In Example 12.5.6, we used a hierarchical model. In that model, the parameters 1, . . . , p were independent random variables with i

In Example 12.5.6, we used a hierarchical model. In that model, the parameters μ1, . . . , μp were independent random variables with μi having the normal distribution with mean ψ and precision λ0τi conditional on ψ and τ1, . . . , τp. To make the model more general, we could also replace λ0 by an unknown parameter λ. That is, let the μi ’s be independent with μi having the normal distribution with mean ψ and precision λτi conditional on ψ, λ, and τ1, . . . , τp. Let λ have the gamma distribution with parameters γ0 and δ0, and let λ be independent of ψ and τ1, . . . , τp. The remaining parameters have the prior distributions stated 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 λ.
b. Find the conditional distributions of each parameter given all of the others. Hint: For all the parameters besides λ, the distributions should be almost identical to those given in Example 12.5.6. Wherever λ0 appears, of course, something will have to change.
c. Use a prior distribution in which α0 = 1, β0 = 0.1, u0 = 0.001, γ0 = δ0 = 1, 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.