To draw from the prior predictive distribution, first sample theta from p(theta) then, for each sampled theta, draw a value from p(y|theta).

For the posterior predictive distribution, sample from the posterior, p(theta|y), then sample new.y from p(new.y|theta).

For example, suppose that Gov only has the support of half the population and that has a std dev of about .07.  this is approx a  Beta(50,50) distribution.

--we observe a poll with 400 respondents, 240 of whom (60%) say they will vote for Gov.

--because the Beta is conjugate prior for the binomial, the posterior is a Beta with a = a+y and b = b+N-y.b

par(mfrow=c(3,2))
draws <- 1000
N <- 400
#####Prior Predictive Dist
prior.theta <- rbeta (draws, 50,50)
y.prior.pred <- rbinom(draws, N, prior.theta)
hist(y.prior.pred,col=2,xlim=c(100,300))

prior.theta.jeff <- rbeta (draws, .5,.5)
y.prior.pred.jeff <- rbinom(draws, N, prior.theta.jeff)
hist(y.prior.pred.jeff,col='blue',xlim=c(0,400))


#####Data
obs.data<-rbinom(draws, N, .6)
hist(obs.data, col=2, xlim=c(190,300))
hist(obs.data, col=2, xlim=c(190,300))


##Posterior Predicitve Dist

post.theta <- rbeta(draws, 290,210)
y.post.pred <- rbinom(draws,N,post.theta) 
hist(y.post.pred, col=2, xlim=c(190,300))


post.theta.jeff <- rbeta(draws, 240.4,160.5)
y.post.pred.jeff <- rbinom(draws,N,post.theta.jeff) 
hist(y.post.pred.jeff, col='blue', xlim=c(190,300))