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\title{Normal approximations}
\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}

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<<setup, cache=FALSE, echo=FALSE>>=
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options(scipen=5)
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@

<<funcs,echo=FALSE>>=
myrnd <- function(x,d) {formatC(x, format="f", digits=d)}
library(shape)
library(diagram)
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@
%\frame{\titlepage}

\begin{frame}[fragile,allowframebreaks]\frametitle{Using the Normal Approximation in Bayesian inference}

We consider a Taylor approximation of the function 
\[
\ell(y;\theta) = \log f_Y(y;\theta) \qquad \theta \in \R^d
\]
with respect to $\theta$ around $\theta_0$.
    \begin{align*}\label{eq:Taylor1}
    \ell(y;\theta) & \bumpeq \ell(y;\theta_0) + \dot{\ell}(y;\theta_0)^\top (\theta-\theta_0) + \frac{1}{2} (\theta-\theta_0)^\top \ddot{\ell} (y;\theta_0) (\theta-\theta_0) 
    \end{align*}
    where
    \[
    \dot{\ell} (y;\theta) = \frac{\partial \ell(y;\theta)}{\partial \theta} \qquad \ddot{\ell} (y;\theta) = \frac{\partial^2 \ell(y;\theta)}{\partial \theta \partial \theta^\top} 
    \]
are $(d \times 1)$ and $(d \times d)$ respectively.

    \framebreak

    Evaluating for each of $y_1,\ldots,y_n$ and summing the result, we have
   \[
    \ell_n(\theta) \bumpeq \ell_n(\theta_0) + \dot{\ell}_n(\theta_0)^\top (\theta-\theta_0) + \frac{1}{2} (\theta-\theta_0)^\top \ddot{\ell}_n (\theta_0) (\theta-\theta_0)
    \]
    Assuming that for $n$ large we can replace $\theta_0$ by $\widehat{\theta}_n$, we have
    \begin{align*}
    \ell_n(\theta) & \bumpeq \ell_n(\widehat{\theta}_n) + \dot{\ell}_n(\widehat{\theta}_n)^\top (\theta-\widehat{\theta}_n) + \frac{1}{2} (\theta-\widehat{\theta}_n)^\top \ddot{\ell}_n (\widehat{\theta}_n) (\theta-\widehat{\theta}_n)\\[6pt]
    & = \ell_n(\widehat{\theta}_n) + \frac{1}{2} (\theta-\widehat{\theta}_n)^\top \ddot{\ell}_n (\widehat{\theta}_n) (\theta-\widehat{\theta}_n)
\end{align*}
as $\ell_n(\widehat{\theta}_n) = \mathbf{0}_d$.

\framebreak

This tells us that the likelihood can be approximated in the vicinity of $\widehat{\theta}_n$ by a quadratic: 
\begin{align*}
\ell_n(\widehat{\theta}_n) - \ell_n(\theta) & = -\frac{1}{2} (\theta-\widehat{\theta}_n)^\top \ddot{\ell}_n (\widehat{\theta}_n) (\theta-\widehat{\theta}_n) \\[6pt]
& = \frac{1}{2} (\theta-\widehat{\theta}_n)^\top \left\{-\ddot{\ell}_n (\widehat{\theta}_n)\right\} (\theta-\widehat{\theta}_n)
\end{align*}
where this quantity is non-negative, as $\widehat{\theta}_n$ is a maximizer of $\ell_n$.
\framebreak

    A quadratic approximation for the likelihood at $\theta$ around $\widehat \theta_n$ is
    \begin{align*}
    \exp\{\ell_n(\theta) \} & \bumpeq   \exp\{\ell_n(\widehat \theta_n) \} \exp\left\{\frac{1}{2} (\widehat \theta_n-\theta)^\top \ddot{\ell}_n (\widehat \theta_n) (\widehat \theta_n -\theta) \right\} \\[6pt]
    & \varpropto \exp\left\{-\frac{1}{2} (\theta - \widehat \theta_n)^\top \{ - \ddot{\ell}_n (\widehat \theta_n) \} (\theta - \widehat \theta_n) \right\}.
    \end{align*}
Note that we can also write the right-hand side as proportional to
\[
\exp\left\{-\frac{1}{2} (\widehat \theta_n-\theta)^\top \{ - \ddot{\ell}_n (\widehat \theta_n) \} (\widehat \theta_n - \theta) \right\}.
\]

\framebreak
Treating $\theta$ as a proposed true data generating parameter, and assuming that $n$ is large enough so that $\widehat \theta_n$ is close to $\theta$, this essentially corresponds to the \emph{frequentist} assumption that
\[
\widehat \theta_n | \theta \approxsim Normal_{d}\left(\theta,\left\{ - \ddot{\ell}_n (\theta) \right\}^{-1} \right)
\]    
or equivalently that
\[
\sqrt{n} (\widehat \theta_n - \theta) \approxsim Normal_{d}\left(\mathbf{0}_d,\left\{ - \frac{1}{n} \ddot{\ell}_n (\theta) \right\}^{-1} \right).
\]
which allows the construction of frequentist \emph{confidence intervals}. 

\framebreak
In the frequentist calculation, it is often assumed that as $n$ is large
\[
- \frac{1}{n} \ddot{\ell}_n (\theta) \bumpeq \mathcal{I}_n(\theta)
\]
where
\[
\mathcal{I}_n(\theta) = \E_{Y|\theta} \left[ - \left. \frac{\partial^2 \ell(Y;\bt)}{\partial \bt \ \partial \bt^\top} \right|_{\bt = \theta};\theta \right]
\]
is the Fisher information.

\framebreak

In the \emph{Bayesian} version, the posterior can be approximated by noting that
\[
\pi_n(\theta) \propto \exp\{\ell_n(\theta)\} \pi_0(\theta)
\]
and assuming that $n$ is large so that $\pi_0(\theta)$ can be treated as constant in the neighbourhood of $\widehat \theta_n$, so that
\[
\pi_n(\theta) \equiv Normal_{d}\left(\widehat \theta_n,\left\{ - \ddot{\ell}_n (\widehat \theta_n) \right\}^{-1} \right).
\]
which allows the construction of Bayesian \emph{credible intervals}.
\end{frame}


\begin{frame}[fragile,allowframebreaks]\frametitle{Example}

Suppose that a Bayesian model is built for realizations for an infinitely exchangeable binary sequence.

\medskip

\begin{itemize}
  \item $Y_i|\theta \sim Bernoulli(\theta)$
  \item $f_Y(y;\theta) = \theta^y (1-\theta)^{1-y}$ for $y=0,1$
  \item Log-mass function and derivatives:
  \begin{align*}
  \ell(y;\theta) & = y \log \theta + (1-y) \log (1-\theta)\\[6pt]
  \dot{\ell}(y;\theta) & = \frac{y}{\theta} - \frac{1-y}{1-\theta} \\[6pt]
  \ddot{\ell}(y;\theta) & = -\frac{y}{\theta^2} - \frac{1-y}{(1-\theta)^2}
  \end{align*}
  
\framebreak

  \item Derivative of the log-likelihood
  \[
  \dot{\ell}_n(\theta) = \frac{s_n}{\theta} - \frac{t_n}{(1-\theta)}
  \]
  where
  \[
  s_n = \sum_{i=1}^n y_i \qquad t_n = \sum_{i=1}^n (1-y_i)
  \]
  yields the maximum likelihood estimate
  \[
  \widehat{\theta}_n = \frac{s_n}{s_n + t_n} = \frac{s_n}{n}
  \]
  
\framebreak

\item This leads to
\begin{align*}
\ddot{\ell}_n(\widehat{\theta}_n) & = - \left[ \frac{s_n}{\widehat{\theta}_n^2} + \frac{t_n}{(1-\widehat{\theta}_n)^2} \right] =  - \left[ \frac{n \widehat{\theta}_n }{\widehat{\theta}_n^2} + \frac{n(1-\widehat{\theta}_n)}{(1-\widehat{\theta}_n)^2} \right] \\[6pt]
& = - \frac{n}{\widehat{\theta}_n (1-\widehat{\theta}_n)}
\end{align*}

\item We also have that
\begin{align*}
\mathcal{I}_n(\theta) & = \E_{Y|\theta} \left[ \frac{Y}{\theta^2} + \frac{1-Y}{(1-\theta)^2} ;\theta \right] = \frac{\theta}{\theta^2} + \frac{1-\theta}{(1-\theta)^2} \\[6pt]
& = \frac{1}{\theta (1-\theta)}
\end{align*}

\framebreak

\item $(1-\gamma)$\% confidence interval:  recall that a confidence interval is the \emph{random} interval $[L_n,U_n]$ such that
\[
P_{Y|\theta}[ \theta \in [L_n,U_n] ; \theta] = 1-\gamma.
\]
that is, \emph{for each given $\theta$},
\[
P_{Y|\theta}[ L_n \leq \theta \leq R_n ; \theta] = 1-\gamma.
\]
To obtain the interval, we usually construct a \emph{pivotal quantity}, that is some quantity
\[
T_n(\theta)
\]
whose distribution does not depend on $\theta$.

\framebreak
Here we use
\[
T_n(\theta) = \dfrac{\widehat{\theta}_n - \theta}{\sqrt{\dfrac{\widehat{\theta}_n (1-\widehat{\theta}_n)}{n}}}
\]
and deduce from the asymptotic theory that for any $\theta$
\[
T_n(\theta) \approxsim Normal(0,1).
\]
Hence for an \emph{equal-tailed} interval, we have
  \[
[L_n, U_n] : \quad  \widehat{\theta}_n \pm z_{1-\gamma/2} \sqrt{\frac{\widehat{\theta}_n (1-\widehat{\theta}_n)}{n}}
  \]
where $z_{a}$ is the $a$th quantile of the standard Normal distribution.

\framebreak

Note that this is a \emph{frequentist} statement:

\begin{itemize}
  \item Suppose we \emph{know} $\theta$; 
  \item \emph{if} the data are generated from $f_Y(y;\theta)$, 
  \item \emph{then} the interval $[L_n,U_n]$ will contain $\theta$ with probability $1-\gamma$ under the $f_Y(y;\theta)$ model.
  
  \medskip
  
  \item The quantities $L_n$ and $U_n$ are functions of the data random variables through the \emph{estimator};
  \item we estimate the interval by computing numerical values $l_n$ and $u_n$ which are functions of the \emph{estimate}.
\end{itemize}

\medskip

Specifically, this is \emph{not} a probability statement about the unknown $\theta$; in the frequentist formulation, such a probability statement is not possible, as the unknown $\theta$ is \emph{not a random quantity}.


\framebreak


\item $(1-\gamma)$\% credible interval:  by the approximation theory
\[
\pi_n(\theta) \bumpeq Normal\left(\widehat \theta_n, \frac{\widehat{\theta}_n (1-\widehat{\theta}_n)}{n} \right).
\] 
Hence we can make the \emph{Bayesian} statement for the unknown $\theta$: in light of the data and the model specification that
\[
P_{\pi_n}[\theta \in [l_n,u_n]] = 1-\gamma.
\]
where $l_n$ and $u_n$ are quantities computed from the \emph{estimate}.

\end{itemize}

\framebreak

But if we consider the standardized parameter
\[
\phi = \frac{\sqrt{n}(\theta - \widehat{\theta}_n)}{\sqrt{\widehat{\theta}_n (1-\widehat{\theta}_n)}}
\]
then it is clear that
\[
\pi_n(\phi) \bumpeq Normal(0,1)
\]
revealing that the Bayesian credible interval will be the same as the frequentist interval.

\framebreak

Numerical example:  $n=20$, $\theta_0=0.65$.
<<fig.height=5.4,fig.width=8>>=
set.seed(39)
n<-20
th0<-0.65
y<-rbinom(n,1,th0)
sn<-sum(y)
tn<-sum(1-y)
c(sn,tn)  #Data
th.n<-sn/(sn+tn)
th.n      #MLE
log.like<-function(xv,sv,tv){
    return(sv*log(xv)+tv*log(1-xv))
}
thv<-seq(0.50,0.85,by=0.001)
ll.n<-log.like(thv,sn,tn)
par(mar=c(4,4,4,2))
plot(thv,ll.n,type='l',
    xlab=expression(theta),ylab=expression(l[n](theta)))
title('Log-likelihood')
abline(v=th.n,lty=2,col='red')
@

<<fig.height=5.4,fig.width=8>>=
#Quadratic approximation
max.val<-log.like(th.n,sn,tn)
qln<-max.val-0.5*(thv-th.n)^2/(th.n*(1-th.n)/n)
par(mar=c(4,4,4,2))
plot(thv,ll.n,type='l',
    xlab=expression(theta),ylab=expression(l[n](theta)))
title('Log-likelihood and quadratic approximation')
lines(thv,qln,col='red')
abline(v=th.n,lty=2,col='red')
legend(0.5,-12.3,c('Exact','Approximate'),lty=1,col=c('black','red'))
@

Assuming a $\pi_0(\theta) \equiv Uniform(0,1)$, we have that
\[
\pi_n(\theta) \equiv Beta(s_n+1,t_n+1)
\]
and using the quadratic approximation we have
\[
\pi_n(\theta) \bumpeq Normal(\widehat{\theta}_n,\widehat{\theta}_n (1-\widehat{\theta})/n)
\]

\framebreak

<<fig.height=5.4,fig.width=8>>=
#Posterior and approximation
thfull<-seq(0,1,by=0.001)
pi.n<-dbeta(thfull,sn+1,tn+1)
qlnfull<--0.5*(thfull-th.n)^2/(th.n*(1-th.n)/n)
const<-1/sqrt(2*pi*(th.n*(1-th.n)/n))
par(mar=c(4,4,4,2))
plot(thfull,pi.n,type='l',
    xlab=expression(theta),ylab=expression(pi[n](theta)))
title('Posterior and quadratic approximation')
lines(thfull,exp(qlnfull)*const,col='red')
abline(v=th.n,lty=2,col='red')
legend(0,4,c('Exact','Approximate'),lty=1,col=c('black','red'))
@


Equal-tail intervals: for $\gamma=0.05$, and a 95\% interval

<<>>=
#Exact
qbeta(c(0.025,0.975),sn+1,tn+1)

#Approximate
qnorm(c(0.025,0.975),th.n,sqrt(th.n*(1-th.n)/n))

@
that is
\begin{align*}
  \textrm{Exact} : & \ [\Sexpr{myrnd(qbeta(c(0.025,0.975),sn+1,tn+1),4)}] \\[6pt]
  \textrm{Approximate} : & \ [\Sexpr{myrnd(qnorm(c(0.025,0.975),th.n,sqrt(th.n*(1-th.n)/n)),4)}] 
\end{align*}

\framebreak

Larger sample size:  $n=200$, $\theta_0=0.65$.
<<fig.height=5.4,fig.width=8,echo=FALSE>>=
set.seed(39)
n<-200
th0<-0.65
y<-rbinom(n,1,th0)
sn<-sum(y)
tn<-sum(1-y)
c(sn,tn)  #Data
th.n<-sn/(sn+tn)
th.n      #MLE
thv<-seq(0.50,0.85,by=0.001)
ll.n<-log.like(thv,sn,tn)
par(mar=c(4,4,4,2))
plot(thv,ll.n,type='l',
    xlab=expression(theta),ylab=expression(l[n](theta)))
title('Log-likelihood')
abline(v=th.n,lty=2,col='red')
@

<<fig.height=5.4,fig.width=8,echo=FALSE>>=
#Quadratic approximation
max.val<-log.like(th.n,sn,tn)
qln<-max.val-0.5*(thv-th.n)^2/(th.n*(1-th.n)/n)
par(mar=c(4,4,4,2))
plot(thv,ll.n,type='l',
    xlab=expression(theta),ylab=expression(l[n](theta)))
title('Log-likelihood and quadratic approximation')
lines(thv,qln,col='red')
abline(v=th.n,lty=2,col='red')
legend(0.5,-140,c('Exact','Approximate'),lty=1,col=c('black','red'))
@


<<fig.height=5.4,fig.width=8,echo=FALSE>>=
#Posterior and approximation
thfull<-seq(0,1,by=0.001)
pi.n<-dbeta(thfull,sn+1,tn+1)
qlnfull<--0.5*(thfull-th.n)^2/(th.n*(1-th.n)/n)
const<-1/sqrt(2*pi*(th.n*(1-th.n)/n))
par(mar=c(4,4,4,2))
plot(thfull,pi.n,type='l',
    xlab=expression(theta),ylab=expression(pi[n](theta)))
title('Posterior and quadratic approximation')
lines(thfull,exp(qlnfull)*const,col='red')
abline(v=th.n,lty=2,col='red')
legend(0,12,c('Exact','Approximate'),lty=1,col=c('black','red'))
@


Equal-tail intervals: for $\gamma=0.05$, and a 95\% interval

<<echo=FALSE>>=
#Exact
qbeta(c(0.025,0.975),sn+1,tn+1)

#Approximate
qnorm(c(0.025,0.975),th.n,sqrt(th.n*(1-th.n)/n))

@
that is
\begin{align*}
  \textrm{Exact} : & \ [\Sexpr{myrnd(qbeta(c(0.025,0.975),sn+1,tn+1),4)}] \\[6pt]
  \textrm{Approximate} : & \ [\Sexpr{myrnd(qnorm(c(0.025,0.975),th.n,sqrt(th.n*(1-th.n)/n)),4)}] 
\end{align*}


\end{frame}

\end{document}