This problem continues Example E of Section 3.5.2. In that example, the prior opinion fo...

This problem continues Example E of Section 3.5.2. In that example, the prior opinion for the value of was represented by the uniform density. Suppose that the prior density had been a beta density with parameters a = b = 3, reflecting a stronger prior belief that the chance of a 1 was near 1/ 2 . Graph this prior density.

Following the reasoning of the example, find the posterior density, plot it, and compare it to the posterior density shown in the example.

Reference

Bayesian Inference A freshly minted coin has a certain probability of coming up heads if it is spun on its edge, but that probability is not necessarily equal to 1 2. Now suppose it is spun n times and comes up heads X times. What has been learned about the chance the coin comes up heads?We will go through a Bayesian treatment of this problem. Let , we might represent our state of knowledge by a uniform density on [0, 1]:

We will see how observing X changes our knowledge about , transforming the prior distribution into a “posterior” distribution.

Given a value θ, X follows a binomial distribution with n trials and probability of success θ: