In this problem, we shall outline a Bayesian solution to the problem described in Example 7.5.10 on page 423. Let τ = 1/σ 2 and use a proper normal-gamma prior of the form described in Sec. 8.6. In addition to the two parameters μ and τ , introduce n additional parameters.
For i = 1, . . . , n, let Yi = 1 if Xi came from the normal distribution with mean μ and precision τ , and let Yi = 0 if Xi came from the standard normal distribution.
a. Find the conditional distribution of μ given τ; Y1,. . . , Yn; and X1, . . . , Xn.
b. Find the conditional distribution of τ given μ; Y1,. . . , Yn; and X1, . . . , Xn.
c. Find the conditional distribution of Yi given μ; τ; X1, . . . , Xn; and the other Yj 's.
d. Describe how to find the posterior distribution of μ and τ using Gibbs sampling.
e. Prove that the posterior mean of Yi is the posterior probability that xi came from the normal distribution with unknown mean and variance.