20. Consider the balanced, additive, one-way ANOVA model, Yij = + i + ij , i = 1,...,I, j = 1, . . .

20. Consider the balanced, additive, one-way ANOVA model, Yij = μ + αi + ij , i = 1,...,I, j = 1, . . . , J, (3.34)

where ij iid

∼ N(0, σ2 e ), μ ∈ , αi ∈ , and σ2 e > 0. We adopt a prior structure that is a product of independent conjugate priors, wherein μ

has a flat prior, αi iid

∼ N(0, σ2

α), and σ2 e ∼ IG

(a, b). Assume that σ2

α, a, and b are known.

(a) Derive the full conditional distributions for μ, αi, and σ2 e , necessary for implementing the Gibbs sampler in this problem.

(b) What is meant by “convergence diagnosis”? Describe some tools you might use to assist in this regard. What might you do to improve a sampler suffering from “slow convergence”?

(c) Suppose for simplicity that σ2 e is also known. What conditions on the data or the priors might lead to slow convergence for μ and the αi?

(Hint: What conditions weaken the identifiability of the parameters?)