#  Function: grid.BetaBin(y,n,a.prir,b.pri)
# 
#  Function to comparge grid approxmation and analytic
#    for a beta prior and binomial likelihood
#
#  Input:  y = number of events
#          n = number of trials
#          a.pri = prior shape parameter 1 of beta
#          b.pri = prior shape parameter 2 of beta 
#
#
#   Output: 3 plots
#           For grid:  theta x prior
#                      theat x likelihood
#                      theta x posterior  
#                      theta x analytic posterior is overlaid in theta x posterior
#
#

grid.BetaBin <- function(y,n,a.pri,b.pri) {
  theta <- seq(0,1,length.out=1000)            # grid values

  analytic.post <- dbeta(theta,y,n-y)          # analytic for comparison
  analytic.post <- analytic.post/sum(analytic.post)

  prior <- dbeta(theta,a.pri,b.pri)          # For the grid approximation
                        
  likelihood <- dbinom(y,n,prob=theta)       # used original data 1928-1996

  posterior.u <- prior*likelihood            # product
  posterior <- posterior.u/sum(posterior.u)  # standardize

  layout(matrix(c(1,1,0,0,1,1,3,3,2,2,3,3,2,2,0,0),4,4,byrow=FALSE))

    plot(theta,prior,type='l',main=paste("Beta(",a.pri,",",b.pri,") Prior") ,col="grey",lwd=2)
plot(theta,likelihood,type='l',main=paste("Binomial Likelihood, y=", y, "& n=", n) ,col="blue",lwd=2)
  plot(theta,analytic.post,type='l',main="Posterior: prior*likelihood/sum(posterior)",col="black",lwd=2)
  lines(theta,posterior,type='l',lwd=2,col='red')
  legend("topleft",
   legend = c( "Grid approximation", "Analytic"),
   col = c("black","red"),
   text.col = c("black","red"), 
   lty=c(1,1),
   bty = "y",
     cex = 1.0)

}