Test of hypothesis that coin is fair

In Section 3.5, I describe a Bayesian test of the hypothesis H that a proportion is equal to 0.5. The R function pbetat.R (in the LearnBayes package) computes the posterior probability of H assuming a beta prior on the alternative hypothesis. We illustrate how the posterior probability depends on the beta parameter a.

Here's a quick way of constructing Figure 3.5 (page 52) using the curve function.

First we revise the function pbetat to accept a matrix argument for the beta parameters where each row of the matrix corresponds to a pair (a, b). The new function is called pbetat.v. Next, we write a short function best that computes the posterior probability of H for our example (5 successes and 15 failures) for a vector of values of log a. (We are assuming a symmetric beta(a, a) prior on the alternative hypothesis.)

best=function(loga)
{p0=.5; prob=.5
data=c(5,15)
AB=exp(cbind(loga,loga))
s=pbetat.v(p0,prob,AB,data)
return(s$post)}

To generate the figure, we can use the curve function.

curve(best,from=-4,to=5,xlab="log a",ylab="Prob(coin is fair)",lwd=3)

We see that for all values of the beta parameter a, the posterior probability of the hypothesis exceeds 0.2.