Exercise 6.1 Derive the posterior distribution (with the same NI priors as in the chapter) of (0,

Exercise 6.1 Derive the posterior distribution (with the same NI priors as in the chapter) of (β0, β1, σ2 ) in the osteoporosis study (data are in ‘osteop.txt’) by:

(a) Gibbs sampler and

(b) the Random Walk Metropolis Sampler with proposal density N(θk , c2 I) on (β0, β1, log(σ )) and vary the value of

c. Compare the performance of the two samplers.

Exercise 6.2 Derive the full conditionals of Example VI.4 and derive the posterior summary measures of all parameters and of derived parameters such as θ − λ. Raftery and Akman (1986) used as priors for θ and λ θ ∼ Gamma(a1, b1 ) and λ ∼ Gamma(a2, b2 ), with a1 = a2 = 0.5, b1 = b2 = 0. Show that this Bayesian model gives virtually the same posterior summary measures as the analysis done by Tanner (1993).