In Example 12.5.6, we modeled the parameters 1, . . . , p as i.i.d. having the

In Example 12.5.6, we modeled the parameters τ1, . . . , τp as i.i.d. having the gamma distribution with parameters α0 and β0.We could have added a level to the hierarchical model that would allow the τi ’s to come from a distribution with an unknown parameter. For example, suppose that we model the τi ’s as conditionally independent having the gamma distribution with parameters α0 and β given β. Let β be independent of ψ and μ1, . . ., μp with β having the gamma distribution with parameters ∊0 and φ0. The rest of the prior distributions are as specified in Example 12.5.6.
a. Write the product of the likelihood and the prior as a function of the parameters μ1, . . . , μp, τ1, . . . , τp, ψ, and β.
b. Find the conditional distributions of each parameter given all of the others. For all the parameters besides β, the distributions should be almost identical to those given in Example 12.5.6. Wherever β0 appears, of course, something will have to change.
c. Use a prior distribution in which α0 = λ0 = 1, u0 = 0.001, ∊0 = 0.3, φ0 = 3.0, and ψ0 = 170. Fit the model to the hot dog calorie data from Example 11.6.2. Compute the posterior means of the four μi’s and 1/τi’s.