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\title{Propensity Score Methods, Models and Adjustment}
\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 = "scriptsize")
options(scipen=5)
options(repos=c(CRAN="https://cloud.r-project.org/"))
@

%\frame{\titlepage}

\begin{frame}[fragile,allowframebreaks]\frametitle{The Metropolis-Hastings Algorithm}

The Metropolis-Hastings algorithm is a Markov chain method for exploring a target distribution $\pi$.

\medskip
The algorithm implements a sequence of moves that

\begin{enumerate}[label=(\roman*)]
  \item propose a new value $z$ from a proposal distribution $q$ that can depend on the current point of the chain $x$; usually this takes the form of a conditional density
      \[
      q(x \rightarrow z) = q(z|x)
      \]
      then
  \item accept $z$ with probability
  \[
  \alpha(x,z) = \min \left\{1,\frac{\pi(z)}{\pi(x)} \frac{q(z \rightarrow x)}{q(x \rightarrow z)} \right\}
  \]
  If the proposal is not accepted, the Markov chain stays at $x$.
  
\end{enumerate}

\framebreak

Suppose that
\[
\pi(x_1,x_2) = \frac{1}{\pi} \qquad x_1^2+x_2^2 < 1
\]
and zero otherwise.  This distribution is Uniform on the unit disk.

\medskip
Suppose that
\[
q((x_1,x_2) \rightarrow (z_1,z_2)) = \frac{1}{\delta^2}
\]
for
\[
\qquad x_1 - \delta/2 < z_1 < x_1 + \delta/2, x_2 - \delta/2 < z_2 < x_2 + \delta/2
\]
that is, the proposal is Uniform on the square of edge length $\delta$ centered at $(x_1,x_2)$.

\framebreak
Then we have that
\[
\alpha((x_1,x_2) , (z_1,z_2)) = 1
\]
provided 
\[
z_1^2+z_2^2 < 1 
\]
otherwise $\alpha((x_1,x_2) , (z_1,z_2)) = 0$.

<<MC1,comment='+', fig.width=6, fig.height=6, fig.align='center',out.width='3in',echo=TRUE>>=
set.seed(32)
nits<-100000
xmat<-matrix(0,nrow=nits,ncol=2)
delta<-0.1
for(iter in 2:nits){
        z<-xmat[iter-1,]+runif(2,-delta/2,delta/2)
        if(sum(z^2) < 1){
            xmat[iter,]<-z
        }else{
            xmat[iter,]<-xmat[iter-1,]
        }
}
th<-seq(-pi,pi,by=0.01)
xv<-cos(th)
yv<-sin(th)
par(pty='s',mar=c(4,4,4,2))
plot(xmat[1:100,],pch=19,cex=0.5,main=paste('100 steps'),
        xlim=range(-1,1),ylim=range(-1,1),
        xlab=expression(x[1]),ylab=expression(x[2]))
lines(xmat[1:100,])
lines(xv,yv,col='red',lwd=2)
@

\framebreak

<<MC2,comment='+', fig.width=6, fig.height=6, fig.align='center',out.width='3in',echo=FALSE>>=
par(pty='s',mar=c(4,4,4,2))
plot(xmat[1:10000,],pch=19,cex=0.5,main=paste('10000 steps'),
        xlim=range(-1,1),ylim=range(-1,1),
        xlab=expression(x[1]),ylab=expression(x[2]))
lines(xv,yv,col='red',lwd=2)
@

\framebreak

<<MC3,comment='+', fig.width=6, fig.height=6, fig.align='center',out.width='3in',echo=FALSE>>=
par(pty='s',mar=c(4,4,4,2))
plot(xmat,pch=19,cex=0.5,main=paste('100000 steps'),
        xlim=range(-1,1),ylim=range(-1,1),
        xlab=expression(x[1]),ylab=expression(x[2]))
lines(xv,yv,col='red',lwd=2)
@

\framebreak

Suppose that
\[
\pi(x_1,x_2) = c \exp\{-\beta h(x_1,x_2)\} \qquad -5 < x_1,x_2 < 5 
\]
and zero otherwise, where
\[
h(x_1,x_2) = (x_1^2+x_2-11)^2 + (x_1+x_2^2-7)^2
\]
and $\beta>0$ is a fixed constant, say $\beta = 0.01$.

\framebreak
<<P0,comment='+', fig.width=8, fig.height=6, out.width='3in', fig.align='center'>>=
test.func<-function(x1,x2,bv=1){
    hval<-(x1^2+x2-11)^2 + (x1+x2^2-7)^2
    fval<-exp(-bv*hval)
    return(fval)
}
f <- Vectorize(test.func,vectorize.args=c("x1","x2"))
x1v<-x2v<-seq(-5,5,by=0.1)
be<-0.01;dmat<-outer(x1v,x2v,f,bv=be)
@

<<P1,comment='+', fig.width=8, fig.height=6,out.width='3.5in', fig.align='center'>>=
library(fields,quietly=TRUE)
par(pty='s',mar=c(4,3,2,2))
cols<-c("blue","lightblue","white","yellow","orange","red")
colfunc <- colorRampPalette(cols)
image.plot(x1v,x2v,dmat,col=colfunc(100),
           xlab=expression(x[1]),ylab=expression(x[2]),cex.axis=0.8)
contour(x1v,x2v,dmat,add=T,nlevels=20)
@

\framebreak

Suppose that as before
\[
q((x_1,x_2) \rightarrow (z_1,z_2)) = \frac{1}{\delta^2}
\]
for
\[
\qquad x_1 - \delta/2 < z_1 < x_1 + \delta/2, x_2 - \delta/2 < z_2 < x_2 + \delta/2
\]
that is, the proposal is Uniform on the square of edge length $\delta$ centered at $(x_1,x_2)$.

\framebreak
Then we have that
\[
\alpha((x_1,x_2) , (z_1,z_2)) = \min \left\{ 1, \frac{\exp\{-\beta h(z_1,z_2)\}}{\exp\{-\beta h(x_1,x_2)\}} \right\}
\]
provided 
\[
-5 < z_1, z_2 < 5
\]
otherwise $\alpha((x_1,x_2) , (z_1,z_2)) = 0$.

<<MC4,comment='+', fig.width=6, fig.height=6,echo=TRUE>>=
set.seed(312)
nits<-10000
xmat<-matrix(0,nrow=nits,ncol=2)
delta<-0.5
be<-0.01
for(iter in 2:nits){
        z<-xmat[iter-1,]+runif(2,-delta/2,delta/2)
        if(abs(z[1])>5 | abs(z[2]) > 5){
            xmat[iter,]<-xmat[iter-1,]
        }else{
            u<-runif(1)
            num<-log(test.func(z[1],z[2],be))
            den<-log(test.func(xmat[iter-1,1],xmat[iter-1,2],be))
            if(log(u) < num-den){
                xmat[iter,]<-z
            }else{
                xmat[iter,]<-xmat[iter-1,]
            }
        }
}
@

\framebreak

<<MC5,comment='+', fig.width=8, fig.height=6, fig.align='center',out.width='3.5in',echo=FALSE>>=
par(pty='s',mar=c(4,3,2,2))
cols<-c("blue","lightblue","white","yellow","orange","red")
colfunc <- colorRampPalette(cols)
image.plot(x1v,x2v,dmat,col=colfunc(100),
           xlab=expression(x[1]),ylab=expression(x[2]),cex.axis=0.8)
contour(x1v,x2v,dmat,add=T,nlevels=20)
points(xmat,pch=19,cex=0.5,main=paste('1000 steps'))
@

\framebreak

<<MC6,comment='+', fig.width=8, fig.height=6, fig.align='center',out.width='4in',echo=FALSE>>=
par(mar=c(4,3,2,2))
plot(1:nits,xmat[,1],type='l',col='red',xlab='Iteration')
lines(1:nits,xmat[,2],col='blue')
@


\end{frame}


\end{document}