%\VignetteIndexEntry{Beta-Binomial Distribution}
%\VignetteKeywords{TailRank,beta binomial distribution}
%\VignetteDepends{TailRank}
%\VignettePackage{TailRank}
\documentclass{article}

\usepackage{hyperref}

\newcommand{\Rfunction}[1]{{\texttt{#1}}}
\newcommand{\Robject}[1]{{\texttt{#1}}}
\newcommand{\Rpackage}[1]{{\textit{#1}}}

\title{The Beta-Binomial Distribution}
\author{Kevin R. Coombes}

\begin{document}

\maketitle
\tableofcontents

\section{Introduction}

This vignette documents the beta-binomial distribution, which is
included in the \Rpackage{TailRank} package 
<<lib>>=
library(TailRank)
@ 
Mathematically, the beta-binomial distribution has parameters $N$,
$u$, and $v$ that determine the density function
$$ {N \choose x} Beta(x+u, N-x+v)/Beta(u,v). $$
Statistically, one can think of this distribution as a hierarchical
model, starting with a binomial distribution $Binom(x, N, \theta)$ whose
success parameter $\theta$ comes from a beta distribution, $\theta
\sim Beta(x, u, v)$. This distribution has a larger variance than the
binomial distribution with a fixed (known) parameter $\theta$.

We provide the usual set of functions to implement a distribution:
\begin{itemize}
\item \Rfunction{dbb} is the distribution function.
\item \Rfunction{pbb} is the cumulative distribution function.
\item \Rfunction{qbb} is the quantile function.
\item \Rfunction{rbb} is the random-sample function.
\end{itemize}  

We start by comparing the distributions of a binomial distribution and
a beta-binomial distribution.
<<compare>>=
N <- 20
u <- 3
v <- 10
p <- u/(u+v)
x <- 0:N
y <- dbinom(x, N, p)
yy <- dbb(x, N, u, v)

@ 

<<fig=TRUE>>=
barplot(t(matrix(c(y, yy), ncol=2)), beside=TRUE, col=c("blue", "red"))
legend("topright", c("Binomial", "Beta-Binomial"), col=c("blue", "red"), pch=15)
@ 

Now we sample data from each of these distributions.
<<>>=
set.seed(561662)
r <- rbinom(1000, N, p)
rr <- rbb(1000, N, u, v)
mean(r)
mean(rr)
var(r)
var(rr)
sd(r)
sd(rr)
@ 

\end{document}
