7.7. Consider a hierarchical nested model Yijk = + i + j(i) + ijk, (7.29) where i = 1,...,I, j = 1,...,Ji, and k

7.7. Consider a hierarchical nested model Yijk = μ + αi + βj(i) + ijk, (7.29)

where i = 1,...,I, j = 1,...,Ji, and k = 1,...,K. After averaging over k for each i and j, we can rewrite the model (7.29) as Yij = μ + αi + βj(i) + ij , i = 1, . . . , I, j = 1,...,Ji, (7.30)

where Yij = K k=1 Yijk/K. Assume that αi ∼ N(0, σ2

α), βj(i) ∼ N(0, σ2

β), and ij ∼

N(0, σ2

), where each set of parameters is independent a priori. Assume that σ2

α, σ2

β, and

σ2 are known. To carry out Bayesian inference for this model, assume an improper flat prior for μ, so f (μ) ∝ 1. We consider two forms of the Gibbs sampler for this problem

[546]:

a. Let n =

i Ji , y·· =

ij yij/n, and yi· =

j yij/Ji hereafter. Show that at iteration t, the conditional distributions necessary to carry out Gibbs sampling for this?