\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.
<<fig=TRUE,echo=TRUE>>=
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.
<<fig=TRUE,echo=TRUE>>=
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.
<<fig=TRUE,echo=TRUE>>=
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}
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