Chapter 16.2, Problem 7E

In Section 16.1 and Exercise 16.6, we considered an example where the number of responders to a treatment for a virulent disease in a sample of size n had a binomial distribution with parameter p and used a beta prior for p with parameters α = 1 and β = 3.

1. a Find the Bayes estimator for p = the proportion of those with the virulent disease who respond to the therapy.
2. b Derive the mean and variance of the Bayes estimator found in part (a).

16.6 Suppose that Y is a binomial random variable based on n trials and success probability p (this is the case for the virulent-disease example in Section 16.1). Use the conjugate beta prior with parameters α and β to derive the posterior distribution of p | y. Compare this posterior with that found in Example 16.1.

EXAMPLE 16.1 Let Y₁, Y₂, …, Y_n denote a random sample from a Bernoulli distribution where P(Y_i = 1) = p and P(Y_i = 0) = 1 − p and assume that the prior distribution for p is beta (α, β). Find the posterior distribution for p.