# The unnormalized posterior for x=-8,8,17 #


postprop<-function(theta){
1/((1+(-8-theta)^2)*(1+(8-theta)^2)*(1+(17-theta)^2))*
exp(-(theta^2)/100)}



# A random walk Metropolis Hastings MCMC #
 
theta<-0
for (i in 1:10000){
 y<-rcauchy(1,theta[i])
 alpha<-min(postprop(y)/postprop(theta[i]),1)
 u<-runif(1)
 if (u<alpha) {theta<-c(theta,y)}
 else {theta<-c(theta,theta[i])}
}



# The estimated posterior #


hist(theta,freq=FALSE,nclass=200)



# The actual posterior #


ppint<-integrate(postprop,-Inf,Inf,subdivisions=1000)$value
th<-(-1000:1000)/30
posterior<-rep(0,2001)
for (i in 1:2001){
 posterior[i]<-postprop(th[i])/ppint}
lines(th,posterior)