Consider again the situation described in Exercise 12. This time, we shall let the prior distribution of μ be more like it was in the conjugate prior. Introduce another parameter γ, whose prior distribution is the gamma distribution with parameters a0 and b0. Let the prior distribution of μ conditional on γ be the normal distribution with mean μ0 and precision γ .
a. Prove that the marginal prior distribution of μ specifies that
b. Suppose that we want the marginal prior distributions of both μ and τ to be the same as they were with the conjugate prior in Sec. 8.6. How must the prior hyperparameters be related in order to make this happen?
c. Show that Table 12.9 specifies the appropriate conditional distribution for each parameter given the others.
Table 12.9 Parameters and conditional distributions for Exercise 13
d. Use the New Mexico nursing home data (Examples 12.5.2 and 12.5.3). Let the prior hyper parameters be α0 = 2, β0 = 6300, μ0 = 200, a0 = 2, and b0 = 3150. Implement a Gibbs sampler to find the posterior distribution of (μ, τ, γ). In particular, calculate an interval containing 95 percent of the posterior distribution of μ.
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