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\title{The Beta-Binomial Model}
\author{Dr David A. Stephens}
\institute{
  Department of Mathematics \& Statistics\\
 McGill University\\
 Montreal, QC, Canada.

  \vspace{0.1 in}

  \texttt{david.stephens@mcgill.ca}\\
  \texttt{\texttt{www.math.mcgill.ca/dstephens/SISCER2020/}}
}

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numberstyle=\tiny, stepnumber=1, numbersep=5pt}


\begin{document}

%\part{NHANES Example}

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<<setup, cache=FALSE, echo=FALSE>>=
library(knitr)
# global chunk options
opts_chunk$set(cache=TRUE, autodep=TRUE, size = "tiny")
options(scipen=5)
options(repos=c(CRAN="https://cloud.r-project.org/"))
@

%\frame{\titlepage}

\begin{frame}[fragile,allowframebreaks]\frametitle{The Beta-Binomial Model}

In the exchangeable binary case, we must have
\[
p_{Y_i}(y_i;\theta) = \theta^{y_i} (1-\theta)^{1-y_i}
\]
for each $0 \leq \theta \leq 1$; however different choices of $\pi_0(d \theta)$ lead to different models for the joint distribution of $Y_1,\ldots,Y_n$.

\framebreak

Suppose that
\[
\pi_0(\theta) \equiv Beta(\alpha_0,\beta_0).
\]
for $\alpha_0, \beta_0 > 0$.   That is, the prior is \emph{continuous}
\[
\pi_0(\theta) = \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \theta^{\alpha_0 - 1} (1-\theta)^{\beta_0 - 1} \qquad 0 < \theta < 1.
\]
and we are assuming that $0 < \theta < 1$ with probability 1.

\framebreak

This model presumes that

\begin{itemize}

\item $\theta \sim Beta(\alpha_0,\beta_0)$,

\item $Y_1,\ldots,Y_n | \theta \sim Bernoulli(\theta)$,

\end{itemize}
with $Y_1,\ldots,Y_n$ conditionally independent.  Let
\[
S_n = \sum_{i=1}^n Y_i \qquad s_n = \sum_{i=1}^n y_i.
\]

\framebreak

By a standard marginalization calculation, we may conclude that the prior predictive takes the form
\begin{align*}
p_{Y_1,\ldots,Y_n} \left(y_1,\ldots,y_n\right) & = \int_{0}^{1}\left\{
\prod_{i=1}^{n}\theta^{y_{i}}\left( 1-\theta \right)
^{1-y_{i}}\right\} \pi_0\left(\theta \right) \ d \theta \\[6pt]
& = \int_{0}^{1} \theta^{s_{n}}\left( 1-\theta \right)
^{n-s_{n}} \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \theta^{\alpha_0 - 1} (1-\theta)^{\beta_0 - 1} \ d \theta \\[6pt]
& = \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \int_{0}^{1} \theta^{s_{n} + \alpha_0 - 1}\left( 1-\theta \right)
^{n-s_{n}+\beta_0 - 1}\ d \theta \\[6pt]
& = \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \frac{\Gamma(\alpha_n)\Gamma(\beta_n) }{\Gamma(\alpha_n + \beta_n)}
\end{align*}
say.

\framebreak

This follows as the integrand is \emph{proportional to a Beta pdf}, where
\[
\alpha_n = s_n+\alpha_0 \qquad \beta_n = n-s_n+\beta_0
\]
that is, in light of the data, the prior \emph{hyperparameters} $(\alpha_0,\beta_0)$ are updated to $(\alpha_n,\beta_n)$ to define the prior predictive marginal model.

\framebreak

There are $n+1$ distinct values for $p_{Y_1,\ldots,Y_n} \left(y_1,\ldots,y_n\right)$ defined by the possible values of $s_n$. We have for $s_n \in \{0,1,\ldots,n\}$,
\[
\Pr[S_n = s_n] = \dbinom{n}{s_n} p_{Y_1,\ldots,Y_n} \left(y_1,\ldots,y_n\right)
\]
where $(y_1,\ldots,y_n)$ is any binary sequence for which
\[
\sum_{i=1}^n y_i = s_n.
\]
This distribution for $S_n$ is termed the \emph{Beta-Binomial} model.

\framebreak

We have for the posterior density
\begin{align*}
\pi_n(\theta) & \ \propto \ \left\{\prod_{i=1}^n \theta^{y_i} (1-\theta)^{1-y_i} \right\} \theta^{\alpha_0 - 1} (1-\theta)^{\beta_0 - 1}\\[6pt]
& \ = \ \theta^{s_n+\alpha_0 - 1} (1-\theta)^{n-s_n + \beta_0 -1 } \\[6pt]
& \ = \ \theta^{\alpha_n - 1} (1-\theta)^{\beta_n - 1}
\end{align*}
Thus the posterior distribution is \emph{also} a Beta distribution.

\framebreak

That is,
\[
\pi_n(\theta) \equiv Beta(s_n+\alpha_0,n-s_n+\beta_0) \equiv Beta(\alpha_n,\beta_n)
\]
say, so that
\[
\pi_n(\theta) = \frac{\Gamma(\alpha_n + \beta_n) }{\Gamma(\alpha_n)\Gamma(\beta_n)} \theta^{\alpha_n - 1} (1-\theta)^{\beta_n -1 }.
\]





\framebreak

Simulation: $n=8$, $\alpha_0 = 2,\beta_0 = 1.5$.

<<Ex01,comment='+', fig.width=6, fig.height=6, fig.align='center',out.width='3in'>>=
set.seed(213)
n<-8
sim.exch01<-function(nv,a0v,b0v){  #Sample the exchangeable variables.
    thv<-rbeta(1,a0v,b0v)
    Yv<-rbinom(nv,1,thv)
}
al0<-2.0
be0<-1.5
nreps<-10000
Ymat<-t(replicate(nreps,sim.exch01(n,al0,be0)))  #2000 replicate draws of Y
@

\framebreak

We have that for $i=1,\ldots,n$, by iterated expectation
\[
\E_{Y_i}[Y_i] = \E_{\pi_0}[\E_{Y_i|\theta}[Y_i|\theta]] = \E_{\pi_0}[\theta]  = \frac{\alpha_0}{\alpha_0+\beta_0}
\]
and
\[
\E_{Y_i}[Y_i^2] = \E_{\pi_0}[\E_{Y_i|\theta}[Y_i^2|\theta]] = \E_{\pi_0}[\E_{Y_i|\theta}[Y_i|\theta]] = \E_{\pi_0}[\theta]  = \frac{\alpha_0}{\alpha_0+\beta_0}
\]
so
\[
\Var_{Y_i}[Y_i] = \frac{\alpha_0}{\alpha_0+\beta_0} - \left(\frac{\alpha_0}{\alpha_0+\beta_0} \right)^2.
\]

\framebreak

<<Ex02,comment='+', fig.width=6, fig.height=6, fig.align='center'>>=
muBetaBinomY<-al0/(al0+be0)
sigBetaBinomY<-sqrt(muBetaBinomY*(1-muBetaBinomY))
c(muBetaBinomY,sigBetaBinomY^2)

apply(Ymat,2,mean)
apply(Ymat,2,var)
@

\framebreak

For the covariances, using iterated expectation we have
\begin{align*}
\E_{Y_i,Y_j}[Y_i Y_j]  & = \E_{\pi_0} \left[ \E_{Y_i,Y_j|\theta}[Y_i Y_j|\theta] \right] \\[6pt]
& = \E_{\pi_0} \left[ \E_{Y_i|\theta}[Y_i|\theta] \E_{Y_j|\theta}[Y_j|\theta] \right]
\end{align*}
as $Y_i$ and $Y_j$ are conditionally independent given $\theta$.

\framebreak

Thus, as $\E_{Y_i|\theta}[Y_i|\theta] = \theta$ for each $i$, we have
\[
\E_{Y_i,Y_j}[Y_i Y_j] = \E_{\pi_0} [ \theta^2 ] = \Var_{\pi_0}[\theta] + \{ \E_{\pi_0}[\theta]\}^2
\]
which, by properties of the Beta distribution, is equal to
\[
\frac{\alpha_0 \beta_0}{(\alpha_0+\beta_0)^2 (\alpha_0+\beta_0+1)} + \left (\frac{\alpha_0}{\alpha_0+\beta_0} \right)^2
\]

\framebreak
Hence, the covariance is
\[
\Cov_{Y_i,Y_j}[Y_i,Y_j] = \frac{\alpha_0 \beta_0}{(\alpha_0+\beta_0)^2 (\alpha_0+\beta_0+1)}
\]
and the correlation
\[
\frac{\Cov_{Y_i,Y_j}[Y_i,Y_j]}{\Var_{Y_i}[Y_i]}
\]
is computed using the previous variance expression.


<<Ex03,comment='+', fig.width=6, fig.height=6, fig.align='center'>>=
covBetaBinomY<-al0*be0/((al0+be0)^2*(al0+be0+1))
corBetaBinomY<-covBetaBinomY/sigBetaBinomY^2
corBetaBinomY
round(cor(Ymat),3)
@

\framebreak
Note: the presumed \emph{data generating model} here is the exchangeable form given by
\[
p_{Y_1,\ldots,Y_n} \left(y_1,\ldots,y_n\right)
\]
We can generate $Y$s from this model directly by using the Beta-Binomial model functions in \texttt{R}.

\medskip
We generate using the \texttt{extraDistr} package
\begin{itemize}
  \item $S_n$ using the function \texttt{rbbinom}
  \item $Y_1,\ldots,Y_n|S_n=s_n$ by selecting uniformly from the set of binary sequences which have $s_n$ ones.
\end{itemize}

\framebreak
The function \texttt{dbbinom} computes the Beta-Binomial pmf
\[
\Pr[S_n = s_n] = \dbinom{n}{s_n} \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \frac{\Gamma(\alpha_0+s_n)\Gamma(\beta_0+n-s_n) }{\Gamma(\alpha_0 + \beta_0 + n)}
\]
for $s_n \in \{0,1,\ldots,n\}$.

\begin{itemize}
  \item $n$ is the \emph{size} parameter,
  \item $(\alpha_0,\beta_0)$ are the \emph{shape} parameters.
\end{itemize}

\framebreak
<<Ex04,comment='+', fig.width=6, fig.height=6, fig.align='center'>>=
library(extraDistr)
Sn<-rbbinom(nreps,size=n,alpha=al0,beta=be0)
table(Sn)/nreps
round(dbbinom(0:n,size=n,alpha=al0,beta=be0),4)
@


\framebreak
<<Ex05,comment='+', fig.width=6, fig.height=6, fig.align='center'>>=
Ymat2<-matrix(0,nrow=nreps,ncol=n)
for(i in 1:nreps){
    Ymat2[i,sample(1:n,size=Sn[i],rep=FALSE)]<-1
}
apply(Ymat2,2,mean)
apply(Ymat2,2,var)
round(cor(Ymat2),3)
@

\framebreak
Let
\[
R_n = \frac{S_n}{n} = \frac{1}{n} \sum_{i=1}^n Y_i.
\]
Then for $r \in \{0,1/n,2/n,\ldots,(n-1)/n, 1\}$
\begin{align*}
\Pr[R_n = r] & \equiv \Pr[S_n = n r] \\[6pt]
& = \dbinom{n}{n r} \frac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \frac{\Gamma(\alpha_0+ n r)\Gamma(\beta_0+n-nr) }{\Gamma(\alpha_0 + \beta_0 + n)}
\end{align*}

We can compute this distribution for $n=10,20,\ldots,100$, and plot the cdf.

\framebreak
<<Ex06,comment='+', fig.width=9, fig.height=6, fig.align='center',out.width='4.2in'>>=
par(mar=c(4,4,2,0))
plot(c(0,1),c(0,1),type='n',ylab='Prob',xlab='r')
for(j in 1:10){
    n<-10*j
    rv<-c(0:n)/n
    pv<-pbbinom(0:n,size=n,alpha=al0,beta=be0)
    lines(rv,pv,type='s')
}
xv<-seq(0,1,length=1000)
lines(xv,pbeta(xv,al0,be0),col='red',lwd=2)
legend(0,1,c(expression(paste('CDF of ',pi[0](theta)))),col='red',lty=1,lwd=2)
@

\framebreak
For the posterior predictive
\begin{align*}
p_{Y_{n+1},\ldots,Y_{n+m} |Y_1,\ldots,Y_n}& \left( y_{n+1},\ldots,y_{n+m} |y_1,\ldots,y_n\right) \\[12pt]
& = \dfrac{p_{Y_{1},\ldots,Y_{n+m}} \left( y_{1},\ldots,y_{n+m}\right)}{p_{Y_{1},\ldots,Y_{n}} \left( y_{1},\ldots,y_{n}\right)} \\[12pt]
& = \dfrac{\dfrac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \dfrac{\Gamma(\alpha_{n+m})\Gamma(\beta_{n+m}) }{\Gamma(\alpha_{n+m} + \beta_{n+m})}}{\dfrac{\Gamma(\alpha_0 + \beta_0)}{ \Gamma(\alpha_0) \Gamma(\beta_0)} \dfrac{\Gamma(\alpha_n)\Gamma(\beta_n) }{\Gamma(\alpha_n + \beta_n)}}
\end{align*}
where
\[
\alpha_{n+m} = s_{n+m} + \alpha_0 \qquad \beta_{n+m} =  (n+m) - s_{n+m} + \beta_0
\]

\framebreak

After cancellation and rearrangement, we have that
\begin{align*}
p_{Y_{n+1},\ldots,Y_{n+m} |Y_1,\ldots,Y_n}& \left( y_{n+1},\ldots,y_{n+m} |y_1,\ldots,y_n\right) \\[12pt]
& = \dfrac{\Gamma(\alpha_n + \beta_n)}{\Gamma(\alpha_n)\Gamma(\beta_n)}\dfrac{\Gamma(\alpha_{n+m})\Gamma(\beta_{n+m}) }{\Gamma(\alpha_{n+m} + \beta_{n+m})} \\[6pt]
& = \dfrac{\Gamma(\alpha_n + \beta_n)}{\Gamma(\alpha_n)\Gamma(\beta_n)}\dfrac{\Gamma(\alpha_{n} + s_{n+1,n+m})\Gamma(\beta_{n}+m-s_{n+1,n+m}) }{\Gamma(\alpha_{n} + \beta_{n}+m)} \\[6pt]
\end{align*}
where
\[
s_{n+1,n+m} = \sum_{i=n+1}^{n+m} y_i.
\]

\framebreak
If we examine the distribution of the random variable
\[
S_{n+1,n+m} = \sum_{i=n+1}^{n+m} Y_i
\]
conditional on $Y_1,\ldots,Y_n$, we see that
\begin{align*}
\Pr[S_{n+1,n+m} = s|&Y_1=y_n,\ldots,Y_n=y_n] \\[6pt]
& = \dbinom{m}{s} \dfrac{\Gamma(\alpha_n + \beta_n)}{\Gamma(\alpha_n)\Gamma(\beta_n)}\dfrac{\Gamma(\alpha_{n} + s)\Gamma(\beta_{n}+m-s) }{\Gamma(\alpha_{n} + \beta_{n}+m)}
\end{align*}
which we identify as the Beta-Binomial distribution with size $m$ and shape parameters $(\alpha_n,\beta_n)$.

\framebreak

We can study the conditional distribution of
\[
R_{n+1,n+m} = \frac{S_{n+1,n+m}}{m} = \frac{1}{m} \sum_{i=n+1}^{n+m} Y_i
\]
given $Y_1 = y_1,\ldots,Y_n = y_n$ as $m=10,20,\ldots,100$ for any fixed $(y_1,\ldots,y_n)$.

\medskip
Note that $\alpha_n$ and $\beta_n$ only depend on $s_n$, so we may simulate that directly using the \texttt{rbbinom} function.

\framebreak

<<Ex07,comment='+', fig.width=9, fig.height=6, fig.align='center',out.width='4.2in'>>=
#Data
n<-10
Sn<-rbbinom(1,size=n,alpha=al0,beta=be0)
aln<-al0+Sn
ben<-be0+n-Sn
print(Sn)

par(mar=c(4,4,2,0))
plot(c(0,1),c(0,1),type='n',ylab='Prob',xlab='r')
for(j in 1:10){
    m<-10*j
    rv<-c(0:m)/m
    pv<-pbbinom(0:m,size=m,alpha=aln,beta=ben)
    lines(rv,pv,type='s')
}
xv<-seq(0,1,length=1000)
lines(xv,pbeta(xv,aln,ben),col='red',lwd=2)
legend(0,1,c(expression(paste('CDF of ',pi[n](theta)))),col='red',lty=1,lwd=2)
@
\end{frame}



\end{document}