Continuing Example E of Section 3.5.2, suppose you had to guess a value of θ. One plausi...

Continuing Example E of Section 3.5.2, suppose you had to guess a value of θ. One plausible guess would be the value of θ that maximizes the posterior density. Find that value. Does the result make intuitive sense?

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: