Let X1, . . . , Xn be i.i.d. with the normal distribution having mean μ and precision τ . Gibbs sampling allows one to use a prior distribution for (μ, τ) in which μ and τ are independent. Let the prior distribution of μ be the normal distribution with mean μ0 and variance γ0. Let the prior distribution of τ be the gamma distribution with parameters α0 and β0.
a. Show that Table 12.8 specifies the appropriate conditional distribution for each parameter given the other.
b. 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, and γ0 = 6.35 × 10ˆ’4. Implement a Gibbs sampler to find the posterior distribution of (μ, Ï„ ). In particular, calculate an interval containing 95 percent of the posterior distribution of μ.
Table 12.8 Parameters and conditional distributions for Exercise 12

Transcribed Image Text:

Paramete Prior times likelihood looks like the p.d.f. of this distribution gamma distribution with parameters Cto 0.5n2 normal distribution with mean (m)μο + ntr) /(r0+ nt) and precision