Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

682 Bayesian Statistics

In this last expression, making the change of variabley=1/bwhich has the Jacobian

db/dy=−y−^2 ,weobtain

g(y|x, λ) ∝ y^3 exp

{

−y

[

1

τ

+λ

]}

y−^2

∝ y^2 −^1 exp

{

−y

[

1+λτ

τ

]}

,

which is easily seen to be the pdf of the Γ(2,τ/[λτ+ 1]) distribution. Therefore,
the Gibbs sampler is, fori=1, 2 ,...,m,wheremis specified,

Λi|x, bi− 1 ∼ Γ(x+1,bi− 1 /[1 +bi− 1 ])

Bi=Yi−^1 ,whereYi|x, λi ∼ Γ(2,τ/[λiτ+1]).

As a numerical illustration of the last example, suppose we setτ =0.05 and
observex= 6. The R function^1 hierarch1.scomputes the Gibbs sampler given in
the example. It requires specification of the value ofiat which the Gibbs sample
commences and the length of the chain beyond this point. We set these values at
m= 1000 andn∗= 4000, respectively, i.e., the length of the chain used in the
estimate is 3000. To see the effect that varyingτ has on the Bayes estimator, we
performed five Gibbs samplers, with these results:

τ 0.040 0.045 0.050 0.055 0.060

̂δ 6.600 6.490 6.530 6.500 6.440

There is some variation. As discussed in Lehmann and Casella (1998), in general,
there is less effect on the Bayes estimator due to variability of the hyperparameter
than in regular Bayes due to the variance of the prior.

11.4.1 EmpiricalBayes

The empirical Bayes model consists of the first two lines of the hierarchical Bayes

model; i.e.,

X|θ ∼ f(x|θ)

Θ|γ ∼ h(θ|γ).

Instead of attempting to model the parameterγwith a pdf as in hierarchical Bayes,

empirical Bayes methodology estimatesγbased on the data as follows. Recall that

g(x,θ|γ)=

g(x,θ,γ)

ψ(γ)

=

f(x|θ)h(θ|γ)ψ(γ)

ψ(γ)

= f(x|θ)h(θ|γ).

(^1) Downloadable at the site listed in the Preface