3.5 Mixture of Conjugate Priors (Biased Coin) (pp.49-52).txt

#Mixture of Conjugate Priors
#(Albert pp. 49-52)
#here we have 2 beta priors each for p(head)=.3 or .7

library(LearnBayes)

#c(,)create vector of mixing probs and shape parameters
#note that mean of beta=a/a+b
#so if p=.3 then a=6 and b=14

probs=c(.5,.5)
beta.par1=c(6,14)
beta.par2=c(14,6)

#rbinds creates a matrix
betapar=rbind(beta.par1, beta.par2)

#suppose n=10, heads=7, tails=3

data=c(7, 3)

#R function binomial.beta.mix computes the posterior distribution
#when the proportion p has a mixture of betas prior distribution

post=binomial.beta.mix(probs, betapar, data)
post

#from the output g(p|data)=.093beta(13,17) + .907beta(21,9)

#initially bias of coin is 50-50 that p(head)=.3 or .7
#since more heads were observed, there is evidence that bias is towards heads
#thus, the posterior density places greater weight on the second component

#prior nd posterior densities

curve(post$probs[1]*dbeta(x, 13, 17)+post$probs[2]*dbeta(x, 21, 9), 
from=0, to=1, lwd=3, xlab="P", ylab="DENSITY")
curve(.5*dbeta(x, 6, 12)+.5*dbeta(x, 12, 6), 0, 1, add=TRUE)
legend("topleft", legend=c("Prior", "Posterior"), lwd=c(1,3))