binwidth=midpts[2]-midpts[1] lo=round(10000*(midpts-binwidth/2))/10000 val=0*p for (i in 1:length(p)) { val[i]=prob[sum(p[i]>=lo)] } return(val) }LearnBayes/R/indepmetrop.R0000644000176200001440000000151311411451754015154 0ustar liggesusersindepmetrop=function (logpost, proposal, start, m, ...) { logmultinorm = function(x, m, v) { return(-0.5 * t(x - m) %*% solve(v) %*% (x - m)) } pb = length(start) Mpar = array(0, c(m, pb)) mu = matrix(proposal$mu) if(diff(dim(mu))>0) mu=t(mu) v = proposal$var a = chol(v) f0 = logpost(start, ...) th0 = matrix(t(start)) accept = 0 for (i in 1:m) { th1 = mu + t(a) %*% array(rnorm(pb), c(pb, 1)) f1 = logpost(t(th1), ...) R = exp(logmultinorm(th0, mu, v) - logmultinorm(th1, mu, v) + f1 - f0) u = runif(1) < R if (u == 1) { th0 = th1 f0 = f1 } Mpar[i, ] = th0 accept = accept + u } accept = accept/m return(list(par = Mpar, accept = accept)) } LearnBayes/R/mycontour.R0000644000176200001440000000126211052362064014662 0ustar liggesusersmycontour=function (logf, limits, data, ...) { LOGF=function(theta, data) { if(is.matrix(theta)==TRUE){ val=matrix(0,c(dim(theta)[1],1)) for (j in 1:dim(theta)[1]) val[j]=logf(theta[j,],data) } else val=logf(theta,data) return(val) } ng = 50 x0 = seq(limits[1], limits[2], len = ng) y0 = seq(limits[3], limits[4], len = ng) X = outer(x0, rep(1, ng)) Y = outer(rep(1, ng), y0) n2 = ng^2 Z = LOGF(cbind(X[1:n2], Y[1:n2]), data) Z = Z - max(Z) Z = matrix(Z, c(ng, ng)) contour(x0, y0, Z, levels = seq(-6.9, 0, by = 2.3), lwd = 2, ...) } LearnBayes/R/dmt.R0000644000176200001440000000120610554302212013400 0ustar liggesusersdmt=function (x, mean = rep(0, d), S, df = Inf, log = FALSE) { if (df == Inf) return(dmnorm(x, mean, S, log = log)) d <- if (is.matrix(S)) ncol(S) else 1 if (d > 1 & is.vector(x)) x <- matrix(x, 1, d) n <- if (d == 1) length(x) else nrow(x) X <- t(matrix(x, nrow = n, ncol = d)) - mean Q <- apply((solve(S) %*% X) * X, 2, sum) logDet <- sum(logb(abs(diag(qr(S)$qr)))) logPDF <- (lgamma((df + d)/2) - 0.5 * (d * logb(pi * df) + logDet) - lgamma(df/2) - 0.5 * (df + d) * logb(1 + Q/df)) if (log) logPDF else exp(logPDF) } LearnBayes/R/bradley.terry.post.R0000644000176200001440000000045711112644224016400 0ustar liggesusersbradley.terry.post=function(theta,data) { N=dim(data)[1]; M=length(theta) sigma=exp(theta[M]) logf=function(k) { i=data[k,1]; j=data[k,2] p=exp(theta[i]-theta[j])/(1+exp(theta[i]-theta[j])) data[k,3]*log(p)+data[k,4]*log(1-p) } sum(sapply(1:N,logf))+sum(dnorm(theta[-M],0,sigma,log=TRUE)) } LearnBayes/R/pdiscp.R0000644000176200001440000000125710506047142014111 0ustar liggesusers"pdiscp" <- function(p,probs,n,s) { # # PDISCP Predictive distribution of number of successes in future binomial # experiment with a discrete prior. PRED = PDISCP(P,PROBS,N,S) returns # vector PRED of predictive probabilities, where P is the vector of # values of the proportion, PROBS is the corresponding vector of # probabilities, N is the future binomial sample size, and S is the vector of # numbers of successes for which predictive probabilities will be computed. #------------------------ # Written by Jim Albert # albert@bgnet.bgsu.edu # November 2004 #------------------------ pred=0*s; for (i in 1:length(p)) { pred=pred+probs[i]*dbinom(s,n,p[i]); } return(pred) } LearnBayes/R/cauchyerrorpost.R0000644000176200001440000000024610706716154016070 0ustar liggesuserscauchyerrorpost=function(theta, data) { logf=function(data,theta) log(dt((data-theta[1])/exp(theta[2]),df=1)/exp(theta[2])) return(sum(logf(data,theta))) } LearnBayes/R/betabinexch0.R0000644000176200001440000000055010706716076015171 0ustar liggesusersbetabinexch0=function (theta, data) { eta = theta[1] K = theta[2] y = data[, 1] n = data[, 2] N = length(y) logf=function(y,n,K,eta) lbeta(K * eta + y, K * (1 - eta) + n - y)-lbeta(K * eta, K * (1 - eta)) val=sum(logf(y,n,K,eta)) val = val - 2 * log(1 + K) - log(eta) - log(1 - eta) return(val) } LearnBayes/R/normnormexch.R0000644000176200001440000000033612324477272015356 0ustar liggesusersnormnormexch=function(theta,data){ y=data[,1] sigma2=data[,2] mu=theta[1] tau=exp(theta[2]) logf=function(mu,tau,y,sigma2) dnorm(y,mu,sqrt(sigma2+tau^2),log=TRUE) sum(logf(mu,tau,y,sigma2))+log(tau) }LearnBayes/R/bayesresiduals.R0000644000176200001440000000041510537417252015650 0ustar liggesusersbayesresiduals=function(lmfit,post,k) { ehat=lmfit$residuals h=hat(model.matrix(lmfit)) prob=0*ehat for (i in 1:length(prob)) { z1=(k-ehat[i]/post$sigma)/sqrt(h[i]) z2=(-k-ehat[i]/post$sigma)/sqrt(h[i]) prob[i]=mean(1-pnorm(z1)+pnorm(z2)) } return(prob) }LearnBayes/R/discrete.bayes.R0000644000176200001440000000052211311777614015536 0ustar liggesusersdiscrete.bayes= function (df, prior, y, ...) { param = as.numeric(names(prior)) lk=function(j) prod(df(y,param[j],...)) likelihood=sapply(1:length(param),lk) pred = sum(prior * likelihood) prob = prior * likelihood/pred obj = list(prob = prob, pred = pred) class(obj) <- "bayes" obj } LearnBayes/R/simcontour.R0000644000176200001440000000164210706716352015037 0ustar liggesuserssimcontour=function (logf, limits, data, m) { LOGF=function(theta, data) { if(is.matrix(theta)==TRUE){ val=matrix(0,c(dim(theta)[1],1)) for (j in 1:dim(theta)[1]) val[j]=logf(theta[j,],data) } else val=logf(theta,data) return(val) } ng = 50 x0 = seq(limits[1], limits[2], len = ng) y0 = seq(limits[3], limits[4], len = ng) X = outer(x0, rep(1, ng)) Y = outer(rep(1, ng), y0) n2 = ng^2 Z = LOGF(cbind(X[1:n2], Y[1:n2]), data) Z = Z - max(Z) Z = matrix(Z, c(ng, ng)) d = cbind(X[1:n2], Y[1:n2], Z[1:n2]) dx = diff(x0[1:2]) dy = diff(y0[1:2]) prob = d[, 3] prob = exp(prob) prob = prob/sum(prob) i = sample(2500, m, replace = TRUE, prob = prob) return(list(x = d[i, 1] + runif(m) * dx - dx/2, y = d[i, 2] + runif(m) * dy - dy/2)) } LearnBayes/R/ctable.R0000644000176200001440000000142210537537544014071 0ustar liggesusersctable=function(y,a) # # C_TABLE Bayes factor for testing independence in a contingency table. # BF=C_TABLE(Y,A) returns the Bayes factor BF against independence in a # 2-way contingency table using uniform priors, where Y is a matrix # containing the 2-way table, and A is a matrix of prior parameters #------------------------ # Written by Jim Albert # albert@bgnet.bgsu.edu # November 2004 #------------------------ { ldirich=function(a) { val=sum(lgamma(a))-lgamma(sum(a)) return(val) } ac=colSums(a); ar=rowSums(a) yc=colSums(y); yr=rowSums(y) d=dim(y); oc=1+0*yc; or=1+0*yr; I=d[1];J=d[2] lbf=ldirich(c(y)+c(a))+ldirich(ar-(J-1)*or)+ldirich(ac-(I-1)*oc)- ldirich(c(a))-ldirich(yr+ar-(J-1)*or)-ldirich(yc+ac-(I-1)*oc) bf=exp(lbf) return(bf) } LearnBayes/R/bprobit.probs.R0000644000176200001440000000064610537403452015421 0ustar liggesusersbprobit.probs=function(X1,fit) { # bprobit.probs Produces a simulated sample from the posterior # distribution of an expected response for a linear regression model # X1 = design matrix of interest # fit = output of bayes.probit function d=dim(X1) n1=d[1] md=dim(fit); m=md[1] m1=array(0,c(m,n1)) for (j in 1:n1) { m1[,j]=pnorm(X1[j,]%*%t(fit)) } return(m1) } LearnBayes/R/ordergibbs.R0000644000176200001440000000402610537773050014756 0ustar liggesusersordergibbs=function(data,m) { # implements Gibbs sampling for table of means # with prior belief in order restriction # input: data = data matrix with two columns [sample mean, sample size] # m = number of iterations of Gibbs sampling # output: matrix of simulated values of means where each row # represents one simulated draw ##################################################### rnormt=function(n,mu,sigma,lo,hi) { # simulates n random variates from a normal(mu,sigma) # distribution truncated on the interval (lo, hi) p=pnorm(c(lo,hi),mu,sigma) return(mu+sigma*qnorm(runif(n)*(p[2]-p[1])+p[1])) } ##################################################### y=data[,1] # sample means n=data[,2] # sample sizes s=.65 # assumed value of sigma for this example I=8; J=5 # number of rows and columns in matrix # placing vectors y, n into matrices y=t(array(y,c(J,I))) n=t(array(n,c(J,I))) y=y[seq(8,1,by=-1),] n=n[seq(8,1,by=-1),] # setting up the matrix of values of the population means mu # two rows and two columns are added that help in the simulation # of individual values of mu from truncated normal distributions mu0=Inf*array(1,c(I+2,J+2)) mu0[1,]=-mu0[1,] mu0[,1]=-mu0[,1] mu0[1,1]=-mu0[1,1] mu=mu0 # starting value of mu that satisfies order restriction m1=c(2.64,3.02,3.02,3.07,3.34) m2=c(2.37,2.63,2.74,2.76,2.91) m3=c(2.37,2.47,2.64,2.66,2.66) m4=c(2.31,2.33,2.33,2.33,2.33) m5=c(2.04,2.11,2.11,2.33,2.33) m6=c(1.85,1.85,1.85,2.10,2.10) m7=c(1.85,1.85,1.85,1.88,1.88) m8=c(1.59,1.59,1.59,1.67,1.88) muint=rbind(m8,m7,m6,m5,m4,m3,m2,m1) mu[2:(I+1),2:(J+1)]=muint MU=array(0,c(m,I*J)) # arry MU stores simulated values of mu ##################### main loop ####################### for (k in 1:m) { for (i in 2:(I+1)) { for (j in 2:(J+1)) { lo=max(c(mu[i-1,j],mu[i,j-1])) hi=min(c(mu[i+1,j],mu[i,j+1])) mu[i,j]=rnormt(1,y[i-1,j-1],s/sqrt(n[i-1,j-1]),lo,hi) } } mm=mu[2:(I+1),2:(J+1)] MU[k,]=array(mm,c(1,I*J)) } return(MU) } LearnBayes/R/transplantpost.R0000644000176200001440000000142510706716370015730 0ustar liggesuserstransplantpost=function (theta, data) { x = data[, 1] y = data[, 3] t = data[, 2] d = data[, 4] tau = exp(theta[1]) lambda = exp(theta[2]) p = exp(theta[3]) xnt = x[t == 0] dnt = d[t == 0] z = x[t == 1] y = y[t == 1] dt = d[t == 1] logf=function(xnt,dnt,lambda,p) (dnt==0)*(p*log(lambda)+log(p)- (p + 1) * log(lambda + xnt)) + (dnt==1)*p*log(lambda/(lambda + xnt)) logg=function(z,y,tau,lambda,p) (dt==0)*(p * log(lambda) + log(p * tau)-(p + 1) * log(lambda + y + tau * z)) + (dt==1) * p * log(lambda/(lambda + y + tau * z)) val=sum(logf(xnt,dnt,lambda,p))+sum(logg(z,y,tau,lambda,p)) val = val + theta[1] + theta[2] + theta[3] return(val) } LearnBayes/R/rtruncated.R0000644000176200001440000000013310735453532015002 0ustar liggesusersrtruncated=function(n,lo,hi,pf,qf,...) qf(pf(lo,...)+runif(n)*(pf(hi,...)-pf(lo,...)),...)LearnBayes/R/careertraj.setup.R0000644000176200001440000000077110556700166016117 0ustar liggesuserscareertraj.setup=function(data) { Player=data[,1] player.names=names(table(Player)) N=length(player.names) m=max(table(Player)) y=array(0,c(N,m)) n=0*y x=0*y T=rep(0,N) for (i in 1:N) { data1=data[Player==player.names[i],] nk=dim(data1) ni=nk[1] for (j in 1:ni) { y[i,j]=data1[j,10] n[i,j]=data1[j,5]-data1[j,13] x[i,j]=data1[j,3] T[i]=T[i]+(n[i,j]>0) } } return(list(player.names=player.names,y=y,n=n,x=x,T=T,N=N)) }LearnBayes/R/robustt.R0000644000176200001440000000111510554303760014326 0ustar liggesusersrobustt=function(y,v,m) { rigamma=function(n,a,b) { # simulates n values from a Inverse Gamma # distribution with shape a and rate b # density x^(-a-1) exp(b/x) return(1/rgamma(n,shape=a,rate=b)) } n=length(y) mu=mean(y); sig2=sd(y)^2; lam=array(1,c(n,1)) M=array(0,c(m,1)); S2=M; LAM=array(0,c(m,n)) for (i in 1:m) { lam=rgamma(n,shape=(v+1)/2,rate=v/2+(y-mu)^2/2/sig2) mu=rnorm(1,mean=sum(y*lam)/sum(lam),sd=sqrt(sig2/sum(lam))) sig2=rigamma(1,n/2,sum(lam*(y-mu)^2)/2) M[i]=mu; S2[i]=sig2; LAM[i,]=lam } par=list(mu=M,s2=S2,lam=LAM) return(par) } LearnBayes/R/sir.R0000644000176200001440000000106310706716362013430 0ustar liggesuserssir=function (logf, tpar, n, data) { k = length(tpar$m) theta = rmt(n, mean = c(tpar$m), S = tpar$var, df = tpar$df) lf=matrix(0,c(dim(theta)[1],1)) for (j in 1:dim(theta)[1]) lf[j]=logf(theta[j,],data) lp = dmt(theta, mean = c(tpar$m), S = tpar$var, df = tpar$df, log = TRUE) md = max(lf - lp) wt = exp(lf - lp - md) probs = wt/sum(wt) indices = sample(1:n, size = n, prob = probs, replace = TRUE) if (k > 1) theta = theta[indices, ] else theta = theta[indices] return(theta) } LearnBayes/R/weibullregpost.R0000644000176200001440000000067210706716376015714 0ustar liggesusersweibullregpost=function (theta, data) { logf=function(t,c,x,sigma,mu,beta) { z=(log(t)-mu-x%*%beta)/sigma f=1/sigma*exp(z-exp(z)) S=exp(-exp(z)) c*log(f)+(1-c)*log(S) } k = dim(data)[2] p = k - 2 t = data[, 1] c = data[, 2] X = data[, 3:k] sigma = exp(theta[1]) mu = theta[2] beta = array(theta[3:k], c(p,1)) return(sum(logf(t,c,X,sigma,mu,beta))) } LearnBayes/R/blinregpred.R0000644000176200001440000000071510537536114015130 0ustar liggesusersblinregpred=function(X1,theta.sample) { #blinregpred Produces a simulated sample from the posterior predictive # distribution of a linear regression model # X1 = design matrix of interest # theta.sample = output of blinreg function d=dim(X1) n1=d[1] m=length(theta.sample$sigma) y1=array(0,c(m,n1)) for (j in 1:n1) { y1[,j]=t(X1[j,]%*%t(theta.sample$beta))+rnorm(m)*theta.sample$sigma } return(y1) } LearnBayes/R/rmnorm.R0000644000176200001440000000033310554300052014126 0ustar liggesusersrmnorm=function(n = 1, mean = rep(0, d), varcov) { d <- if (is.matrix(varcov)) ncol(varcov) else 1 z <- matrix(rnorm(n * d), n, d) %*% chol(varcov) y <- t(mean + t(z)) return(y) } LearnBayes/R/bayes.model.selection.R0000644000176200001440000000331511054044500017004 0ustar liggesusersbayes.model.selection=function (y, X, c, constant = TRUE) { base2 = function(s, k) { r = rep(0, k) for (j in seq(k, 1, by = -1)) { r[j] = floor(s/(2^(j - 1))) s = s - r[j] * (2^(j - 1)) } return(r) } regpost.mod = function(theta, stuff) { y = stuff$y X = stuff$X c = stuff$c beta = theta[-length(theta)] sigma = exp(theta[length(theta)]) if (length(beta) > 1) loglike = sum(dnorm(y, mean = X %*% as.vector(beta), sd = sigma, log = TRUE)) else loglike = sum(dnorm(y, mean = X * beta, sd = sigma, log = TRUE)) logprior = dmnorm(beta, mean = 0 * beta, varcov = c * sigma^2 * solve(t(X) %*% X), log = TRUE) return(loglike + logprior) } require(LearnBayes) X = as.matrix(X) if (constant == FALSE) X = cbind(1, X) p = dim(X)[2] - 1 GAM = array(TRUE, c(2^p, p + 1)) for (k in 1:(2^p)) GAM[k, ] = as.logical(c(1, base2(k - 1, p))) gof = rep(0, 2^p) converge = rep(TRUE, 2^p) for (j in 1:2^p) { X0 = X[, GAM[j, ]] fit = lm(y ~ 0 + X0) beta = fit$coef s = sqrt(sum(fit$residuals^2)/fit$df.residual) theta = c(beta, log(s)) S = list(X = X0, y = y, c = c) fit = laplace(regpost.mod, theta, S) gof[j] = fit$int converge[j] = fit$converge } Prob=exp(gof-max(gof))/sum(exp(gof-max(gof))) mod.prob=data.frame(GAM[, -1], round(gof,2), round(Prob,5)) names(mod.prob)=c(dimnames(X)[[2]][-1],"log.m","Prob") return(list(mod.prob=mod.prob, converge = converge)) } LearnBayes/R/normpostsim.R0000644000176200001440000000144011317415102015207 0ustar liggesusersnormpostsim=function (data, prior=NULL, m = 1000) { if (length(prior)==0) { S = sum((data - mean(data))^2) xbar = mean(data) n = length(data) SIGMA2 = S/rchisq(m, n - 1) MU = rnorm(m, mean = xbar, sd = sqrt(SIGMA2)/sqrt(n)) } else { a=prior$sigma2[1] b=prior$sigma2[2] mu0=prior$mu[1] tau2=prior$mu[2] S = sum((data - mean(data))^2) xbar = mean(data) n = length(data) SIGMA2=rep(0,m) MU=rep(0,m) sigma2=S/n for (j in 1:m) { prec=n/sigma2+1/tau2 mu1=(xbar*n/sigma2+mu0/tau2)/prec v1=1/prec mu=rnorm(1,mu1,sqrt(v1)) a1=a+n/2 b1=b+sum((data-mu)^2)/2 sigma2=rigamma(1,a1,b1) SIGMA2[j]=sigma2 MU[j]=mu } } return(list(mu = MU, sigma2 = SIGMA2)) } LearnBayes/R/hiergibbs.R0000644000176200001440000000373510537773172014605 0ustar liggesusershiergibbs=function(data,m) { ############################################################### # Implements Gibbs sampling algorithm for posterior of table # of means with hierarchical regression prior # # INPUT # data: 40 by 4 matrix where the observed sample means are # in column 1, sample sizes are in column 2, and values of # two covariates in columns 3 and 4. # m: number of cycles of Gibbs sampling # # OUTPUT # a list with # -- beta: matrix of simulated values of beta with each row a simulated value # -- mu: matrix of simulated values of cell means # -- var: vector of simulated values of second-stage variance sigma^2_pi ############################################################### y=data[,1] # n=data[,2] # x1=data[,3] # x2=data[,4] # defines variables y,n,x1,x2,a X=cbind(1+0*x1,x1,x2) # s2=.65^2/n # p=3; N=length(y) # mbeta=array(0,c(m,p)) # mmu=array(0,c(m,length(n))) # sets up arrays to store simulated draws ms2pi=array(0,c(m,1)) # ######################################## defines prior parameters b1=array(c(.55,.018,.033),c(3,1)) bvar=array(c(8.49e-03,-1.94e-05, -2.88e-04, -1.94e-05, 7.34e-07, -1.52e-06, -2.88e-04,-1.52e-06, 1.71e-05),c(3,3)) ibvar=solve(bvar) s=.02; v=16; mu=y; s2pi=.006 # starting values of mu and s2pi in Gibbs sampling for (j in 1:m) { pvar=solve(ibvar+t(X)%*%X/s2pi) # pmean=pvar%*%(ibvar%*%b1+t(X)%*%mu/s2pi) # simulates beta beta=t(chol(pvar))%*%array(rnorm(p),c(p,1))+pmean # s2pi=(sum((mu-X%*%beta)^2)/2+s/2)/rgamma(1,shape=(N+v)/2) # simulates s2pi postvar=1/(1/s2+1/s2pi) # postmean=(y/s2+X%*%beta/s2pi)*postvar # simulates mu mu=rnorm(n,postmean,sqrt(postvar)) # mbeta[j,]=t(beta) # mmu[j,]=t(mu) # stores simulated draws ms2pi[j]=s2pi # } return(list(beta=mbeta,mu=mmu,var=ms2pi)) } LearnBayes/R/normal.select.R0000644000176200001440000000037711063057620015400 0ustar liggesusersnormal.select=function (quantile1, quantile2) { p1 = quantile1$p x1 = quantile1$x p2 = quantile2$p x2 = quantile2$x sigma=(x1-x2)/diff(qnorm(c(p2,p1))) mu=x1-sigma*qnorm(p1) return(list(mu=mu,sigma=sigma)) } LearnBayes/R/mnormt.onesided.R0000644000176200001440000000144710537550542015744 0ustar liggesusersmnormt.onesided=function(m0,normpar,data) { # # mnormt.onesided Performs a test that a normal mean is <= certain value. # m0 = value to be tested # normpar = mean and standard deviation of normal prior on mu # data = (sample mean, sample size, known sampling standard deviation) xbar=data[1]; n=data[2]; s=data[3] prior.mean=normpar[1] prior.sd=normpar[2] prior.var=prior.sd^2 priorH=pnorm(m0,prior.mean,prior.sd) priorA=1-priorH prior.odds=priorH/priorA post.precision=1/prior.var+n/s^2 post.var=1/post.precision post.sd=sqrt(post.var) post.mean=(xbar*n/s^2+prior.mean/prior.var)/post.precision postH=pnorm(m0,post.mean,post.sd) postA=1-postH post.odds=postH/postA BF=post.odds/prior.odds return(list(BF=BF,prior.odds=prior.odds,post.odds=post.odds,postH=postH)) } LearnBayes/R/plot.bayes.R0000644000176200001440000000006111311715564014704 0ustar liggesusersplot.bayes=function(x,...) barplot(x$prob,...)LearnBayes/R/logisticpost.R0000644000176200001440000000053310706716306015355 0ustar liggesuserslogisticpost=function (beta, data) { x = data[, 1] n = data[, 2] y = data[, 3] beta0 = beta[1] beta1 = beta[2] logf=function(x,n,y,beta0,beta1) { lp = beta0 + beta1 * x p = exp(lp)/(1 + exp(lp)) y * log(p) + (n - y) * log(1 - p) } return(sum(logf(x,n,y,beta0,beta1))) } LearnBayes/R/gibbs.R0000644000176200001440000000073311453352334013717 0ustar liggesusersgibbs=function(logpost,start,m,scale,...) { p=length(start) vth=array(0,dim=c(m,p)) f0=logpost(start,...) arate=array(0,dim=c(1,p)) th0=start for (i in 1:m) { for (j in 1:p) { th1=th0 th1[j]=th0[j]+rnorm(1)*scale[j] f1=logpost(th1,...) u=runif(1)=prob]; j=j[1] eprob=cp[j]; set=sort(xs[1:j]) v=list(prob=eprob,set=set) return(v) } LearnBayes/R/poissgamexch.R0000644000176200001440000000066510710103714015316 0ustar liggesusers poissgamexch=function (theta, datapar) { y = datapar$data[, 2] e = datapar$data[, 1] z0 = datapar$z0 alpha = exp(theta[1]) mu = exp(theta[2]) beta = alpha/mu logf=function(y,e,alpha,beta) lgamma(alpha + y) - (y + alpha) * log(e + beta) + alpha * log(beta)-lgamma(alpha) val=sum(logf(y,e,alpha,beta)) val = val + log(alpha) - 2 * log(alpha + z0) return(val) }LearnBayes/R/normal.normal.mix.R0000644000176200001440000000066310732110664016203 0ustar liggesusersnormal.normal.mix=function(probs,normalpar,data) { N=length(probs) y=data[1]; sigma2=data[2] prior.mean=normalpar[,1] prior.var=normalpar[,2] post.precision=1/prior.var+1/sigma2 post.var=1/post.precision post.mean=(y/sigma2+prior.mean/prior.var)/post.precision m.prob=dnorm(y,prior.mean,sqrt(sigma2+prior.var)) post.probs=probs*m.prob/sum(probs*m.prob) return(list(probs=post.probs,normalpar=cbind(post.mean,post.var))) }LearnBayes/R/laplace.R0000644000176200001440000000066511411450002014217 0ustar liggesuserslaplace=function (logpost, mode, ...) { options(warn=-1) fit=optim(mode, logpost, gr = NULL, ..., hessian=TRUE, control=list(fnscale=-1)) options(warn=0) mode=fit$par h=-solve(fit$hessian) p=length(mode) int = p/2 * log(2 * pi) + 0.5 * log(det(h)) + logpost(mode, ...) stuff = list(mode = mode, var = h, int = int, converge=fit$convergence==0) return(stuff) }LearnBayes/R/prior.two.parameters.R0000644000176200001440000000035111311763630016730 0ustar liggesusersprior.two.parameters = function(parameter1, parameter2) { prior = matrix(1, length(parameter1), length(parameter2)) prior = prior/sum(prior) dimnames(prior)[[1]] = parameter1 dimnames(prior)[[2]] = parameter2 prior }LearnBayes/R/rwmetrop.R0000644000176200001440000000127211411451462014503 0ustar liggesusersrwmetrop=function (logpost, proposal, start, m, ...) { pb = length(start) Mpar = array(0, c(m, pb)) b = matrix(t(start)) lb = logpost(start, ...) a = chol(proposal$var) scale = proposal$scale accept = 0 for (i in 1:m) { bc = b + scale * t(a) %*% array(rnorm(pb), c(pb, 1)) lbc = logpost(t(bc), ...) prob = exp(lbc - lb) if (is.na(prob) == FALSE) { if (runif(1) < prob) { lb = lbc b = bc accept = accept + 1 } } Mpar[i, ] = b } accept = accept/m stuff = list(par = Mpar, accept = accept) return(stuff) } LearnBayes/R/reg.gprior.post.R0000644000176200001440000000056011054061512015661 0ustar liggesusersreg.gprior.post=function(theta,dataprior) { y=dataprior$data$y; X=dataprior$data$X c0=dataprior$prior$c0; beta0=dataprior$prior$b0 beta=theta[-length(theta)]; sigma=exp(theta[length(theta)]) loglike=sum(dnorm(y,mean=X%*%as.vector(beta),sd=sigma,log=TRUE)) logprior=dmnorm(beta,mean=beta0,varcov=c0*sigma^2*solve(t(X)%*%X),log=TRUE) return(loglike+logprior) }LearnBayes/R/pbetap.R0000644000176200001440000000123310156424012014070 0ustar liggesuserspbetap=function(ab,n,s) { # # PBETAP Predictive distribution of number of successes in future binomial # experiment with a beta prior. PRED = PBETAP(AB,N,S) returns a vector # PRED of predictive probabilities, where AB is the vector of beta # parameters, N is the future binomial sample size, and S is the vector of # numbers of successes for which predictive probabilities will be computed. #------------------------ # Written by Jim Albert # albert@bgnet.bgsu.edu # November 2004 #------------------------ pred=0*s; a=ab[1]; b=ab[2]; lcon=lgamma(n+1)-lgamma(s+1)-lgamma(n-s+1); pred=exp(lcon+lbeta(s+a,n-s+b)-lbeta(a,b)); return(pred) } LearnBayes/R/logctablepost.R0000644000176200001440000000060010706716300015461 0ustar liggesuserslogctablepost=function (theta, data) { theta1 = theta[1] theta2 = theta[2] s1 = data[1] f1 = data[2] s2 = data[3] f2 = data[4] logitp1 = (theta1 + theta2)/2 logitp2 = (theta2 - theta1)/2 term1 = s1 * logitp1 - (s1 + f1) * log(1 + exp(logitp1)) term2 = s2 * logitp2 - (s2 + f2) * log(1 + exp(logitp2)) return(term1 + term2) } LearnBayes/R/discrete.bayes.2.R0000644000176200001440000000117411311757622015677 0ustar liggesusersdiscrete.bayes.2=function(df,prior,y=NULL,...) { like=function(i,...) if(is.matrix(y)==TRUE) df(y[i,],param1,param2,...) else df(y[i],param1,param2,...) n.rows=dim(prior)[1] n.cols=dim(prior)[2] param1=as.numeric(dimnames(prior)[[1]]) param2=as.numeric(dimnames(prior)[[2]]) param1=outer(param1,rep(1,n.cols)) param2=outer(rep(1,n.rows),param2) likelihood=1 if(length(y)>0) { n=ifelse(is.matrix(y)==FALSE,length(y),dim(y)[1]) for(j in 1:n) likelihood=likelihood*like(j,...) } product=prior*likelihood pred=sum(prior*likelihood) prob=prior*likelihood/pred obj=list(prob=prob,pred=pred) class(obj)<-"bayes2" obj }LearnBayes/R/bfexch.R0000644000176200001440000000053410710253300014053 0ustar liggesusersbfexch=function (theta, datapar) { y = datapar$data[, 1] n = datapar$data[, 2] K = datapar$K eta = exp(theta)/(1 + exp(theta)) logf=function(K,eta,y,n) lbeta(K*eta+y, K*(1-eta)+n-y)-lbeta(K*eta, K*(1-eta)) sum(logf(K,eta,y,n)) + log(eta * (1 - eta))- lbeta(sum(y) + 1, sum(n - y) + 1) } LearnBayes/R/print.bayes.R0000644000176200001440000000004511311722366015062 0ustar liggesusersprint.bayes=function(x,...) x$probLearnBayes/R/lbinorm.R0000644000176200001440000000041510706716272014275 0ustar liggesusers lbinorm=function (xy, par) { m = par$m v = par$v x = xy[1] y = xy[2] zx = (x - m[1])/sqrt(v[1, 1]) zy = (y - m[2])/sqrt(v[2, 2]) r = v[1, 2]/sqrt(v[1, 1] * v[2, 2]) return(-0.5/(1 - r^2) * (zx^2 - 2 * r * zx * zy + zy^2)) } LearnBayes/R/rmt.R0000644000176200001440000000041710554301160013422 0ustar liggesusersrmt=function (n = 1, mean = rep(0, d), S, df = Inf) { d <- if (is.matrix(S)) ncol(S) else 1 if (df == Inf) x <- 1 else x <- rchisq(n, df)/df z <- rmnorm(n, rep(0, d), S) y <- t(mean + t(z/sqrt(x))) return(y) } LearnBayes/R/bayes.influence.R0000644000176200001440000000103610554311106015671 0ustar liggesusersbayes.influence=function(theta,data) { y=data[,1]; n=data[,2] N=length(y) summary=quantile(theta[,2],c(.05,.5,.95)) summary.obs=array(0,c(N,3)) K=exp(theta[,2]) eta=exp(theta[,1])/(1+exp(theta[,1])) m=length(K) for (i in 1:N) { weight=exp(lbeta(K*eta,K*(1-eta))-lbeta(K*eta+y[i],K*(1-eta)+n[i]-y[i])) probs=weight/sum(weight) indices=sample(1:m,size=m,prob=probs,replace=TRUE) theta.s=theta[indices,] summary.obs[i,]=quantile(theta.s[,2],c(.05,.5,.95)) } return(list(summary=summary,summary.obs=summary.obs)) } LearnBayes/R/groupeddatapost.R0000644000176200001440000000037210706716164016042 0ustar liggesusersgroupeddatapost=function (theta, data) { dj=function(f,int.lo,int.hi,mu,sigma) f*log(pnorm(int.hi,mu,sigma)-pnorm(int.lo,mu,sigma)) mu = theta[1] sigma = exp(theta[2]) sum(dj(data$f,data$int.lo,data$int.hi,mu,sigma)) } LearnBayes/R/logpoissgamma.R0000644000176200001440000000036710576544442015506 0ustar liggesuserslogpoissgamma=function(theta,datapar) { y=datapar$data npar=datapar$par lambda=exp(theta) loglike=log(dgamma(lambda,shape=sum(y)+1,rate=length(y))) logprior=log(dgamma(lambda,shape=npar[1],rate=npar[2])*lambda) return(loglike+logprior) } LearnBayes/R/poisson.gamma.mix.R0000644000176200001440000000102110735454006016170 0ustar liggesuserspoisson.gamma.mix=function(probs,gammapar,data) { N=length(probs) y=data$y; t=data$t; n=length(y) post.gammapar=gammapar+outer(rep(1,N),c(sum(y),sum(t))) L=post.gammapar[,1]/post.gammapar[,2] loglike=0 for (j in 1:n) loglike=loglike+dpois(y[j],L*t[j],log=TRUE) m.prob=exp(loglike+ dgamma(L,shape=gammapar[,1],rate=gammapar[,2],log=TRUE) - dgamma(L,shape=post.gammapar[,1],rate=post.gammapar[,2],log=TRUE)) post.probs=probs*m.prob/sum(probs*m.prob) return(list(probs=post.probs,gammapar=post.gammapar)) }LearnBayes/R/dmnorm.R0000644000176200001440000000101510616346554014126 0ustar liggesusersdmnorm=function (x, mean = rep(0, d), varcov, log = FALSE) { d <- if (is.matrix(varcov)) ncol(varcov) else 1 if (d > 1 & is.vector(x)) x <- matrix(x, 1, d) n <- if (d == 1) length(x) else nrow(x) X <- t(matrix(x, nrow = n, ncol = d)) - mean Q <- apply((solve(varcov) %*% X) * X, 2, sum) logDet <- sum(logb(abs(diag(qr(varcov)[[1]])))) logPDF <- as.vector(Q + d * logb(2 * pi) + logDet)/(-2) if (log) logPDF else exp(logPDF) } LearnBayes/R/mnormt.twosided.R0000644000176200001440000000055510601630322015756 0ustar liggesusersmnormt.twosided <- function (m0, prob, t, data) { xbar = data[1] n = data[2] h = data[3] num = 0.5 * log(n) - log(h) - 0.5 * n/h^2 * (xbar - m0)^2 den = -0.5 * log(h^2/n + t^2) - 0.5/(h^2/n + t^2) * (xbar - m0)^2 bf = exp(num - den) post = prob * bf/(prob * bf + 1 - prob) return(list(bf = bf, post = post)) } LearnBayes/R/bfindep.R0000644000176200001440000000202510537532462014240 0ustar liggesusersbfindep=function(y,K,m) { # compute Bayes factor against independence # using Albert and Gupta independence priors # ymat - I x J matrix # K - Dirichlet precision parameter # m - number of iterations rdirichlet=function (n, alpha) { l <- length(alpha) x <- matrix(rgamma(l * n, alpha), ncol = l, byrow = TRUE) sm <- x %*% rep(1, l) return(x/as.vector(sm)) } ldirichlet=function(alpha) { # log dirichlet function # for multiple values stored in matrix alpha return(rowSums(lgamma(alpha))-lgamma(rowSums(alpha))) } yc=colSums(y); yr=rowSums(y); n=sum(yc) d=dim(y); I=d[1]; J=d[2] etaA=rdirichlet(m,yr+1) etaB=rdirichlet(m,yc+1) Keta=c(); KetaY=c() for (i in 1:I) { for (j in 1:J) { Keta=cbind(Keta,K*etaA[,i]*etaB[,j]) KetaY=cbind(KetaY,K*etaA[,i]*etaB[,j]+y[i,j]) }} logint=ldirichlet(KetaY)-ldirichlet(Keta) for (i in 1:I) logint=logint-yr[i]*log(etaA[,i]) for (j in 1:J) logint=logint-yc[j]*log(etaB[,j]) int=exp(logint) return(list(bf=mean(int),nse=sd(int)/sqrt(m))) } LearnBayes/R/normpostpred.R0000644000176200001440000000052510735454626015375 0ustar liggesusersnormpostpred=function(parameters,sample.size,f=min) { normalsample=function(j,parameters,sample.size) rnorm(sample.size,mean=parameters$mu[j],sd=sqrt(parameters$sigma2[j])) m=length(parameters$mu) post.pred.samples=sapply(1:m,normalsample,parameters,sample.size) stat=apply(post.pred.samples,2,f) return(stat) }LearnBayes/R/pdisc.R0000644000176200001440000000133210506047142013723 0ustar liggesusers"pdisc" <- function(p,prior,data) { # PDISC Posterior distribution for a proportion with discrete models. # POST = PDISC(P,PRIOR,DATA) returns a vector of posterior probabilities. # P is the vector of values of the proportion, PRIOR is the corresponding # vector of prior probabilities and DATA is the vector of data (number of # successes and failures in set of independent Bernoulli trials #------------------------ # Written by Jim Albert # albert@bgnet.bgsu.edu # November 2004 #------------------------ s=data[1]; f=data[2] p1=p+.5*(p==0)-.5*(p==1) like=s*log(p1)+f*log(1-p1) like=like*(p>0)*(p<1)-999*((p==0)*(s>0)+(p==1)*(f>0)) like=exp(like-max(like)) product=like*prior post=product/sum(product) return(post) } LearnBayes/R/rejectsampling.R0000644000176200001440000000104110706734160015632 0ustar liggesusersrejectsampling=function (logf, tpar, dmax, n, data) { d = length(tpar$m) theta = rmt(n, mean = c(tpar$m), S = tpar$var, df = tpar$df) lf = matrix(0, c(dim(theta)[1], 1)) for (j in 1:dim(theta)[1]) lf[j] = logf(theta[j, ], data) lg = dmt(theta, mean = c(tpar$m), S = tpar$var, df = tpar$df, log = TRUE) if (d == 1) { prob = exp(c(lf) - lg - dmax) return(theta[runif(n) < prob]) } else { prob = exp(lf - lg - dmax) return(theta[runif(n) < prob, ]) } } LearnBayes/R/plot.bayes2.R0000644000176200001440000000040411311761100014752 0ustar liggesusersplot.bayes2=function(x,marginal=0,...) if(marginal==0)image(as.numeric(dimnames(x$prob)[[1]]), as.numeric(dimnames(x$prob)[[2]]),x$prob, col=gray(1-(0:32)/32),...) else if(marginal==1) barplot(apply(x$prob,1,sum),...) else barplot(apply(x$prob,2,sum),...)LearnBayes/R/summary.bayes.R0000644000176200001440000000071611311725252015425 0ustar liggesuserssummary.bayes=function(object,coverage=.9,...) { x = as.numeric(names(object$prob)) p = object$prob post.mean=sum(x*p) post.sd=sqrt(sum((x-post.mean)^2*p)) names(p)=NULL n = length(x) sp = sort(p, index.return = TRUE) ps = sp$x i = sp$ix[seq(n, 1, -1)] ps = p[i] xs = x[i] cp = cumsum(ps) ii = 1:n j = ii[cp >= coverage] j = j[1] eprob = cp[j] set = sort(xs[1:j]) v = list(mean=post.mean,sd=post.sd,coverage = eprob, set = set) return(v) } LearnBayes/R/rigamma.R0000644000176200001440000000013010707363334014240 0ustar liggesusersrigamma = function(n, a, b) { return(1/rgamma(n, shape = a, rate = b)) } LearnBayes/R/pbetat.R0000644000176200001440000000141510540516266014110 0ustar liggesuserspbetat=function(p0,prob,ab,data) { # # PBETAT Performs a test that a proportion is equal to a specific value. # PBETAT(P0,PROB,AB,DATA) gives a vector of the Bayes factor and # the probability of the hypothesis P=P0, where P0 is the proportion # value to be tested, PROB is the prior probability of the hypothesis, # AB is the vector of parameters of the beta density under the # alternative hypothesis, and DATA is the vector of numbers of # successes and failures. #------------------------ # Written by Jim Albert # albert@bgnet.bgsu.edu # November 2004 #------------------------ a=ab[1]; b=ab[2] s=data[1]; f=data[2] lbf=s*log(p0)+f*log(1-p0)+lbeta(a,b)-lbeta(a+s,b+f) bf=exp(lbf) post=prob*bf/(prob*bf+1-prob) return(list(bf=bf,post=post)) } LearnBayes/R/blinregexpected.R0000644000176200001440000000070410537535376016006 0ustar liggesusersblinregexpected=function(X1,theta.sample) { #blinregpred Produces a simulated sample from the posterior # distribution of an expected response for a linear regression model # X1 = design matrix of interest # theta.sample = output of blinreg function d=dim(X1) n1=d[1] m=length(theta.sample$sigma) m1=array(0,c(m,n1)) for (j in 1:n1) { m1[,j]=t(X1[j,]%*%t(theta.sample$beta)) } return(m1) } LearnBayes/R/triplot.R0000644000176200001440000000110011130237650014306 0ustar liggesuserstriplot=function(prior,data,where="topright") { a=prior[1]; b=prior[2] s=data[1]; f=data[2] p = seq(0.005, 0.995, length = 500) prior=dbeta(p,a,b) like=dbeta(p,s+1,f+1) post=dbeta(p,a+s, b+f) m=max(c(prior,like,post)) plot(p,post,type="l", ylab="Density", lty=2, lwd=3, main=paste("Bayes Triplot, beta(",a,",",b,") prior, s=",s,", f=",f), ylim=c(0,m),col="red") lines(p,like,lty=1, lwd=3,col="blue") lines(p,prior,lty=3, lwd=3,col="green") legend(where,c("Prior","Likelihood","Posterior"), lty=c(3,1,2), lwd=c(3,3,3), col=c("green","blue","red")) }LearnBayes/R/normchi2post.R0000644000176200001440000000034010706716326015257 0ustar liggesusersnormchi2post=function(theta,data) { mu = theta[1] sig2 = theta[2] logf=function(y,mu,sig2) -(y-mu)^2/2/sig2-log(sig2)/2 z=sum(logf(data,mu,sig2)) z = z - log(sig2) return(z) } LearnBayes/R/logpoissnormal.R0000644000176200001440000000035710537423222015700 0ustar liggesuserslogpoissnormal=function(theta,datapar) { y=datapar$data npar=datapar$par lambda=exp(theta) loglike=log(dgamma(lambda,shape=sum(y)+1,scale=1/length(y))) logprior=log(dnorm(theta,mean=npar[1],sd=npar[2])) return(loglike+logprior) } LearnBayes/R/beta.select.R0000644000176200001440000000125211050664616015021 0ustar liggesusersbeta.select=function(quantile1,quantile2) { betaprior1=function(K,x,p) # suppose one is given a beta(K*m, K*(1-m)) prior # where the pth quantile is given by x # function outputs the prior mean m { m.lo=0; m.hi=1; flag=0 while(flag==0) { m0=(m.lo+m.hi)/2 p0=pbeta(x,K*m0,K*(1-m0)) if(p00)&(prob2<1)) app=approx(prob2[ind],logK[ind],p2) K0=exp(app$y) m0=betaprior1(K0,x1,p1) return(round(K0*c(m0,(1-m0)),2)) }LearnBayes/R/betabinexch.R0000644000176200001440000000060310706716146015106 0ustar liggesusersbetabinexch=function (theta, data) { eta = exp(theta[1])/(1 + exp(theta[1])) K = exp(theta[2]) y = data[, 1] n = data[, 2] N = length(y) logf=function(y,n,K,eta) lbeta(K * eta + y, K * (1 - eta) + n - y)-lbeta(K * eta, K * (1 - eta)) val=sum(logf(y,n,K,eta)) val = val + theta[2] - 2 * log(1 + exp(theta[2])) return(val) } LearnBayes/R/predplot.R0000644000176200001440000000055610735451614014470 0ustar liggesuserspredplot=function(prior,n,yobs) { y=0:n; a=prior[1]; b=prior[2] probs=pbetap(prior,n,y) m=max(probs)*1.05 plot(y,probs,type="h",ylab="Probability",ylim=c(0,m), main=paste("Predictive Dist., beta(",a,",",b,") prior, n=",n, ", yobs=",yobs),lwd=2,col="blue") points(yobs,0,pch=19,cex=2.5,col="red") text(yobs,m/8,"yobs",col="red")}LearnBayes/R/regroup.R0000644000176200001440000000044010577472442014320 0ustar liggesusersregroup=function(data,g) { d=dim(data); n=d[1]; m=d[2] N=floor(n/g) dataG=array(0,c(N,m)) k=0 for (j in seq(1,(N-1)*g+1,g)) { k=k+1 for (i in 0:(g-1)) dataG[k,]=dataG[k,]+data[j+i,] } if (n>N*g) { for (i in (N*g+1):n) dataG[N,]=dataG[N,]+data[i,] } return(dataG) } LearnBayes/R/impsampling.R0000644000176200001440000000106710706716200015146 0ustar liggesusersimpsampling=function (logf, tpar, h, n, data) { theta = rmt(n, mean = c(tpar$m), S = tpar$var, df = tpar$df) lf=matrix(0,c(dim(theta)[1],1)) for (j in 1:dim(theta)[1]) lf[j]=logf(theta[j,],data) H=lf for (j in 1:dim(theta)[1]) H[j]=h(theta[j,]) lp = dmt(theta, mean = c(tpar$m), S = tpar$var, df = tpar$df, log = TRUE) md = max(lf - lp) wt = exp(lf - lp - md) est = sum(wt * H)/sum(wt) SEest = sqrt(sum((H - est)^2 * wt^2))/sum(wt) return(list(est = est, se = SEest, theta = theta, wt = wt)) } LearnBayes/vignettes/0000755000176200001440000000000012341620471014307 5ustar liggesusersLearnBayes/vignettes/BinomialInference.Rnw0000644000176200001440000000531612341503364020356 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Learning About a Binomial Proportion} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Learning About a Binomial Proportion} \author{Jim Albert} \maketitle \section*{Constructing a Beta Prior} Suppose we are interested in the proportion $p$ on sunny days in my town. The function {\tt bayes.select} is a convenient tool for specifying a beta prior based on knowledge of two prior quantiles. Suppose my prior median for the proportion of sunny days is $.2$ and my 75th percentile is $.28$. <<>>= library(LearnBayes) beta.par <- beta.select(list(p=0.5, x=0.2), list(p=0.75, x=.28)) beta.par @ A beta(2.95, 10.82) prior matches this prior information \section*{Updating with Data} Next, I observe the weather for 10 days and observe 6 sunny days. (There are 6 ``successes" and 4 ``failures".) The posterior distribution is beta with shape parameters 2.95 + 6 and 10.82 + 4. \section*{Triplot} The {\tt triplot} function shows the prior, likelihood, and posterior on the same display; the inputs are the vector of prior parameters and the data vector. <>= triplot(beta.par, c(6, 4)) @ \section*{Simulating from Posterior to Perform Inference} One can perform inference about the proportion $p$ by simulating a large number of draws from the posterior and summarizing the simulated sample. Here the {\tt rbeta} function is used to simulate from the beta posterior and the {\tt quantile} function is used to construct a 90 percent probability interval for $p$. <<>>= beta.post.par <- beta.par + c(6, 4) post.sample <- rbeta(1000, beta.post.par[1], beta.post.par[2]) quantile(post.sample, c(0.05, 0.95)) @ \section*{Predictive Checking} One can check the suitability of this model by means of a predictive check. The function {\tt predplot} displays the prior predictive density for the number of successes and overlays the observed number of successes. <>= predplot(beta.par, 10, 6) @ The observed data is in the tail of the predictive distribution suggesting some incompability of the prior information and the sample. \section*{Prediction of a Future Sample} Suppose we want to predict the number of sunny days in the future 20 days. The function {\tt pbetap} computes the posterior predictive distribution with a beta prior. The inputs are the vector of beta prior parameters, the future sample size, and the vector of number of successes in the future experiment. <>= n <- 20 s <- 0:n pred.probs <- pbetap(beta.par, n, s) plot(s, pred.probs, type="h") discint(cbind(s, pred.probs), 0.90) @ The probability that we will observe between 0 and 8 successes in the future sample is .92. \end{document}LearnBayes/vignettes/DiscreteBayes.Rnw0000644000176200001440000000736712341503177017545 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Bayes using Discrete Priors} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Bayes using Discrete Priors} \author{Jim Albert} \maketitle \section*{Learning About a Proportion} \subsection*{A Discrete Prior} Consider a population of ``successes" and ``failures" where the proportion of successes is $p$. Suppose $p$ takes on the discrete set of values 0, .01, ..., .99, 1 and one assigns a uniform prior on these values. We enter the values of $p$ and the associated probabilities into the vectors {\tt p} and {\tt prior}, respectively. <<>>= p <- seq(0, 1, by = 0.01) prior <- 1 / 101 + 0 * p @ <>= plot(p, prior, type="h", main="Prior Distribution") @ \subsection*{Posterior Distribution} Suppose one takes a random sample from the population without replacement and observes 20 successes and 12 failiures. The function {\tt pdisc} in the {\tt LearnBayes} package computes the associated posterior probabilities for $p$. The inputs to {\tt pdisc} are the prior (vector of values of $p$ and vector of prior probabilities) and a vector containing the number of successes and failures. <<>>= library(LearnBayes) post <- pdisc(p, prior, c(20, 12)) @ <>= plot(p, post, type="h", main="Posterior Distribution") @ A highest probability interval for a discrete distribution is obtained using the {\tt discint} function. This function has two inputs: the probability distribution matrix where the first column contains the values and the second column contains the probabilities, and the desired probability content. To illustrate, we compute a 90 percent probability interval for $p$ from the posterior distribution. <<>>= discint(cbind(p, post), 0.90) @ The probability that $p$ falls in the interval (0.49, 0.75) is approximately 0.90. \subsection*{Prediction} Suppose a new sample of size 20 is to be taken and we're interested in predicting the number of successes. The current opinion about the proportion is reflected in the posterior distribution stored in the vectors {\tt p} and {\tt post}. We store the possible number of successes in the future sample in {\tt s} and the function {\tt pdiscp} computes the corresponding predictive probabilities. <<>>= n <- 20 s <- 0:20 pred.probs <- pdiscp(p, post, n, s) @ <>= plot(s, pred.probs, type="h", main="Predictive Distribution") @ \section*{Learning About a Poisson Mean} Discrete models can be used for other sampling distributions using the {\tt discrete.bayes} function. To illustrate, suppose the number of accidents in a particular year is Poisson with mean $\lambda$. A priori one believes that $\lambda$ is equally likely to take on the values 20, 21, ..., 30. We put the prior probabilities 1/11, ..., 1/11 in the vector {\tt prior} and use the {\tt names} function to name the components of this vector with the values of $\lambda$. <<>>= prior <- rep(1/11, 11) names(prior) <- 20:30 @ One observes the number of accidents for ten weeks -- these values are placed in the vector {\tt y}: <<>>= y <- c(24, 25, 31, 31, 22, 21, 26, 20, 16, 22) @ To compute the posterior probabilities, we use the function {\tt discrete.bayes}; the inputs are the Poisson sampling density {\tt dpois}, the vector of prior probabilities {\tt prior}, and the vector of observations {\tt y}. <<>>= post <- discrete.bayes(dpois, prior, y) @ One can display the posterior probabilities by use of the {\tt print} method, one displays the posterior probabilites by the {\tt plot} method, and one summarizes the posterior distribution by the {\tt summary} method. <<>>= print(post) @ <>= plot(post) @ <<>>= summary(post) @ \end{document}LearnBayes/vignettes/MultilevelModeling.Rnw0000644000176200001440000001321112341620141020570 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Multilevel Modeling} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Multilevel Modeling} \author{Jim Albert} \maketitle \section*{Efron and Morris Baseball Data} Efron and Morris, in a famous 1975 JASA paper, introduced the problem of estimating the true batting averages for 18 players during the 1971 baseball season. In the table, we observe the number of hits for each player in the first 35 batting opportunities in the season. <<>>= d <- data.frame(Name=c("Clemente", "Robinson", "Howard", "Johnstone", "Berry", "Spencer", "Kessinger", "Alvarado", "Santo", "Swaboda", "Petrocelli", "Rodriguez", "Scott", "Unser", "Williams", "Campaneris", "Munson", "Alvis"), Hits=c(18, 17, 16, 15, 14, 14, 13, 12, 11, 11, 10, 10, 10, 10, 10, 9, 8, 7), At.Bats=45) @ \section*{The Multilevel Model} One can simultaneously estimate the true batting averages by the following multilevel model. We assume the hits for the $j$th player $y_j$ has a binomial distribution with sample size $n_j$ and probability of success $p_j$, $j = 1, ..., 18$. The true batting averages $p_1, .., p_{18}$ are assumed to be a random sample from a beta($a, b$) distribution. It is convenient to reparameterize $a$ and $b$ into the mean $\eta = a / (a + b)$ and precision $K = a + b$. We assign $(\eta, K)$ the noninformative prior $$ g(\eta, K) \propto \frac{1}{\eta (1 - \eta)}\frac{1}{(1 + K)^2} $$ After data $y$ is observed, the posterior distribution of the parameters $(\{p_j\}, \eta, K)$ has the convenient representation $$ g(\{p_j\}, \eta, K | y) = g(\eta, K | y) \times g(\{p_j\} | \eta, K, y). $$ Conditional on $\eta$ and $K$, the posterior distributions of $p_1, ..., p_{18}$ are independent, where $$ p_j \sim Beta(y_j + K \eta, n_j - y_j + K ( 1 - \eta)). $$ The posterior density of $(\eta, K)$ is given by $$ g(\eta, K| y) \propto \prod_{j=1}^{18} \left(\frac{B(y_j + K \eta, n_j - y_j + K (1 - \eta))} {B(K \eta, n_j - y_j + K (1 - \eta))}\right) \frac{1}{\eta (1 - \eta)}\frac{1}{(1 + K)^2}. $$ \section*{Simulation of the Posterior of $(\eta, K)$} For computational purposes, it is convenient to reparameterize $\eta$ and $K$ to the real-valued parameters $$ \theta_1 = \log \frac{\eta}{1 - \eta}, \theta_2 = \log K. $$ The log posterior of the vector $\theta = (\theta_1, \theta_2)$ is programmed in the function {\tt betaabinexch}. We initially use the {\tt laplace} function to find the posterior mode and associated variance-covariance matrix. The inputs are the log posterior function, an initial guess at the mode, and the data. <<>>= library(LearnBayes) laplace.fit <- laplace(betabinexch, c(0, 0), d[, c("Hits", "At.Bats")]) laplace.fit @ The outputs from {\tt laplace} are used to inform the inputs of a random walk Metropolis algorithm in the function {\tt rwmetrop}. The inputs are the function defining the log posterior, the estimate of the variance-covarance matrix and scale for the proposal density, the starting value in the Markov Chain, and the data. <<>>= mcmc.fit <- rwmetrop(betabinexch, list(var=laplace.fit$var, scale=2), c(0, 0), 5000, d[, c("Hits", "At.Bats")]) @ To demonstrate that this MCMC algorithm produces a reasonable sample from the posterior, the {\tt mycontour} function displays a contour graph of the exact posterior density and the {\tt points} function is used to overlay 5000 draws from the MCMC algorithm. <>= mycontour(betabinexch, c(-1.5, -0.5, 2, 12), d[, c("Hits", "At.Bats")], xlab="Logit ETA", ylab="Log K") with(mcmc.fit, points(par)) @ \section*{Simulation of the Posterior of the Probabilities} One can simulate from the joint posterior of $(\{p_j\}, \eta, K)$, by (1) simulating $(\eta, K)$ from its marginal posterior, and (2) simulating $p_1, ..., p_{18}$ from the conditional distribution $[\{p_j\} | \eta, K]$. In the R script, I store the simulated draws from the posterior of $K$ and $\eta$ in the vectors {\tt K} and {\tt eta}. Then the function {\tt p.estimate} simulates draws from the posterior of the $j$th probability and computes a 90\% probability interval by extracting the 5th and 95th percentiles. I repeat this process for all 18 players by the {\tt sapply} function and display the 90\% intervals for all players. <<>>= eta <- with(mcmc.fit, exp(par[, 1]) / (1 + exp(par[, 1]))) K <- exp(mcmc.fit$par[, 2]) p.estimate <- function(j, eta, K){ yj <- d[j, "Hits"] nj <- d[j, "At.Bats"] p.sim <- rbeta(5000, yj + K * eta, nj - yj + K * (1 - eta)) quantile(p.sim, c(0.05, 0.50, 0.95)) } E <- t(sapply(1:18, p.estimate, eta, K)) rownames(E) <- d[, "Name"] round(E, 3) @ The following graph displays the 90 percent probability intervals for the players' true batting averages. The blue line represents {\it individual estimates} where each batting probability is estimated by the observed batting average. The red line represents the {\it combined estimate} where one combines all of the data. The multilevel estimate represented by the dot is a compromise between the individual estimate and the combined estimate. <>= plot(d$Hits / 45, E[, 2], pch=19, ylim=c(.15, .40), xlab="Observed AVG", ylab="True Probability", main="90 Percent Probability Intervals") for (j in 1:18) lines(d$Hits[j] / 45 * c(1, 1), E[j, c(1, 3)]) abline(a=0, b=1, col="blue") abline(h=mean(d$Hits) / 45, col="red") legend("topleft", legend=c("Individual", "Combined"), lty=1, col=c("blue", "red")) @ \end{document}LearnBayes/vignettes/BayesFactors.Rnw0000644000176200001440000001232112341613722017364 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Bayes Factors} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Bayes Factors} \author{Jim Albert} \maketitle \section*{Models for Fire Calls} To motivate the discussion of plausible models, the website \newline {\tt http://www.franklinvillefire.org/callstatistics.htm} gives the number of fire calls for each month in Franklinville, NC for the last several years. Suppose we observe the fire call counts $y_1, ..., y_N$ for $N$ consecutive months. Here is a general model for these data. \begin{itemize} \item $y_1, ..., y_N$ are independent $f(y | \theta)$ \item $\theta$ has a prior $g(\theta)$ \end{itemize} Also suppose we have some prior beliefs about the mean fire count $E(y)$. We believe that this mean is about 70 and the standard deviation of this guess is 10. Given this general model structure, we have to think of possible choices for $f$, the sampling density. We think of the popular distributions, say Poisson, normal, exponential, etc. Also we should think about different choices for the prior density. For the prior, there are many possible choices -- we typically choose one that can represent my prior information. Once we decide on several plausible choices of sampling density and prior, then we'll compare the models by Bayes factors. To do this, we compute the prior predictive density of the actual data for each possible model. The Laplace method provides a convenient and accurate approximation to the logarithm of the predictive density and we'll use the function {\tt laplace} from the {\tt LearnBayes} package. Continuing our example, suppose our prior beliefs about the mean count of fire calls $\theta$ is Gamma(280, 4). (Essentially this says that our prior guess at $\theta$ is 70 and the prior standard deviation is about 4.2.) But we're unsure about the sampling model -- it could be (model $M_1$) Poisson($\theta$), (model $M_2$) normal with mean $\theta$ and standard deviation 12, or (model $M_3$) normal with mean $\theta$ and standard deviation 6. To get some sense about the best sampling model, a histogram of the fire call counts are graphed below. I have overlaid fitted Poisson and normal distributions where I estimate $\theta$ by the sample mean. The Poisson model appears to be the best fit, followed by the Normal model with standard deviation 6, and the Normal model with standard deviation 12. We want to formalize this comparison by computation of Bayes factors. <>= fire.counts <- c(75, 88, 84, 99, 79, 68, 86, 109, 73, 85, 101, 85, 75, 81, 64, 77, 83, 83, 88, 83, 78, 83, 78, 80, 82, 90, 74, 72, 69, 72, 76, 76, 104, 86, 92, 88) hist(fire.counts, probability=TRUE, ylim=c(0, .08)) x <- 60:110 lines(x, dpois(x, lambda=mean(fire.counts)), col="red") lines(x, dnorm(x, mean=mean(fire.counts), sd=12), col="blue") lines(x, dnorm(x, mean=mean(fire.counts), sd=6), col="green") legend("topright", legend=c("M1: Poisson(theta)", "M2: N(theta, 12)", "M3: N(theta, 6)"), col=c("red", "blue", "green"), lty=1) @ \section*{Bayesian Model Comparison} Under the general model, the predictive density of $y$ is given by the integral $$ f(y) = \int \prod_{j=1}^N f(y_j | \theta) g(\theta) d\theta. $$ This density can be approximated by the Laplace method implemented in the {\tt laplace} function. One compares the suitability of two Bayesian models by comparing the corresponding values of the predictive density. The Bayes factor in support of model $M_1$ over model $M_2$ is given by the ratio $$ BF_{12} = \frac{f_1(y)}{f_2(y)}. $$ Computationally, it is convenient to compute the predictive densities on the log scale, so the Bayes factor can be expressed as $$ BF_{12} = \exp \left(\log f_1(y) - \log f_2(y)\right). $$ To compute the predictive density for a model, say model $M_1$, we initially define a function {\tt model.1} which gives the log posterior. <<>>= model.1 <- function(theta, y){ sum(log(dpois(y, theta))) + dgamma(theta, shape=280, rate=4) } @ Then the log predictive density at $y$ is computed by using the {\tt laplace} function with inputs the function name, a guess at the posterior mode, and the data (vector of fire call counts). The component {\tt int} gives the log of $f(y)$ <<>>= library(LearnBayes) log.pred.1 <- laplace(model.1, 80, fire.counts)$int log.pred.1 @ We similarly find the predictive densities of the models $M_2$ and $M_3$ by defining functions for the corresponding posteriors and using {\tt laplace}: <<>>= model.2 <- function(theta, y){ sum(log(dnorm(y, theta, 6))) + dgamma(theta, shape=280, rate=4) } model.3 <- function(theta, y){ sum(log(dnorm(y, theta, 12))) + dgamma(theta, shape=280, rate=4) } log.pred.2 <- laplace(model.2, 80, fire.counts)$int log.pred.3 <- laplace(model.3, 80, fire.counts)$int @ Displaying the three models and predictive densities, we see that model $M_1$ is preferred to $M_3$ which is preferred to model $M_2$. <<>>= data.frame(Model=1:3, log.pred=c(log.pred.1, log.pred.2, log.pred.3)) @ The Bayes factor in support of model $M_1$ over model $M_3$ is given by <<>>= exp(log.pred.1 - log.pred.3) @ \end{document}LearnBayes/vignettes/MCMCintro.Rnw0000644000176200001440000001201512341503775016600 0ustar liggesusers\documentclass{article} %\VignetteIndexEntry{Introduction to Markov Chain Monte Carlo} %\VignetteDepends{LearnBayes} \begin{document} \SweaveOpts{concordance=TRUE} \title{Introduction to Markov Chain Monte Carlo} \author{Jim Albert} \maketitle \section*{A Selected Data Problem} Here is an interesting problem with ``selected data". Suppose you are measuring the speeds of cars driving on an interstate. You assume the speeds are normally distributed with mean $\mu$ and standard deviation $\sigma$. You see 10 cars pass by and you only record the minimum and maximum speeds. What have you learned about the normal parameters? First we focus on the construction of the likelihood. Given values of the normal parameters, what is the probability of observing minimum = $x$ and the maximum = $y$ in a sample of size n? Essentially we're looking for the joint density of two order statistics which is a standard result. Let $f$ and $F $denote the density and cdf of a normal density with mean $\mu$ and standard deviation $\sigma$. Then the joint density of $(x, y)$ is given by $$f(x, y | \mu, \sigma) \propto f(x) f(y) [F(y) - F(x)]^{n-2}, x < y$$ After we observe data, the likelihood is this sampling density viewed as function of the parameters. Suppose we take a sample of size 10 and we observe $x = 52, y = 84$. Then the likelihood is given by $$ L(\mu, \sigma) \propto f(52) f(84) [F(84) - F(52)]^{8} $$ \section*{Defining the log posterior} First I write a short function {\tt minmaxpost} that computes the logarithm of the posterior density. The arguments to this function are $\theta = (\mu, \log \sigma)$ and data which is a list with components {\tt n}, {\tt min}, and {\tt max}. I'd recommend using the R functions {\tt pnorm} and {\tt dnorm} in computing the density -- it saves typing errors. <<>>= minmaxpost <- function(theta, data){ mu <- theta[1] sigma <- exp(theta[2]) dnorm(data$min, mu, sigma, log=TRUE) + dnorm(data$max, mu, sigma, log=TRUE) + (data$n - 2) * log(pnorm(data$max, mu, sigma) - pnorm(data$min, mu, sigma)) } @ \section*{Normal approximation to posterior} We work with the parameterization $(\mu, \log \sigma)$ which will give us a better normal approximation. A standard noninformative prior is uniform on $(\mu, \log \sigma)$. The function {\tt laplace} is used to summarize this posterior. The arguments to {\tt laplace} are the name of the log posterior function, an initial estimate at $\theta$, and the data that is used in the log posterior function. The output of laplace includes mode, the posterior mode, and var, the corresponding estimate at the variance-covariance matrix. <<>>= data <- list(n=10, min=52, max=84) library(LearnBayes) fit <- laplace(minmaxpost, c(70, 2), data) fit @ In this example, this gives a pretty good approximation in this situation. The {\tt mycontour} function is used to display contours of the exact posterior and overlay the matching normal approximation using a second application of {\tt mycontour}. <>= mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) mycontour(lbinorm, c(45, 95, 1.5, 4), list(m=fit$mode, v=fit$var), add=TRUE, col="red") @ \section*{Random Walk Metropolis Sampling} The {\tt rwmetrop} function implements the M-H random walk algorithm. There are four inputs: (1) the function defining the log posterior, (2) a list containing var, the estimated var-cov matrix, and scale, the M-H random walk scale constant, (3) the starting value in the Markov Chain simulation, (4) the number of iterations of the algorithm, and (5) any data and prior parameters used in the log posterior density. Here we use {\tt fit\$v} as our estimated var-cov matrix, use a scale value of 3, start the simulation at $(\mu, \log \sigma) = (70, 2)$ and try 10,000 iterations. <<>>= mcmc.fit <- rwmetrop(minmaxpost, list(var=fit$v, scale=3), c(70, 2), 10000, data) @ I display the acceptance rate -- here it is 19\% which is a reasonable value. <<>>= mcmc.fit$accept @ We display the contours of the exact posterior and overlay the simulated draws. <>= mycontour(minmaxpost, c(45, 95, 1.5, 4), data, xlab=expression(mu), ylab=expression(paste("log ",sigma))) points(mcmc.fit$par) @ It appears like we have been successful in getting a good sample from this posterior distribution. \section*{Random Walk Metropolis Sampling} To illustrate simulation-based inference, suppose one is interested in learning about the upper quartile $$ P.75 = \mu + 0.674 \times \sigma $$ of the car speed distribution. For each simulated draw of $(\mu, \sigma)$ from the posterior, we compute the upper quartile $P.75$. 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LazyData: yes Description: LearnBayes contains a collection of functions helpful in learning the basic tenets of Bayesian statistical inference. It contains functions for summarizing basic one and two parameter posterior distributions and predictive distributions. It contains MCMC algorithms for summarizing posterior distributions defined by the user. It also contains functions for regression models, hierarchical models, Bayesian tests, and illustrations of Gibbs sampling. License: GPL (>= 2) Packaged: 2014-05-29 11:59:53 UTC; albert NeedsCompilation: no Repository: CRAN Date/Publication: 2014-05-29 17:33:08 LearnBayes/man/0000755000176200001440000000000012341466566013067 5ustar liggesusersLearnBayes/man/prior.two.parameters.Rd0000644000176200001440000000114711311764316017454 0ustar liggesusers\name{prior.two.parameters} \alias{prior.two.parameters} \title{Construct discrete uniform prior for two parameters} \description{ Constructs a discrete uniform prior distribution for two parameters } \usage{ prior.two.parameters(parameter1, parameter2) } \arguments{ \item{parameter1}{vector of values of first parameter} \item{parameter2}{vector of values of second parameter} } \value{ matrix of uniform probabilities where the rows and columns are labelled with the parameter values } \author{Jim Albert} \examples{ prior.two.parameters(c(1,2,3,4),c(2,4,7)) } \keyword{models} LearnBayes/man/pbetat.Rd0000644000176200001440000000136010735604276014632 0ustar liggesusers\name{pbetat} \alias{pbetat} \title{Bayesian test of a proportion} \description{ Bayesian test that a proportion is equal to a specified value using a beta prior} \usage{ pbetat(p0,prob,ab,data) } \arguments{ \item{p0}{value of the proportion to be tested } \item{prob}{prior probability of the hypothesis} \item{ab}{vector of parameter values of the beta prior under the alternative hypothesis} \item{data}{vector containing the number of successes and number of failures} } \value{ \item{bf}{the Bayes factor in support of the null hypothesis} \item{post}{the posterior probability of the null hypothesis} } \author{Jim Albert} \examples{ p0=.5 prob=.5 ab=c(10,10) data=c(5,15) pbetat(p0,prob,ab,data) } \keyword{models} LearnBayes/man/logpoissnormal.Rd0000644000176200001440000000127710735604076016430 0ustar liggesusers\name{logpoissnormal} \alias{logpoissnormal} \title{Log posterior with Poisson sampling and normal prior} \description{ Computes the logarithm of the posterior density of a Poisson log mean with a normal prior } \usage{ logpoissnormal(theta,datapar) } \arguments{ \item{theta}{vector of values of the log mean parameter} \item{datapar}{list with components data, vector of observations, and par, vector of parameters of the normal prior} } \value{ vector of values of the log posterior for all values in theta } \author{Jim Albert} \examples{ data=c(2,4,3,6,1,0,4,3,10,2) par=c(0,1) datapar=list(data=data,par=par) theta=c(-1,0,1,2) logpoissnormal(theta,datapar) } \keyword{models} LearnBayes/man/blinreg.Rd0000644000176200001440000000171211054057172014766 0ustar liggesusers\name{blinreg} \alias{blinreg} \title{Simulation from Bayesian linear regression model} \description{ Gives a simulated sample from the joint posterior distribution of the regression vector and the error standard deviation for a linear regression model with a noninformative or g prior. } \usage{ blinreg(y,X,m,prior=NULL) } \arguments{ \item{y}{vector of responses} \item{X}{design matrix} \item{m}{number of simulations desired} \item{prior}{list with components c0 and beta0 of Zellner's g prior} } \value{ \item{beta}{matrix of simulated draws of beta where each row corresponds to one draw} \item{sigma}{vector of simulated draws of the error standard deviation} } \author{Jim Albert} \examples{ chirps=c(20,16.0,19.8,18.4,17.1,15.5,14.7,17.1,15.4,16.2,15,17.2,16,17,14.1) temp=c(88.6,71.6,93.3,84.3,80.6,75.2,69.7,82,69.4,83.3,78.6,82.6,80.6,83.5,76.3) X=cbind(1,chirps) m=1000 s=blinreg(temp,X,m) } \keyword{models} LearnBayes/man/normnormexch.Rd0000644000176200001440000000132612341467421016066 0ustar liggesusers\name{normnormexch} \alias{normnormexch} \title{Log posterior of mean and log standard deviation for Normal/Normal exchangeable model} \description{ Computes the log posterior density of mean and log standard deviation for a Normal/Normal exchangeable model where (mean, log sd) is given a uniform prior. } \usage{ normnormexch(theta,data) } \arguments{ \item{theta}{vector of parameter values of mu and log tau} \item{data}{a matrix with columns y (observations) and v (sampling variances)} } \value{ value of the log posterior } \author{Jim Albert} \examples{ s.var <- c(0.05, 0.05, 0.05, 0.05, 0.05) y.means <- c(1, 4, 3, 6,10) data=cbind(y.means, s.var) theta=c(-1, 0) normnormexch(theta,data) } \keyword{models} LearnBayes/man/cauchyerrorpost.Rd0000644000176200001440000000126610735603554016612 0ustar liggesusers\name{cauchyerrorpost} \alias{cauchyerrorpost} \title{Log posterior of median and log scale parameters for Cauchy sampling} \description{ Computes the log posterior density of (M,log S) when a sample is taken from a Cauchy density with location M and scale S and a uniform prior distribution is taken on (M, log S) } \usage{ cauchyerrorpost(theta,data) } \arguments{ \item{theta}{vector of parameter values of M and log S} \item{data}{vector containing sample of observations} } \value{ value of the log posterior } \author{Jim Albert} \examples{ data=c(108, 51, 7, 43, 52, 54, 53, 49, 21, 48) theta=c(40,1) cauchyerrorpost(theta,data) } \keyword{models} LearnBayes/man/impsampling.Rd0000644000176200001440000000223410706727350015671 0ustar liggesusers\name{impsampling} \alias{impsampling} \title{Importance sampling using a t proposal density} \description{ Implements importance sampling to compute the posterior mean of a function using a multivariate t proposal density } \usage{ impsampling(logf,tpar,h,n,data) } \arguments{ \item{logf}{function that defines the logarithm of the density of interest} \item{tpar}{list of parameters of t proposal density including the mean m, scale matrix var, and degrees of freedom df} \item{h}{function that defines h(theta)} \item{n}{number of simulated draws from proposal density} \item{data}{data and or parameters used in the function logf} } \value{ \item{est}{estimate at the posterior mean} \item{se}{simulation standard error of estimate} \item{theta}{matrix of simulated draws from proposal density} \item{wt}{vector of importance sampling weights} } \author{Jim Albert} \examples{ data(cancermortality) start=c(-7,6) fit=laplace(betabinexch,start,cancermortality) tpar=list(m=fit$mode,var=2*fit$var,df=4) myfunc=function(theta) return(theta[2]) theta=impsampling(betabinexch,tpar,myfunc,1000,cancermortality) } \keyword{models} LearnBayes/man/predplot.Rd0000644000176200001440000000122610735465516015207 0ustar liggesusers\name{predplot} \alias{predplot} \title{Plot of predictive distribution for binomial sampling with a beta prior} \description{ For a proportion problem with a beta prior, plots the prior predictive distribution of the number of successes in n trials and displays the observed number of successes. } \usage{ predplot(prior,n,yobs) } \arguments{ \item{prior}{vector of parameters for beta prior} \item{n}{sample size} \item{yobs}{observed number of successes} } \author{Jim Albert} \examples{ prior=c(3,10) # proportion has a beta(3, 10) prior n=20 # sample size yobs=10 # observed number of successes predplot(prior,n,yobs) } \keyword{models} LearnBayes/man/pbetap.Rd0000644000176200001440000000117512341465557014634 0ustar liggesusers\name{pbetap} \alias{pbetap} \title{Predictive distribution for a binomial sample with a beta prior} \description{ Computes predictive distribution for number of successes of future binomial experiment with a beta prior distribution for the proportion. } \usage{ pbetap(ab, n, s) } \arguments{ \item{ab}{vector of parameters of the beta prior} \item{n}{size of future binomial sample} \item{s}{vector of number of successes for future binomial experiment} } \value{ vector of predictive probabilities for the values in the vector s } \author{Jim Albert} \examples{ ab=c(3,12) n=10 s=0:10 pbetap(ab,n,s) } \keyword{models} LearnBayes/man/normal.normal.mix.Rd0000644000176200001440000000206410735604134016721 0ustar liggesusers\name{normal.normal.mix} \alias{normal.normal.mix} \title{Computes the posterior for normal sampling and a mixture of normals prior} \description{ Computes the parameters and mixing probabilities for a normal sampling problem, variance known, where the prior is a discrete mixture of normal densities. } \usage{ normal.normal.mix(probs,normalpar,data) } \arguments{ \item{probs}{vector of probabilities of the normal components of the prior} \item{normalpar}{matrix where each row contains the mean and variance parameters for a normal component of the prior} \item{data}{vector of observation and sampling variance} } \value{ \item{probs}{vector of probabilities of the normal components of the posterior} \item{normalpar}{matrix where each row contains the mean and variance parameters for a normal component of the posterior} } \author{Jim Albert} \examples{ probs=c(.5, .5) normal.par1=c(0,1) normal.par2=c(2,.5) normalpar=rbind(normal.par1,normal.par2) y=1; sigma2=.5 data=c(y,sigma2) normal.normal.mix(probs,normalpar,data) } \keyword{models} LearnBayes/man/schmidt.Rd0000644000176200001440000000155211126707736015011 0ustar liggesusers\name{schmidt} \alias{schmidt} \docType{data} \title{Batting data for Mike Schmidt} \description{ Batting statistics for the baseball player Mike Schmidt during all the seasons of his career. } \usage{ schmidt } \format{ A data frame with 18 observations on the following 14 variables. \describe{ \item{Year}{year of the season} \item{Age}{Schmidt's age that season} \item{G}{games played} \item{AB}{at-bats} \item{R}{runs scored} \item{H}{number of hits} \item{X2B}{number of doubles} \item{X3B}{number of triples} \item{HR}{number of home runs} \item{RBI}{number of runs batted in} \item{SB}{number of stolen bases} \item{CS}{number of times caught stealing} \item{BB}{number of walks} \item{SO}{number of strikeouts} } } \source{Sean Lahman's baseball database from www.baseball1.com.} \keyword{datasets} LearnBayes/man/achievement.Rd0000644000176200001440000000134411126663170015636 0ustar liggesusers\name{achievement} \alias{achievement} \docType{data} \title{School achievement data} \description{ Achievement data for a group of Austrian school children } \usage{ achievement } \format{ A data frame with 109 observations on the following 7 variables. \describe{ \item{Gen}{gender of child where 0 is male and 1 is female} \item{Age}{age in months} \item{IQ}{iq score} \item{math1}{test score on mathematics computation} \item{math2}{test score on mathematics problem solving} \item{read1}{test score on reading speed} \item{read2}{test score on reading comprehension} } } \source{ Abraham, B., and Ledolter, J. (2006), Introduction to Regression Modeling, Duxbury. } \keyword{datasets} LearnBayes/man/discint.Rd0000644000176200001440000000127210735603572015010 0ustar liggesusers\name{discint} \alias{discint} \title{Highest probability interval for a discrete distribution} \description{ Computes a highest probability interval for a discrete probability distribution } \usage{ discint(dist, prob) } \arguments{ \item{dist}{probability distribution written as a matrix where the first column contain the values and the second column the probabilities} \item{prob}{probability content of interest} } \value{ \item{prob}{exact probability content of interval} \item{set}{set of values of the probability interval} } \author{Jim Albert} \examples{ x=0:10 probs=dbinom(x,size=10,prob=.3) dist=cbind(x,probs) pcontent=.8 discint(dist,pcontent) } \keyword{models} LearnBayes/man/studentdata.Rd0000644000176200001440000000172311126712224015662 0ustar liggesusers\name{studentdata} \alias{studentdata} \docType{data} \title{Student dataset} \description{ Answers to a sheet of questions given to a large number of students in introductory statistics classes } \usage{ studentdata } \format{ A data frame with 657 observations on the following 11 variables. \describe{ \item{Student}{student number} \item{Height}{height in inches} \item{Gender}{gender} \item{Shoes}{number of pairs of shoes owned} \item{Number}{number chosen between 1 and 10} \item{Dvds}{name of movie dvds owned} \item{ToSleep}{time the person went to sleep the previous night (hours past midnight)} \item{WakeUp}{time the person woke up the next morning} \item{Haircut}{cost of last haircut including tip} \item{Job}{number of hours working on a job per week} \item{Drink}{usual drink at suppertime among milk, water, and pop} } } \source{Collected by the author during the Fall 2006 semester.} \keyword{datasets} LearnBayes/man/blinregexpected.Rd0000644000176200001440000000203610537535326016517 0ustar liggesusers\name{blinregexpected} \alias{blinregexpected} \title{Simulates values of expected response for linear regression model} \description{ Simulates draws of the posterior distribution of an expected response for a linear regression model with a noninformative prior} \usage{ blinregexpected(X1,theta.sample) } \arguments{ \item{X1}{matrix where each row corresponds to a covariate set} \item{theta.sample}{list with components beta, matrix of simulated draws of regression vector, and sigma, vector of simulated draws of sampling error standard deviation} } \value{ matrix where a column corresponds to the simulated draws of the expected response for a given covariate set } \author{Jim Albert} \examples{ chirps=c(20,16.0,19.8,18.4,17.1,15.5,14.7,17.1,15.4,16.2,15,17.2,16,17,14.1) temp=c(88.6,71.6,93.3,84.3,80.6,75.2,69.7,82,69.4,83.3,78.6,82.6,80.6,83.5,76.3) X=cbind(1,chirps) m=1000 theta.sample=blinreg(temp,X,m) covset1=c(1,15) covset2=c(1,20) X1=rbind(covset1,covset2) blinregexpected(X1,theta.sample) } \keyword{models} LearnBayes/man/bayesresiduals.Rd0000644000176200001440000000174410735603250016367 0ustar liggesusers\name{bayesresiduals} \alias{bayesresiduals} \title{Computation of posterior residual outlying probabilities for a linear regression model} \description{ Computes the posterior probabilities that Bayesian residuals exceed a cutoff value for a linear regression model with a noninformative prior } \usage{ bayesresiduals(lmfit,post,k) } \arguments{ \item{lmfit}{output of the regression function lm} \item{post}{list with components beta, matrix of simulated draws of regression parameter, and sigma, vector of simulated draws of sampling standard deviation} \item{k}{cut-off value that defines an outlier} } \value{ vector of posterior outlying probabilities } \author{Jim Albert} \examples{ chirps=c(20,16.0,19.8,18.4,17.1,15.5,14.7,17.1,15.4,16.2,15,17.2,16,17,14.1) temp=c(88.6,71.6,93.3,84.3,80.6,75.2,69.7,82,69.4,83.3,78.6,82.6,80.6,83.5,76.3) X=cbind(1,chirps) lmfit=lm(temp~X) m=1000 post=blinreg(temp,X,m) k=2 bayesresiduals(lmfit,post,k) } \keyword{models} LearnBayes/man/footballscores.Rd0000644000176200001440000000146411126676114016374 0ustar liggesusers\name{footballscores} \alias{footballscores} \docType{data} \title{Game outcomes and point spreads for American football} \description{ Game outcomes and point spreads for 672 professional American football games. } \usage{ footballscores } \format{ A data frame with 672 observations on the following 8 variables. \describe{ \item{year}{year of game} \item{home}{indicates if favorite is the home team} \item{favorite}{score of favorite team} \item{underdog}{score of underdog team} \item{spread}{point spread} \item{favorite.name}{name of favorite team} \item{underdog.name}{name of underdog team} \item{week}{week number of the season} } } \source{Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2003), Bayesian Data Analysis, Chapman and Hall.} \keyword{datasets} LearnBayes/man/baseball.1964.Rd0000644000176200001440000000141711126663514015520 0ustar liggesusers\name{baseball.1964} \alias{baseball.1964} \docType{data} \title{Team records in the 1964 National League baseball season} \description{ Head to head records for all teams in the 1964 National League baseball season. Teams are coded as Cincinnati (1), Chicago (2), Houston (3), Los Angeles (4), Milwaukee (5), New York (6), Philadelphia (7), Pittsburgh (8), San Francisco (9), and St. Louis (10). } \usage{ baseball.1964 } \format{ A data frame with 45 observations on the following 4 variables. \describe{ \item{Team.1}{Number of team 1} \item{Team.2}{Number of team 2} \item{Wins.Team1}{Number of games won by team 1} \item{Wins.Team2}{Number of games won by team 2} } } \source{ www.baseball-reference.com website. } \keyword{datasets} LearnBayes/man/rigamma.Rd0000644000176200001440000000103611126477044014764 0ustar liggesusers\name{rigamma} \alias{rigamma} \title{Random number generation for inverse gamma distribution} \description{ Simulates from a inverse gamma (a, b) distribution with density proportional to $y^(-a-1) exp(-b/y)$ } \usage{ rigamma(n, a, b) } \arguments{ \item{n}{number of random numbers to be generated} \item{a}{inverse gamma shape parameter} \item{b}{inverse gamma rate parameter} } \value{ vector of n simulated draws } \author{Jim Albert} \examples{ a=10 b=5 n=20 rigamma(n,a,b) } \keyword{models} LearnBayes/man/discrete.bayes.Rd0000644000176200001440000000175211311755722016256 0ustar liggesusers\name{discrete.bayes} \alias{discrete.bayes} \alias{print.bayes} \alias{plot.bayes} \alias{summary.bayes} \title{Posterior distribution with discrete priors} \description{ Computes the posterior distribution for an arbitrary one parameter distribution for a discrete prior distribution. } \usage{ discrete.bayes(df,prior,y,...) } \arguments{ \item{df}{name of the function defining the sampling density} \item{prior}{vector defining the prior density; names of the vector define the parameter values and entries of the vector define the prior probabilities} \item{y}{vector of data values} \item{...}{any further fixed parameter values used in the sampling density function} } \value{ \item{prob}{vector of posterior probabilities} \item{pred}{scalar with prior predictive probability} } \author{Jim Albert} \examples{ prior=c(.25,.25,.25,.25) names(prior)=c(.2,.25,.3,.35) y=5 n=10 discrete.bayes(dbinom,prior,y,size=n) } \keyword{models} LearnBayes/man/logctablepost.Rd0000644000176200001440000000135410735604030016204 0ustar liggesusers\name{logctablepost} \alias{logctablepost} \title{Log posterior of difference and sum of logits in a 2x2 table} \description{ Computes the log posterior density for the difference and sum of logits in a 2x2 contingency table for independent binomial samples and uniform prior placed on the logits } \usage{ logctablepost(theta,data) } \arguments{ \item{theta}{vector of parameter values "difference of logits" and "sum of logits")} \item{data}{vector containing number of successes and failures for first sample, and then second sample} } \value{ value of the log posterior } \author{Jim Albert} \examples{ s1=6; f1=2; s2=3; f2=10 data=c(s1,f1,s2,f2) theta=c(2,4) logctablepost(theta,data) } \keyword{models} LearnBayes/man/discrete.bayes.2.Rd0000644000176200001440000000224411311761434016410 0ustar liggesusers\name{discrete.bayes.2} \alias{discrete.bayes.2} \alias{plot.bayes2} \title{Posterior distribution of two parameters with discrete priors} \description{ Computes the posterior distribution for an arbitrary two parameter distribution for a discrete prior distribution. } \usage{ discrete.bayes.2(df,prior,y=NULL,...) } \arguments{ \item{df}{name of the function defining the sampling density of two parameters} \item{prior}{matrix defining the prior density; the row names and column names of the matrix define respectively the values of parameter 1 and values of parameter 2 and the entries of the matrix give the prior probabilities} \item{y}{y is a matrix of data values, where each row corresponds to a single observation} \item{...}{any further fixed parameter values used in the sampling density function} } \value{ \item{prob}{matrix of posterior probabilities} \item{pred}{scalar with prior predictive probability} } \author{Jim Albert} \examples{ p1 = seq(0.1, 0.9, length = 9) p2 = p1 prior = matrix(1/81, 9, 9) dimnames(prior)[[1]] = p1 dimnames(prior)[[2]] = p2 discrete.bayes.2(twoproplike,prior) } \keyword{models} LearnBayes/man/dmt.Rd0000644000176200001440000000152010554302720014121 0ustar liggesusers\name{dmt} \alias{dmt} \title{Probability density function for multivariate t} \description{ Computes the density of a multivariate t distribution } \usage{ dmt(x, mean = rep(0, d), S, df = Inf, log=FALSE) } \arguments{ \item{x}{vector of length d or matrix with d columns, giving the coordinates of points where density is to evaluated} \item{mean}{numeric vector giving the location parameter of the distribution} \item{S}{a positive definite matrix representing the scale matrix of the distribution} \item{df}{degrees of freedom} \item{log}{a logical value; if TRUE, the logarithm of the density is to be computed} } \value{ vector of density values } \author{Jim Albert} \examples{ mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) df <- 4 x <- c(2,14,0) f <- dmt(x, mu, Sigma, df) } \keyword{models} LearnBayes/man/rdirichlet.Rd0000644000176200001440000000074410735604612015503 0ustar liggesusers\name{rdirichlet} \alias{rdirichlet} \title{Random draws from a Dirichlet distribution} \description{ Simulates a sample from a Dirichlet distribution } \usage{ rdirichlet(n,par) } \arguments{ \item{n}{number of simulations required} \item{par}{vector of parameters of the Dirichlet distribution} } \value{ matrix of simulated draws where each row corresponds to a single draw } \author{Jim Albert} \examples{ par=c(2,5,4,10) n=10 rdirichlet(n,par) } \keyword{models} LearnBayes/man/reg.gprior.post.Rd0000644000176200001440000000143211054061564016405 0ustar liggesusers\name{reg.gprior.post} \alias{reg.gprior.post} \title{Computes the log posterior of a normal regression model with a g prior.} \description{ Computes the log posterior of (beta, log sigma) for a normal regression model with a g prior with parameters beta0 and c0. } \usage{ reg.gprior.post(theta, dataprior) } \arguments{ \item{theta}{vector of components of beta and log sigma} \item{dataprior}{list with components data and prior; data is a list with components y and X, prior is a list with components b0 and c0} } \value{ value of the log posterior } \author{Jim Albert} \examples{ data(puffin) data=list(y=puffin$Nest, X=cbind(1,puffin$Distance)) prior=list(b0=c(0,0), c0=10) reg.gprior.post(c(20,-.5,1),list(data=data,prior=prior)) } \keyword{models} LearnBayes/man/bprobit.probs.Rd0000644000176200001440000000152111070706026016124 0ustar liggesusers\name{bprobit.probs} \alias{bprobit.probs} \title{Simulates fitted probabilities for a probit regression model} \description{ Gives a simulated sample for fitted probabilities for a binary response regression model with a probit link and noninformative prior. } \usage{ bprobit.probs(X1,fit) } \arguments{ \item{X1}{matrix where each row corresponds to a covariate set} \item{fit}{simulated matrix of draws of the regression vector} } \value{ matrix of simulated draws of the fitted probabilities, where a column corresponds to a particular covariate set } \author{Jim Albert} \examples{ response=c(0,1,0,0,0,1,1,1,1,1) covariate=c(1,2,3,4,5,6,7,8,9,10) X=cbind(1,covariate) m=1000 fit=bayes.probit(response,X,m) x1=c(1,3) x2=c(1,8) X1=rbind(x1,x2) fittedprobs=bprobit.probs(X1,fit$beta) } \keyword{models} LearnBayes/man/rmnorm.Rd0000644000176200001440000000120410554300672014652 0ustar liggesusers\name{rmnorm} \alias{rmnorm} \title{Random number generation for multivariate normal} \description{ Simulates from a multivariate normal distribution } \usage{ rmnorm(n = 1, mean = rep(0, d), varcov) } \arguments{ \item{n}{number of random numbers to be generated} \item{mean}{numeric vector giving the mean of the distribution} \item{varcov}{a positive definite matrix representing the variance-covariance matrix of the distribution} } \value{ matrix of n rows of random vectors } \author{Jim Albert} \examples{ mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) x <- rmnorm(10, mu, Sigma) } \keyword{models} LearnBayes/man/betabinexch.Rd0000644000176200001440000000120610735603306015617 0ustar liggesusers\name{betabinexch} \alias{betabinexch} \title{Log posterior of logit mean and log precision for Binomial/beta exchangeable model} \description{ Computes the log posterior density of logit mean and log precision for a Binomial/beta exchangeable model } \usage{ betabinexch(theta,data) } \arguments{ \item{theta}{vector of parameter values of logit eta and log K} \item{data}{a matrix with columns y (counts) and n (sample sizes)} } \value{ value of the log posterior } \author{Jim Albert} \examples{ n=c(20,20,20,20,20) y=c(1,4,3,6,10) data=cbind(y,n) theta=c(-1,0) betabinexch(theta,data) } \keyword{models} LearnBayes/man/donner.Rd0000644000176200001440000000127511126671224014635 0ustar liggesusers\name{donner} \alias{donner} \docType{data} \title{Donner survival study} \description{ Data contains the age, gender and survival status for 45 members of the Donner Party who experienced difficulties in crossing the Sierra Nevada mountains in California. } \usage{ donner } \format{ A data frame with 45 observations on the following 3 variables. \describe{ \item{age}{age of person} \item{male}{gender that is 1 (0) if person is male (female)} \item{survival}{survival status, 1 or 0 if person survived or died} } } \source{Grayson, D. (1960), Donner party deaths: a demographic assessment, Journal of Anthropological Assessment, 46, 223-242.} \keyword{datasets} LearnBayes/man/logisticpost.Rd0000644000176200001440000000137610735604044016076 0ustar liggesusers\name{logisticpost} \alias{logisticpost} \title{Log posterior for a binary response model with a logistic link and a uniform prior} \description{ Computes the log posterior density of (beta0, beta1) when yi are independent binomial(ni, pi) and logit(pi)=beta0+beta1*xi and a uniform prior is placed on (beta0, beta1) } \usage{ logisticpost(beta,data) } \arguments{ \item{beta}{vector of parameter values beta0 and beta1} \item{data}{matrix of columns of covariate values x, sample sizes n, and number of successes y} } \value{ value of the log posterior } \author{Jim Albert} \examples{ x = c(-0.86,-0.3,-0.05,0.73) n = c(5,5,5,5) y = c(0,1,3,5) data = cbind(x, n, y) beta=c(2,10) logisticpost(beta,data) } \keyword{models} LearnBayes/man/poissgamexch.Rd0000644000176200001440000000126610735604342016044 0ustar liggesusers\name{poissgamexch} \alias{poissgamexch} \title{Log posterior of Poisson/gamma exchangeable model} \description{ Computes the log posterior density of log alpha and log mu for a Poisson/gamma exchangeable model } \usage{ poissgamexch(theta,datapar) } \arguments{ \item{theta}{vector of parameter values of log alpha and log mu} \item{datapar}{list with components data, a matrix with columns e and y, and z0, prior hyperparameter} } \value{ value of the log posterior } \author{Jim Albert} \examples{ e=c(532,584,672,722,904) y=c(0,0,2,1,1) data=cbind(e,y) theta=c(-4,0) z0=.5 datapar=list(data=data,z0=z0) poissgamexch(theta,datapar) } \keyword{models} LearnBayes/man/hearttransplants.Rd0000644000176200001440000000155411126677202016747 0ustar liggesusers\name{hearttransplants} \alias{hearttransplants} \docType{data} \title{Heart transplant mortality data} \description{ The number of deaths within 30 days of heart transplant surgery for 94 U.S. hospitals that performed at least 10 heart transplant surgeries. Also the exposure, the expected number of deaths, is recorded for each hospital.} \usage{ hearttransplants } \format{ A data frame with 94 observations on the following 2 variables. \describe{ \item{e}{expected number of deaths (the exposure)} \item{y}{observed number of deaths within 30 days of heart transplant surgery} } } \source{Christiansen, C. and Morris, C. (1995), Fitting and checking a two-level Poisson model: modeling patient mortality rates in heart transplant patients, in Berry, D. and Stangl, D., eds, Bayesian Biostatistics, Marcel Dekker.} \keyword{datasets} LearnBayes/man/soccergoals.Rd0000644000176200001440000000073211126710352015645 0ustar liggesusers\name{soccergoals} \alias{soccergoals} \docType{data} \title{Goals scored by professional soccer team} \description{ Number of goals scored by a single professional soccer team during the 2006 Major League Soccer season} \usage{ soccergoals } \format{ A data frame with 35 observations on the following 1 variable. \describe{ \item{goals}{number of goals scored} } } \source{Collected by author from the www.espn.com website.} \keyword{datasets} LearnBayes/man/beta.select.Rd0000644000176200001440000000155111063057650015537 0ustar liggesusers\name{beta.select} \alias{beta.select} \title{Selection of Beta Prior Given Knowledge of Two Quantiles} \description{ Finds the shape parameters of a beta density that matches knowledge of two quantiles of the distribution. } \usage{ beta.select(quantile1, quantile2) } \arguments{ \item{quantile1}{list with components p, the value of the first probability, and x, the value of the first quantile} \item{quantile2}{list with components p, the value of the second probability, and x, the value of the second quantile} } \value{ vector of shape parameters of the matching beta distribution } \author{Jim Albert} \examples{ # person believes the median of the prior is 0.25 # and the 90th percentile of the prior is 0.45 quantile1=list(p=.5,x=0.25) quantile2=list(p=.9,x=0.45) beta.select(quantile1,quantile2) } \keyword{models} LearnBayes/man/bermuda.grass.Rd0000644000176200001440000000115311126663762016110 0ustar liggesusers\name{bermuda.grass} \alias{bermuda.grass} \docType{data} \title{Bermuda grass experiment data} \description{ Yields of bermuda grass for a factorial design of nutrients nitrogen, phosphorus, and potassium. } \usage{ bermuda.grass } \format{ A data frame with 64 observations on the following 4 variables. \describe{ \item{y}{yield of bermuda grass in tons per acre} \item{Nit}{level of nitrogen} \item{Phos}{level of phosphorus} \item{Pot}{level of potassium} } } \source{ McCullagh, P., and Nelder, J. (1989), Generalized Linear Models, Chapman and Hall. } \keyword{datasets} LearnBayes/man/cancermortality.Rd0000644000176200001440000000105311126670250016540 0ustar liggesusers\name{cancermortality} \alias{cancermortality} \docType{data} \title{Cancer mortality data} \description{ Number of cancer deaths and number at risk for 20 cities in Missouri. } \usage{ cancermortality } \format{ A data frame with 20 observations on the following 2 variables. \describe{ \item{y}{number of cancer deaths} \item{n}{number at risk} } } \source{Tsutakawa, R., Shoop, G., and Marienfeld, C. (1985), Empirical Bayes Estimation of Cancer Mortality Rates, Statistics in Medicine, 4, 201-212. } \keyword{datasets} LearnBayes/man/bayes.probit.Rd0000644000176200001440000000217511070713204015742 0ustar liggesusers\name{bayes.probit} \alias{bayes.probit} \title{Simulates from a probit binary response regression model using data augmentation and Gibbs sampling} \description{ Gives a simulated sample from the joint posterior distribution of the regression vector for a binary response regression model with a probit link and a informative normal(beta, P) prior. Also computes the log marginal likelihood when a subjective prior is used. } \usage{ bayes.probit(y,X,m,prior=list(beta=0,P=0)) } \arguments{ \item{y}{vector of binary responses} \item{X}{covariate matrix} \item{m}{number of simulations desired} \item{prior}{list with components beta, the prior mean, and P, the prior precision matrix} } \value{ \item{beta}{matrix of simulated draws of regression vector beta where each row corresponds to one draw} \item{log.marg}{simulation estimate at log marginal likelihood of the model} } \author{Jim Albert} \examples{ response=c(0,1,0,0,0,1,1,1,1,1) covariate=c(1,2,3,4,5,6,7,8,9,10) X=cbind(1,covariate) prior=list(beta=c(0,0),P=diag(c(.5,10))) m=1000 s=bayes.probit(response,X,m,prior) } \keyword{models} LearnBayes/man/gibbs.Rd0000644000176200001440000000201311411452202014413 0ustar liggesusers\name{gibbs} \alias{gibbs} \title{Metropolis within Gibbs sampling algorithm of a posterior distribution} \description{ Implements a Metropolis-within-Gibbs sampling algorithm for an arbitrary real-valued posterior density defined by the user } \usage{ gibbs(logpost,start,m,scale,...) } \arguments{ \item{logpost}{function defining the log posterior density} \item{start}{array with a single row that gives the starting value of the parameter vector} \item{m}{the number of iterations of the chain} \item{scale}{vector of scale parameters for the random walk Metropolis steps} \item{...}{data that is used in the function logpost} } \value{ \item{par}{a matrix of simulated values where each row corresponds to a value of the vector parameter} \item{accept}{vector of acceptance rates of the Metropolis steps of the algorithm} } \author{Jim Albert} \examples{ data=c(6,2,3,10) start=array(c(1,1),c(1,2)) m=1000 scale=c(2,2) s=gibbs(logctablepost,start,m,scale,data) } \keyword{models} LearnBayes/man/strikeout.Rd0000644000176200001440000000145711126712302015374 0ustar liggesusers\name{strikeout} \alias{strikeout} \docType{data} \title{Baseball strikeout data} \description{ For all professional baseball players in the 2004 season, dataset gives the number of strikeouts and at-bats when runners are in scoring position and when runners are not in scoring position. } \usage{ strikeout } \format{ A data frame with 438 observations on the following 4 variables. \describe{ \item{r}{number of strikeouts of player when runners are not in scoring position} \item{n}{number of at-bats of player when runners are not in scoring position} \item{s}{number of strikeouts of player when runners are in scoring position} \item{m}{number of at-bats of player when runners are in scoring position} } } \source{Collected from www.espn.com website.} \keyword{datasets} LearnBayes/man/birthweight.Rd0000644000176200001440000000122011126665020015653 0ustar liggesusers\name{birthweight} \alias{birthweight} \docType{data} \title{Birthweight regression study} \description{ Dobson describes a study where one is interested in predicting a baby's birthweight based on the gestational age and the baby's gender. } \usage{ birthweight } \format{ A data frame with 24 observations on the following 3 variables. \describe{ \item{age}{gestational age in weeks} \item{gender}{gender of the baby where 0 (1) is male (female)} \item{weight}{birthweight of baby in grams} } } \source{Dobson, A. (2001), An Introduction to Generalized Linear Models, New York: Chapman and Hall.} \keyword{datasets} LearnBayes/man/dmnorm.Rd0000644000176200001440000000154710616350474014652 0ustar liggesusers\name{dmnorm} \alias{dmnorm} \title{The probability density function for the multivariate normal (Gaussian) probability distribution } \description{ Computes the density of a multivariate normal distribution } \usage{ dmnorm(x, mean = rep(0, d), varcov, log = FALSE) } \arguments{ \item{x}{vector of length d or matrix with d columns, giving the coordinates of points where density is to evaluated} \item{mean}{numeric vector giving the location parameter of the distribution} \item{varcov}{a positive definite matrix representing the scale matrix of the distribution} \item{log}{a logical value; if TRUE, the logarithm of the density is to be computed} } \value{ vector of density values } \author{Jim Albert} \examples{ mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) x <- c(2,14,0) f <- dmnorm(x, mu, Sigma) } \keyword{models} LearnBayes/man/sir.Rd0000644000176200001440000000156610706731044014150 0ustar liggesusers\name{sir} \alias{sir} \title{Sampling importance resampling} \description{ Implements sampling importance resampling for a multivariate t proposal density. } \usage{ sir(logf,tpar,n,data) } \arguments{ \item{logf}{function defining logarithm of density of interest} \item{tpar}{list of parameters of multivariate t proposal density including the mean m, the scale matrix var, and the degrees of freedom df} \item{n}{number of simulated draws from the posterior} \item{data}{data and parameters used in the function logf} } \value{ matrix of simulated draws from the posterior where each row corresponds to a single draw } \author{Jim Albert} \examples{ data(cancermortality) start=c(-7,6) fit=laplace(betabinexch,start,cancermortality) tpar=list(m=fit$mode,var=2*fit$var,df=4) theta=sir(betabinexch,tpar,1000,cancermortality) } \keyword{models} LearnBayes/man/blinregpred.Rd0000644000176200001440000000201610537536302015641 0ustar liggesusers\name{blinregpred} \alias{blinregpred} \title{Simulates values of predicted response for linear regression model} \description{ Simulates draws of the predictive distribution of a future response for a linear regression model with a noninformative prior} \usage{ blinregpred(X1,theta.sample) } \arguments{ \item{X1}{matrix where each row corresponds to a covariate set} \item{theta.sample}{list with components beta, matrix of simulated draws of regression vector, and sigma, vector of simulated draws of sampling error standard deviation} } \value{ matrix where a column corresponds to the simulated draws of the predicted response for a given covariate set } \author{Jim Albert} \examples{ chirps=c(20,16.0,19.8,18.4,17.1,15.5,14.7,17.1,15.4,16.2,15,17.2,16,17,14.1) temp=c(88.6,71.6,93.3,84.3,80.6,75.2,69.7,82,69.4,83.3,78.6,82.6,80.6,83.5,76.3) X=cbind(1,chirps) m=1000 theta.sample=blinreg(temp,X,m) covset1=c(1,15) covset2=c(1,20) X1=rbind(covset1,covset2) blinregpred(X1,theta.sample) } \keyword{models} LearnBayes/man/normpostsim.Rd0000644000176200001440000000173411317415030015733 0ustar liggesusers\name{normpostsim} \alias{normpostsim} \title{Simulation from Bayesian normal sampling model} \description{ Gives a simulated sample from the joint posterior distribution of the mean and variance for a normal sampling prior with a noninformative or informative prior. The prior assumes mu and sigma2 are independent with mu assigned a normal prior with mean mu0 and variance tau2, and sigma2 is assigned a inverse gamma prior with parameters a and b. } \usage{ normpostsim(data,prior=NULL,m=1000) } \arguments{ \item{data}{vector of observations} \item{prior}{list with components mu, a vector with the prior mean and variance, and sigma2, a vector of the inverse gamma parameters} \item{m}{number of simulations desired} } \value{ \item{mu}{vector of simulated draws of normal mean} \item{sigma2}{vector of simulated draws of normal variance} } \author{Jim Albert} \examples{ data(darwin) s=normpostsim(darwin$difference) } \keyword{models} LearnBayes/man/howardprior.Rd0000644000176200001440000000111010735603724015701 0ustar liggesusers\name{howardprior} \alias{howardprior} \title{Logarithm of Howard's dependent prior for two proportions} \description{ Computes the logarithm of a dependent prior on two proportions proposed by Howard in a Statistical Science paper in 1998. } \usage{ howardprior(xy,par) } \arguments{ \item{xy}{vector of proportions p1 and p2} \item{par}{vector containing parameter values alpha, beta, gamma, delta, sigma} } \value{ value of the log posterior } \author{Jim Albert} \examples{ param=c(1,1,1,1,2) p=c(.1,.5) howardprior(p,param) } \keyword{models} LearnBayes/man/ordergibbs.Rd0000644000176200001440000000112510735604236015470 0ustar liggesusers\name{ordergibbs} \alias{ordergibbs} \title{Gibbs sampling for a hierarchical regression model} \description{ Implements Gibbs sampling for estimating a two-way table of means under a order restriction. } \usage{ ordergibbs(data,m) } \arguments{ \item{data}{data matrix with first two columns observed sample means and sample sizes} \item{m}{number of cycles of Gibbs sampling} } \value{ matrix of simulated draws of the normal means where each row represents one simulated draw } \author{Jim Albert} \examples{ data(iowagpa) m=1000 s=ordergibbs(iowagpa,m) } \keyword{models} LearnBayes/man/stanfordheart.Rd0000644000176200001440000000143111126712202016176 0ustar liggesusers\name{stanfordheart} \alias{stanfordheart} \docType{data} \title{Data from Stanford Heart Transplanation Program} \description{ Heart transplant data for 82 patients from Stanford Heart Transplanation Program} \usage{ stanfordheart } \format{ A data frame with 82 observations on the following 4 variables. \describe{ \item{survtime}{survival time in months} \item{transplant}{variable that is 1 or 0 if patient had transplant or not} \item{timetotransplant}{time a transplant patient waits for operation} \item{state}{variable that is 1 or 0 if time is censored or not} } } \source{Turnbull, B., Brown, B. and Hu, M. (1974), Survivorship analysis of heart transplant data, Journal of the American Statistical Association, 69, 74-80.} \keyword{datasets} LearnBayes/man/lbinorm.Rd0000644000176200001440000000106010735604006015001 0ustar liggesusers\name{lbinorm} \alias{lbinorm} \title{Logarithm of bivariate normal density} \description{ Computes the logarithm of a bivariate normal density } \usage{ lbinorm(xy,par) } \arguments{ \item{xy}{vector of values of two variables x and y} \item{par}{list with components m, a vector of means, and v, a variance-covariance matrix} } \value{ value of the kernel of the log density } \author{Jim Albert} \examples{ mean=c(0,0) varcov=diag(c(1,1)) value=c(1,1) param=list(m=mean,v=varcov) lbinorm(value,param) } \keyword{models} LearnBayes/man/betabinexch0.Rd0000644000176200001440000000115310735603272015702 0ustar liggesusers\name{betabinexch0} \alias{betabinexch0} \title{Log posterior of mean and precision for Binomial/beta exchangeable model} \description{ Computes the log posterior density of mean and precision for a Binomial/beta exchangeable model } \usage{ betabinexch0(theta,data) } \arguments{ \item{theta}{vector of parameter values of eta and K} \item{data}{a matrix with columns y (counts) and n (sample sizes)} } \value{ value of the log posterior} \author{Jim Albert} \examples{ n=c(20,20,20,20,20) y=c(1,4,3,6,10) data=cbind(y,n) theta=c(.1,10) betabinexch0(theta,data) } \keyword{models} LearnBayes/man/laplace.Rd0000644000176200001440000000203111411450074014733 0ustar liggesusers\name{laplace} \alias{laplace} \title{Summarization of a posterior density by the Laplace method} \description{ For a general posterior density, computes the posterior mode, the associated variance-covariance matrix, and an estimate at the logarithm at the normalizing constant. } \usage{ laplace(logpost,mode,...) } \arguments{ \item{logpost}{function that defines the logarithm of the posterior density} \item{mode}{vector that is a guess at the posterior mode} \item{...}{vector or list of parameters associated with the function logpost} } \value{ \item{mode}{current estimate at the posterior mode} \item{var}{current estimate at the associated variance-covariance matrix} \item{int}{estimate at the logarithm of the normalizing constant} \item{converge}{indication (TRUE or FALSE) if the algorithm converged} } \author{Jim Albert} \examples{ logpost=function(theta,data) { s=5 sum(-log(1+(data-theta)^2/s^2)) } data=c(10,12,14,13,12,15) start=10 laplace(logpost,start,data) } \keyword{models} LearnBayes/man/logpoissgamma.Rd0000644000176200001440000000126710735604056016217 0ustar liggesusers\name{logpoissgamma} \alias{logpoissgamma} \title{Log posterior with Poisson sampling and gamma prior} \description{ Computes the logarithm of the posterior density of a Poisson log mean with a gamma prior } \usage{ logpoissgamma(theta,datapar) } \arguments{ \item{theta}{vector of values of the log mean parameter} \item{datapar}{list with components data, vector of observations, and par, vector of parameters of the gamma prior} } \value{ vector of values of the log posterior for all values in theta } \author{Jim Albert} \examples{ data=c(2,4,3,6,1,0,4,3,10,2) par=c(1,1) datapar=list(data=data,par=par) theta=c(-1,0,1,2) logpoissgamma(theta,datapar) } \keyword{models} LearnBayes/man/normal.select.Rd0000644000176200001440000000170411063060372016107 0ustar liggesusers\name{normal.select} \alias{normal.select} \title{Selection of Normal Prior Given Knowledge of Two Quantiles} \description{ Finds the mean and standard deviation of a normal density that matches knowledge of two quantiles of the distribution. } \usage{ normal.select(quantile1, quantile2) } \arguments{ \item{quantile1}{list with components p, the value of the first probability, and x, the value of the first quantile} \item{quantile2}{list with components p, the value of the second probability, and x, the value of the second quantile} } \value{ \item{mean}{mean of the matching normal distribution} \item{sigma}{standard deviation of the matching normal distribution} } \author{Jim Albert} \examples{ # person believes the 15th percentile of the prior is 100 # and the 70th percentile of the prior is 150 quantile1=list(p=.15,x=100) quantile2=list(p=.7,x=150) normal.select(quantile1,quantile2) } \keyword{models} LearnBayes/man/election.2008.Rd0000644000176200001440000000122311126675622015541 0ustar liggesusers\name{election.2008} \alias{election.2008} \docType{data} \title{Poll data from 2008 U.S. Presidential Election} \description{ Results of recent state polls in the 2008 United States Presidential Election between Barack Obama and John McCain. } \usage{ election.2008 } \format{ A data frame with 51 observations on the following 4 variables. \describe{ \item{State}{name of the state} \item{M.pct}{percentage of poll survey for McCain} \item{O.pct}{precentage of poll survey for Obama} \item{EV}{number of electoral votes} } } \source{Data collected by author in November 2008 from www.cnn.com website.} \keyword{datasets} LearnBayes/man/mnormt.onesided.Rd0000644000176200001440000000173710537550666016473 0ustar liggesusers\name{mnormt.onesided} \alias{mnormt.onesided} \title{Bayesian test of one-sided hypothesis about a normal mean} \description{ Computes a Bayesian test of the hypothesis that a normal mean is less than or equal to a specified value} \usage{ mnormt.onesided(m0,normpar,data) } \arguments{ \item{m0}{value of the normal mean to be tested} \item{normpar}{vector of mean and standard deviation of the normal prior distribution} \item{data}{vector of sample mean, sample size, and known value of the population standard deviation} } \value{ \item{BF}{Bayes factor in support of the null hypothesis} \item{prior.odds}{prior odds of the null hypothesis} \item{post.odds}{posterior odds of the null hypothesis} \item{postH}{posterior probability of the null hypothesis} } \author{Jim Albert} \examples{ y=c(182,172,173,176,176,180,173,174,179,175) pop.s=3 data=c(mean(y),length(data),pop.s) m0=175 normpar=c(170,1000) mnormt.onesided(m0,normpar,data) } \keyword{models} LearnBayes/man/rmt.Rd0000644000176200001440000000123410554301460014141 0ustar liggesusers\name{rmt} \alias{rmt} \title{Random number generation for multivariate t} \description{ Simulates from a multivariate t distribution } \usage{ rmt(n = 1, mean = rep(0, d), S, df = Inf) } \arguments{ \item{n}{number of random numbers to be generated} \item{mean}{numeric vector giving the location parameter of the distribution} \item{S}{a positive definite matrix representing the scale matrix of the distribution} \item{df}{degrees of freedom} } \value{ matrix of n rows of random vectors } \author{Jim Albert} \examples{ mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) df <- 4 x <- rmt(10, mu, Sigma, df) } \keyword{models} LearnBayes/man/robustt.Rd0000644000176200001440000000131710537412500015042 0ustar liggesusers\name{robustt} \alias{robustt} \title{Gibbs sampling for a robust regression model} \description{ Implements Gibbs sampling for a robust t sampling model with location mu, scale sigma, and degrees of freedom v } \usage{ robustt(y,v,m) } \arguments{ \item{y}{vector of data values} \item{v}{degrees of freedom for t model} \item{m}{the number of cycles of the Gibbs sampler} } \value{ \item{mu}{vector of simulated values of mu} \item{s2}{vector of simulated values of sigma2} \item{lam}{matrix of simulated draws of lambda, where each row corresponds to a single draw} } \author{Jim Albert} \examples{ data=c(-67,-48,6,8,14,16,23,24,28,29,41,49,67,60,75) fit=robustt(data,4,1000) } \keyword{models} LearnBayes/man/normchi2post.Rd0000644000176200001440000000120610735604160015771 0ustar liggesusers\name{normchi2post} \alias{normchi2post} \title{Log posterior density for mean and variance for normal sampling} \description{ Computes the log of the posterior density of a mean M and a variance S2 when a sample is taken from a normal density and a standard noninformative prior is used. } \usage{ normchi2post(theta,data) } \arguments{ \item{theta}{vector of parameter values M and S2} \item{data}{vector containing the sample observations} } \value{ value of the log posterior } \author{Jim Albert} \examples{ parameter=c(25,5) data=c(20, 32, 21, 43, 33, 21, 32) normchi2post(parameter,data) } \keyword{models} LearnBayes/man/rejectsampling.Rd0000644000176200001440000000176310706734130016360 0ustar liggesusers\name{rejectsampling} \alias{rejectsampling} \title{Rejecting sampling using a t proposal density} \description{ Implements a rejection sampling algorithm for a probability density using a multivariate t proposal density } \usage{ rejectsampling(logf,tpar,dmax,n,data) } \arguments{ \item{logf}{function that defines the logarithm of the density of interest} \item{tpar}{list of parameters of t proposal density including the mean m, scale matrix var, and degrees of freedom df} \item{dmax}{logarithm of the rejection sampling constant} \item{n}{number of simulated draws from proposal density} \item{data}{data and or parameters used in the function logf} } \value{ matrix of simulated draws from density of interest } \author{Jim Albert} \examples{ data(cancermortality) start=c(-7,6) fit=laplace(betabinexch,start,cancermortality) tpar=list(m=fit$mode,var=2*fit$var,df=4) theta=rejectsampling(betabinexch,tpar,-569.2813,1000,cancermortality) } \keyword{models} LearnBayes/man/poisson.gamma.mix.Rd0000644000176200001440000000210010735604416016707 0ustar liggesusers\name{poisson.gamma.mix} \alias{poisson.gamma.mix} \title{Computes the posterior for Poisson sampling and a mixture of gammas prior} \description{ Computes the parameters and mixing probabilities for a Poisson sampling problem where the prior is a discrete mixture of gamma densities. } \usage{ poisson.gamma.mix(probs,gammapar,data) } \arguments{ \item{probs}{vector of probabilities of the gamma components of the prior} \item{gammapar}{matrix where each row contains the shape and rate parameters for a gamma component of the prior} \item{data}{list with components y, vector of counts, and t, vector of time intervals} } \value{ \item{probs}{vector of probabilities of the gamma components of the posterior} \item{gammapar}{matrix where each row contains the shape and rate parameters for a gamma component of the posterior} } \author{Jim Albert} \examples{ probs=c(.5, .5) gamma.par1=c(1,1) gamma.par2=c(10,2) gammapar=rbind(gamma.par1,gamma.par2) y=c(1,3,2,4,10); t=c(1,1,1,1,1) data=list(y=y,t=t) poisson.gamma.mix(probs,gammapar,data)} \keyword{models} LearnBayes/man/binomial.beta.mix.Rd0000644000176200001440000000175210735603470016653 0ustar liggesusers\name{binomial.beta.mix} \alias{binomial.beta.mix} \title{Computes the posterior for binomial sampling and a mixture of betas prior} \description{ Computes the parameters and mixing probabilities for a binomial sampling problem where the prior is a discrete mixture of beta densities. } \usage{ binomial.beta.mix(probs,betapar,data) } \arguments{ \item{probs}{vector of probabilities of the beta components of the prior} \item{betapar}{matrix where each row contains the shape parameters for a beta component of the prior} \item{data}{vector of number of successes and number of failures} } \value{ \item{probs}{vector of probabilities of the beta components of the posterior} \item{betapar}{matrix where each row contains the shape parameters for a beta component of the posterior} } \author{Jim Albert} \examples{ probs=c(.5, .5) beta.par1=c(15,5) beta.par2=c(10,10) betapar=rbind(beta.par1,beta.par2) data=c(20,15) binomial.beta.mix(probs,betapar,data) } \keyword{models} LearnBayes/man/careertraj.setup.Rd0000644000176200001440000000154310556700130016622 0ustar liggesusers\name{careertraj.setup} \alias{careertraj.setup} \title{Setup for Career Trajectory Application} \description{ Setups the data matrices for the use of WinBUGS in the career trajectory application. } \usage{ careertraj.setup(data) } \arguments{ \item{data}{data matrix for ballplayers with variables Player, Year, Age, G, AB, R, H, X2B, X3B, HR, RBI, BB, SO} } \value{ \item{player.names}{vector of player names} \item{y}{matrix of home runs for players where a row corresponds to the home runs for a player during all the years of his career} \item{n}{matrix of AB-SO for all players} \item{x}{matrix of ages for all players for all years of their careers} \item{T}{vector of number of seasons for all players} \item{N}{number of players} } \author{Jim Albert} \examples{ data(sluggerdata) careertraj.setup(sluggerdata) } \keyword{models} LearnBayes/man/regroup.Rd0000644000176200001440000000071110735604634015033 0ustar liggesusers\name{regroup} \alias{regroup} \title{Collapses a matrix by summing over rows} \description{ Collapses a matrix by summing over a specific number of rows } \usage{ regroup(data,g) } \arguments{ \item{data}{a matrix} \item{g}{a positive integer beween 1 and the number of rows of data} } \value{ reduced matrix found by summing over rows } \author{Jim Albert} \examples{ data=matrix(c(1:20),nrow=4,ncol=5) g=2 regroup(data,2) } \keyword{models} LearnBayes/man/birdextinct.Rd0000644000176200001440000000143111126670044015660 0ustar liggesusers\name{birdextinct} \alias{birdextinct} \docType{data} \title{Bird measurements from British islands} \description{ Measurements on breedings pairs of landbird species were collected from 16 islands about Britain over several decades. } \usage{ birdextinct } \format{ A data frame with 62 observations on the following 5 variables. \describe{ \item{species}{name of bird species} \item{time}{average time of extinction on the islands} \item{nesting}{average number of nesting pairs} \item{size}{size of the species, 1 or 0 if large or small} \item{status}{staus of the species, 1 or 0 if resident or migrant} } } \source{ Pimm, S., Jones, H., and Diamond, J. (1988), On the risk of extinction, American Naturalists, 132, 757-785. } \keyword{datasets} LearnBayes/man/bayes.model.selection.Rd0000644000176200001440000000171211054044050017521 0ustar liggesusers\name{bayes.model.selection} \alias{bayes.model.selection} \title{Bayesian regression model selection using G priors} \description{ Using Zellner's G priors, computes the log marginal density for all possible regression models } \usage{ bayes.model.selection(y, X, c, constant=TRUE) } \arguments{ \item{y}{vector of response values} \item{X}{matrix of covariates} \item{c}{parameter of the G prior} \item{constant}{logical variable indicating if a constant term is in the matrix X} } \value{ \item{mod.prob}{data frame specifying the model, the value of the log marginal density and the value of the posterior model probability} \item{converge}{logical vector indicating if the laplace algorithm converged for each model} } \author{Jim Albert} \examples{ data(birdextinct) logtime=log(birdextinct$time) X=cbind(1,birdextinct$nesting,birdextinct$size,birdextinct$status) bayes.model.selection(logtime,X,100) } \keyword{models} LearnBayes/man/marathontimes.Rd0000644000176200001440000000065611126700330016214 0ustar liggesusers\name{marathontimes} \alias{marathontimes} \docType{data} \title{Marathon running times} \description{ Running times in minutes for twenty male runners between the ages 20 and 29 who ran the New York Marathon. } \usage{ marathontimes } \format{ A data frame with 20 observations on the following 1 variable. \describe{ \item{time}{running time} } } \source{www.nycmarathon.org website.} \keyword{datasets} LearnBayes/man/jeter2004.Rd0000644000176200001440000000140511126741172014762 0ustar liggesusers\name{jeter2004} \alias{jeter2004} \docType{data} \title{Hitting data for Derek Jeter} \description{ Batting data for the baseball player Derek Jeter for all 154 games in the 2004 season.} \usage{ jeter2004 } \format{ A data frame with 154 observations on the following 10 variables. \describe{ \item{Game}{the game number} \item{AB}{the number of at-bats} \item{R}{the number of runs scored} \item{H}{the number of hits} \item{X2B}{the number of doubles} \item{X3B}{the number of triples} \item{HR}{the number of home runs} \item{RBI}{the number of runs batted in} \item{BB}{the number of walks} \item{SO}{the number of strikeouts} } } \source{Collected from game log data from www.retrosheet.org.} \keyword{datasets} LearnBayes/man/weibullregpost.Rd0000644000176200001440000000127510735605002016413 0ustar liggesusers\name{weibullregpost} \alias{weibullregpost} \title{Log posterior of a Weibull proportional odds model for survival data} \description{ Computes the log posterior density of (log sigma, mu, beta) for a Weibull proportional odds regression model } \usage{ weibullregpost(theta,data) } \arguments{ \item{theta}{vector of parameter values log sigma, mu, and beta} \item{data}{data matrix with columns survival time, censoring variable, and covariate matrix} } \value{ value of the log posterior } \author{Jim Albert} \examples{ data(chemotherapy) attach(chemotherapy) d=cbind(time,status,treat-1,age) theta=c(-.6,11,.6,0) weibullregpost(theta,d) } \keyword{models} LearnBayes/man/pdisc.Rd0000644000176200001440000000111710506615730014446 0ustar liggesusers\name{pdisc} \alias{pdisc} \title{Posterior distribution for a proportion with discrete priors} \description{ Computes the posterior distribution for a proportion for a discrete prior distribution. } \usage{ pdisc(p, prior, data) } \arguments{ \item{p}{vector of proportion values} \item{prior}{vector of prior probabilities} \item{data}{vector consisting of number of successes and number of failures} } \value{ vector of posterior probabilities } \author{Jim Albert} \examples{ p=c(.2,.25,.3,.35) prior=c(.25,.25,.25,.25) data=c(5,10) pdisc(p,prior,data) } \keyword{models} LearnBayes/man/triplot.Rd0000644000176200001440000000120610735457374015054 0ustar liggesusers\name{triplot} \alias{triplot} \title{Plot of prior, likelihood and posterior for a proportion} \description{ For a proportion problem with a beta prior, plots the prior, likelihood and posterior on one graph. } \usage{ triplot(prior,data,where="topright") } \arguments{ \item{prior}{vector of parameters for beta prior} \item{data}{vector consisting of number of successes and number of failures} \item{where}{the location of the legend for the plot} } \author{Jim Albert} \examples{ prior=c(3,10) # proportion has a beta(3, 10) prior data=c(10,6) # observe 10 successes and 6 failures triplot(prior,data) } \keyword{models} LearnBayes/man/bfindep.Rd0000644000176200001440000000130010616136652014750 0ustar liggesusers\name{bfindep} \alias{bfindep} \title{Bayes factor against independence assuming alternatives close to independence} \description{ Computes a Bayes factor against independence for a two-way contingency table assuming a "close to independence" alternative model} \usage{ bfindep(y,K,m) } \arguments{ \item{y}{matrix of counts} \item{K}{Dirichlet precision hyperparameter} \item{m}{number of simulations} } \value{ \item{bf}{value of the Bayes factor against hypothesis of independence} \item{nse}{estimate of the simulation standard error of the computed Bayes factor} } \author{Jim Albert} \examples{ y=matrix(c(10,4,6,3,6,10),c(2,3)) K=20 m=1000 bfindep(y,K,m) } \keyword{models} LearnBayes/man/bayes.influence.Rd0000644000176200001440000000201710705645352016422 0ustar liggesusers\name{bayes.influence} \alias{bayes.influence} \title{Observation sensitivity analysis in beta-binomial model} \description{ Computes probability intervals for the log precision parameter K in a beta-binomial model for all "leave one out" models using sampling importance resampling } \usage{ bayes.influence(theta,data) } \arguments{ \item{theta}{matrix of simulated draws from the posterior of (logit eta, log K)} \item{data}{matrix with columns of counts and sample sizes} } \value{ \item{summary}{vector of 5th, 50th, 95th percentiles of log K for complete sample posterior} \item{summary.obs}{matrix where the ith row contains the 5th, 50th, 95th percentiles of log K for posterior when the ith observation is removed} } \author{Jim Albert} \examples{ data(cancermortality) start=array(c(-7,6),c(1,2)) fit=laplace(betabinexch,start,cancermortality) tpar=list(m=fit$mode,var=2*fit$var,df=4) theta=sir(betabinexch,tpar,1000,cancermortality) intervals=bayes.influence(theta,cancermortality) } \keyword{models} LearnBayes/man/breastcancer.Rd0000644000176200001440000000141111126665214015776 0ustar liggesusers\name{breastcancer} \alias{breastcancer} \docType{data} \title{Survival experience of women with breast cancer under treatment} \description{ Collett (1994) describes a study to evaluate the effectiveness of a histochemical marker in predicting the survival experience of women with breast cancer. } \usage{ breastcancer } \format{ A data frame with 45 observations on the following 3 variables. \describe{ \item{time}{survival time in months} \item{status}{censoring indicator where 1 (0) indicates a complete (censored) survival time} \item{stain}{indicates by a 0 (1) if tumor was negatively (positively) stained} } } \source{Collett, D. (1994), Modelling Survival Data in Medical Research, London: Chapman and Hall.} \keyword{datasets} LearnBayes/man/groupeddatapost.Rd0000644000176200001440000000140610735603644016557 0ustar liggesusers\name{groupeddatapost} \alias{groupeddatapost} \title{Log posterior of normal parameters when data is in grouped form} \description{ Computes the log posterior density of (M,log S) for normal sampling where the data is observed in grouped form } \usage{ groupeddatapost(theta,data) } \arguments{ \item{theta}{vector of parameter values M and log S} \item{data}{list with components int.lo, a vector of left endpoints, int.hi, a vector of right endpoints, and f, a vector of bin frequencies} } \value{ value of the log posterior } \author{Jim Albert} \examples{ int.lo=c(-Inf,10,15,20,25) int.hi=c(10,15,20,25,Inf) f=c(2,5,8,4,2) data=list(int.lo=int.lo,int.hi=int.hi,f=f) theta=c(20,1) groupeddatapost(theta,data) } \keyword{models} LearnBayes/man/mnormt.twosided.Rd0000644000176200001440000000170210616137316016502 0ustar liggesusers\name{mnormt.twosided} \alias{mnormt.twosided} \title{Bayesian test of a two-sided hypothesis about a normal mean} \description{ Bayesian test that a normal mean is equal to a specified value using a normal prior} \usage{ mnormt.twosided(m0, prob, t, data) } \arguments{ \item{m0}{value of the mean to be tested } \item{prob}{prior probability of the hypothesis} \item{t}{vector of values of the prior standard deviation under the alternative hypothesis} \item{data}{vector containing the sample mean, the sample size, and the known value of the population standard deviation} } \value{ \item{bf}{vector of values of the Bayes factor in support of the null hypothesis} \item{post}{vector of posterior probabilities of the null hypothesis} } \author{Jim Albert} \examples{ m0=170 prob=.5 tau=c(.5,1,2,4,8) samplesize=10 samplemean=176 popsd=3 data=c(samplemean,samplesize,popsd) mnormt.twosided(m0,prob,tau,data) } \keyword{models} LearnBayes/man/hiergibbs.Rd0000644000176200001440000000132010735603670015302 0ustar liggesusers\name{hiergibbs} \alias{hiergibbs} \title{Gibbs sampling for a hierarchical regression model} \description{ Implements Gibbs sampling for estimating a two-way table of means under a hierarchical regression model. } \usage{ hiergibbs(data,m) } \arguments{ \item{data}{data matrix with columns observed sample means, sample sizes, and values of two covariates} \item{m}{number of cycles of Gibbs sampling} } \value{ \item{beta}{matrix of simulated values of regression vector} \item{mu}{matrix of simulated values of cell means} \item{var}{vector of simulated values of second-stage prior variance} } \author{Jim Albert} \examples{ data(iowagpa) m=1000 s=hiergibbs(iowagpa,m) } \keyword{models} LearnBayes/man/iowagpa.Rd0000644000176200001440000000137211126677760015010 0ustar liggesusers\name{iowagpa} \alias{iowagpa} \docType{data} \title{Admissions data for an university} \description{ Students at a major university are categorized with respect to their high school rank and their ACT score. For each combination of high school rank and ACT score, one records the mean grade point average (GPA). } \usage{ iowagpa } \format{ A data frame with 40 observations on the following 4 variables. \describe{ \item{gpa}{mean grade point average} \item{n}{sample size} \item{HSR}{high school rank} \item{ACT}{act score} } } \source{Albert, J. (1994), A Bayesian approach to estimation of GPA's of University of Iowa freshmen under order restrictions, Journal of Educational Statistics, 19, 1-22.} \keyword{datasets} LearnBayes/man/chemotherapy.Rd0000644000176200001440000000172611126670444016044 0ustar liggesusers\name{chemotherapy} \alias{chemotherapy} \docType{data} \title{Chemotherapy treatment effects on ovarian cancer} \description{ Edmunson et al (1979) studied the effect of different chemotherapy treatments following surgical treatment of ovarian cancer. } \usage{ chemotherapy } \format{ A data frame with 26 observations on the following 5 variables. \describe{ \item{patient}{patient number} \item{time}{survival time in days following treatment} \item{status}{indicates if time is censored (0) or actually observed (1)} \item{treat}{control group (0) or treatment group (1)} \item{age}{age of the patient} } } \source{Edmonson, J., Felming, T., Decker, D., Malkasian, G., Jorgensen, E., Jefferies, J.,Webb, M., and Kvols, L. (1979), Different chemotherapeutic sensitivities and host factors affecting prognosis in advanced ovarian carcinoma versus minimal residual disease, Cancer Treatment Reports, 63, 241-247. } \keyword{datasets} LearnBayes/man/mycontour.Rd0000644000176200001440000000152011052362304015372 0ustar liggesusers\name{mycontour} \alias{mycontour} \title{Contour plot of a bivariate density function} \description{ For a general two parameter density, draws a contour graph where the contour lines are drawn at 10 percent, 1 percent, and .1 percent of the height at the mode. } \usage{ mycontour(logf,limits,data,...) } \arguments{ \item{logf}{function that defines the logarithm of the density} \item{limits}{limits (xlo, xhi, ylo, yhi) where the graph is to be drawn} \item{data}{vector or list of parameters associated with the function logpost} \item{...}{further arguments to pass to contour} } \value{ A contour graph of the density is drawn } \author{Jim Albert} \examples{ m=array(c(0,0),c(2,1)) v=array(c(1,.6,.6,1),c(2,2)) normpar=list(m=m,v=v) mycontour(lbinorm,c(-4,4,-4,4),normpar) } \keyword{models} LearnBayes/man/rwmetrop.Rd0000644000176200001440000000205711411451514015221 0ustar liggesusers\name{rwmetrop} \alias{rwmetrop} \title{Random walk Metropolis algorithm of a posterior distribution} \description{ Simulates iterates of a random walk Metropolis chain for an arbitrary real-valued posterior density defined by the user } \usage{ rwmetrop(logpost,proposal,start,m,...) } \arguments{ \item{logpost}{function defining the log posterior density} \item{proposal}{a list containing var, an estimated variance-covariance matrix, and scale, the Metropolis scale factor} \item{start}{vector containing the starting value of the parameter} \item{m}{the number of iterations of the chain} \item{...}{data that is used in the function logpost} } \value{ \item{par}{a matrix of simulated values where each row corresponds to a value of the vector parameter} \item{accept}{the acceptance rate of the algorithm} } \author{Jim Albert} \examples{ data=c(6,2,3,10) varcov=diag(c(1,1)) proposal=list(var=varcov,scale=2) start=array(c(1,1),c(1,2)) m=1000 s=rwmetrop(logctablepost,proposal,start,m,data) } \keyword{models} LearnBayes/man/puffin.Rd0000644000176200001440000000133011126701356014627 0ustar liggesusers\name{puffin} \alias{puffin} \docType{data} \title{Bird measurements from British islands} \description{ Measurements on breedings of the common puffin on different habits at Great Island, Newfoundland. } \usage{ puffin } \format{ A data frame with 38 observations on the following 5 variables. \describe{ \item{Nest}{nesting frequency (burrows per 9 square meters)} \item{Grass}{grass cover (percentage)} \item{Soil}{mean soil depth (in centimeters)} \item{Angle}{angle of slope (in degrees)} \item{Distance}{distance from cliff edge (in meters)} } } \source{Peck, R., Devore, J., and Olsen, C. (2005), Introduction to Statistics And Data Analysis, Thomson Learning.} \keyword{datasets} LearnBayes/man/calculus.grades.Rd0000644000176200001440000000112711126666230016424 0ustar liggesusers\name{calculus.grades} \alias{calculus.grades} \docType{data} \title{Calculus grades dataset} \description{ Grades and other variables collected for a sample of calculus students. } \usage{ calculus.grades } \format{ A data frame with 100 observations on the following 3 variables. \describe{ \item{grade}{indicates if student received a A or B in class} \item{prev.grade}{indicates if student received a A in prerequisite math class} \item{act}{score on the ACT math test} } } \source{Collected by a colleague of the author at his university.} \keyword{datasets} LearnBayes/man/histprior.Rd0000644000176200001440000000123711316472766015404 0ustar liggesusers\name{histprior} \alias{histprior} \title{Density function of a histogram distribution} \description{ Computes the density of a probability distribution defined on a set of equal-width intervals } \usage{ histprior(p,midpts,prob) } \arguments{ \item{p}{vector of values for which density is to be computed} \item{midpts}{vector of midpoints of the intervals} \item{prob}{vector of probabilities of the intervals} } \value{ vector of values of the probability density } \author{Jim Albert} \examples{ midpts=c(.1,.3,.5,.7,.9) prob=c(.2,.2,.4,.1,.1) p=seq(.01,.99,by=.01) plot(p,histprior(p,midpts,prob),type="l") } \keyword{models} LearnBayes/man/bfexch.Rd0000644000176200001440000000135210735603446014611 0ustar liggesusers\name{bfexch} \alias{bfexch} \title{Logarithm of integral of Bayes factor for testing homogeneity of proportions} \description{ Computes the logarithm of the integral of the Bayes factor for testing homogeneity of a set of proportions } \usage{ bfexch(theta,datapar) } \arguments{ \item{theta}{value of the logit of the prior mean hyperparameter} \item{datapar}{list with components data, matrix with columns y (counts) and n (sample sizes), and K, prior precision hyperparameter} } \value{ value of the logarithm of the integral } \author{Jim Albert} \examples{ y=c(1,3,2,4,6,4,3) n=c(10,10,10,10,10,10,10) data=cbind(y,n) K=20 datapar=list(data=data,K=K) theta=1 bfexch(theta,datapar) } \keyword{models} LearnBayes/man/election.Rd0000644000176200001440000000144211126675312015150 0ustar liggesusers\name{election} \alias{election} \docType{data} \title{Florida election data} \description{ For each of the Florida counties in the 2000 presidential election, the number of votes for George Bush, Al Gore, and Pat Buchanan is recorded. Also the number of votes for the minority candidate Ross Perot in the 1996 presidential election is recorded. } \usage{ election } \format{ A data frame with 67 observations on the following 5 variables. \describe{ \item{county}{name of Florida county} \item{perot}{number of votes for Ross Perot in 1996 election} \item{gore}{number of votes for Al Gore in 2000 election} \item{bush}{number of votes for George Bush in 2000 election} \item{buchanan}{number of votes for Pat Buchanan in 2000 election} } } \keyword{datasets} LearnBayes/man/simcontour.Rd0000644000176200001440000000156110735604720015552 0ustar liggesusers\name{simcontour} \alias{simcontour} \title{Simulated draws from a bivariate density function on a grid} \description{ For a general two parameter density defined on a grid, simulates a random sample. } \usage{ simcontour(logf,limits,data,m) } \arguments{ \item{logf}{function that defines the logarithm of the density} \item{limits}{limits (xlo, xhi, ylo, yhi) that cover the joint probability density} \item{data}{vector or list of parameters associated with the function logpost} \item{m}{size of simulated sample} } \value{ \item{x}{vector of simulated draws of the first parameter} \item{y}{vector of simulated draws of the second parameter} } \author{Jim Albert} \examples{ m=array(c(0,0),c(2,1)) v=array(c(1,.6,.6,1),c(2,2)) normpar=list(m=m,v=v) s=simcontour(lbinorm,c(-4,4,-4,4),normpar,1000) plot(s$x,s$y) } \keyword{models} LearnBayes/man/indepmetrop.Rd0000644000176200001440000000216711411451674015701 0ustar liggesusers\name{indepmetrop} \alias{indepmetrop} \title{Independence Metropolis independence chain of a posterior distribution} \description{ Simulates iterates of an independence Metropolis chain with a normal proposal density for an arbitrary real-valued posterior density defined by the user} \usage{ indepmetrop(logpost,proposal,start,m,...) } \arguments{ \item{logpost}{function defining the log posterior density} \item{proposal}{a list containing mu, an estimated mean and var, an estimated variance-covariance matrix, of the normal proposal density} \item{start}{vector containing the starting value of the parameter} \item{m}{the number of iterations of the chain} \item{...}{data that is used in the function logpost} } \value{ \item{par}{a matrix of simulated values where each row corresponds to a value of the vector parameter} \item{accept}{the acceptance rate of the algorithm} } \author{Jim Albert} \examples{ data=c(6,2,3,10) proposal=list(mu=array(c(2.3,-.1),c(2,1)),var=diag(c(1,1))) start=array(c(0,0),c(1,2)) m=1000 fit=indepmetrop(logctablepost,proposal,start,m,data) } \keyword{models} LearnBayes/man/ctable.Rd0000644000176200001440000000101410537537516014603 0ustar liggesusers\name{ctable} \alias{ctable} \title{Bayes factor against independence using uniform priors} \description{ Computes a Bayes factor against independence for a two-way contingency table assuming uniform prior distributions} \usage{ ctable(y,a) } \arguments{ \item{y}{matrix of counts} \item{a}{matrix of prior hyperparameters} } \value{ value of the Bayes factor against independence } \author{Jim Albert} \examples{ y=matrix(c(10,4,6,3,6,10),c(2,3)) a=matrix(rep(1,6),c(2,3)) ctable(y,a) } \keyword{models} LearnBayes/man/pdiscp.Rd0000644000176200001440000000136210735604320014625 0ustar liggesusers\name{pdiscp} \alias{pdiscp} \title{Predictive distribution for a binomial sample with a discrete prior} \description{ Computes predictive distribution for number of successes of future binomial experiment with a discrete distribution for the proportion. } \usage{ pdiscp(p, probs, n, s) } \arguments{ \item{p}{vector of proportion values} \item{probs}{vector of probabilities} \item{n}{size of future binomial sample} \item{s}{vector of number of successes for future binomial experiment} } \value{ vector of predictive probabilities for the values in the vector s } \author{Jim Albert} \examples{ p=c(.1,.2,.3,.4,.5,.6,.7,.8,.9) prob=c(0.05,0.10,0.10,0.15,0.20,0.15,0.10,0.10,0.05) n=10 s=0:10 pdiscp(p,prob,n,s) } \keyword{models} LearnBayes/man/sluggerdata.Rd0000644000176200001440000000152411126710140015636 0ustar liggesusers\name{sluggerdata} \alias{sluggerdata} \docType{data} \title{Hitting statistics for ten great baseball players} \description{ Career hitting statistics for ten great baseball players } \usage{ sluggerdata } \format{ A data frame with 199 observations on the following 13 variables. \describe{ \item{Player}{names of the ballplayer} \item{Year}{season played} \item{Age}{age of the player during the season} \item{G}{games played} \item{AB}{number of at-bats} \item{R}{number of runs scored} \item{H}{number of hits} \item{X2B}{number of doubles} \item{X3B}{number of triples} \item{HR}{number of home runs} \item{RBI}{runs batted in} \item{BB}{number of base on balls} \item{SO}{number of strikeouts} } } \source{Sean Lahman's baseball database from www.baseball1.com.} \keyword{datasets} LearnBayes/man/rtruncated.Rd0000644000176200001440000000171410735604660015526 0ustar liggesusers\name{rtruncated} \alias{rtruncated} \title{Simulates from a truncated probability distribution} \description{ Simulates a sample from a truncated distribution where the functions for the cdf and inverse cdf are available. } \usage{ rtruncated(n,lo,hi,pf,qf,...) } \arguments{ \item{n}{size of simulated sample} \item{lo}{low truncation point} \item{hi}{high truncation point} \item{pf}{function containing cdf of untruncated distribution} \item{qf}{function containing inverse cdf of untruncated distribution} \item{...}{parameters used in the functions pf and qf} } \value{ vector of simulated draws from distribution} \author{Jim Albert} \examples{ # want a sample of 10 from normal(2, 1) distribution truncated below by 3 n=10 lo=3 hi=Inf rtruncated(n,lo,hi,pnorm,qnorm,mean=2,sd=1) # want a sample of 20 from beta(2, 5) distribution truncated to (.3, .8) n=20 lo=0.3 hi=0.8 rtruncated(n,lo,hi,pbeta,qbeta,2,5) } \keyword{models} LearnBayes/man/normpostpred.Rd0000644000176200001440000000166510735604174016114 0ustar liggesusers\name{normpostpred} \alias{normpostpred} \title{Posterior predictive simulation from Bayesian normal sampling model} \description{ Given simulated draws from the posterior from a normal sampling model, outputs simulated draws from the posterior predictive distribution of a statistic of interest. } \usage{ normpostpred(parameters,sample.size,f=min) } \arguments{ \item{parameters}{list of simulated draws from the posterior where mu contains the normal mean and sigma2 contains the normal variance} \item{sample.size}{size of sample of future sample} \item{f}{function defining the statistic} } \value{ simulated sample of the posterior predictive distribution of the statistic} \author{Jim Albert} \examples{ # finds posterior predictive distribution of the min statistic of a future sample of size 15 data(darwin) s=normpostsim(darwin$difference) sample.size=15 sim.stats=normpostpred(s,sample.size,min) } \keyword{models} LearnBayes/man/bradley.terry.post.Rd0000644000176200001440000000133111126476654017125 0ustar liggesusers\name{bradley.terry.post} \alias{bradley.terry.post} \title{Log posterior of a Bradley Terry random effects model} \description{ Computes the log posterior density of the talent parameters and the log standard deviation for a Bradley Terry model with normal random effects } \usage{ bradley.terry.post(theta,data) } \arguments{ \item{theta}{vector of talent parameters and log standard deviation} \item{data}{data matrix with columns team1, team2, wins by team1, and wins by team2} } \value{value of the log posterior} \author{Jim Albert} \examples{ data(baseball.1964) team.strengths=rep(0,10) log.sigma=0 bradley.terry.post(c(team.strengths,log.sigma),baseball.1964) } \keyword{models} LearnBayes/man/transplantpost.Rd0000644000176200001440000000123210735604766016451 0ustar liggesusers\name{transplantpost} \alias{transplantpost} \title{Log posterior of a Pareto model for survival data} \description{ Computes the log posterior density of (log tau, log lambda, log p) for a Pareto model for survival data } \usage{ transplantpost(theta,data) } \arguments{ \item{theta}{vector of parameter values of log tau, log lambda, and log p} \item{data}{data matrix with columns survival time, transplant indicator, time to transplant, and censoring indicator} } \value{ value of the log posterior } \author{Jim Albert} \examples{ data(stanfordheart) theta=c(0,3,-1) transplantpost(theta,stanfordheart) } \keyword{models} LearnBayes/man/darwin.Rd0000644000176200001440000000076211126676440014641 0ustar liggesusers\name{darwin} \alias{darwin} \docType{data} \title{Darwin's data on plants} \description{ Fifteen differences of the heights of cross and self fertilized plants quoted by Fisher (1960)} \usage{ darwin } \format{ A data frame with 15 observations on the following 1 variable. \describe{ \item{difference}{difference of heights of two types of plants} } } \source{Fisher, R. (1960), Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd.} \keyword{datasets}