11.4. Modern Bayesian Methods 67911.4ModernBayesianMethods........................
The prior pdf has an important influence in Bayesian inference. We need only
consider the different Bayes estimators for the normal model based on different
priors, as shown in Examples 11.1.3 and 11.2.1. One way of having more control
over the prior is to model the prior in terms of another random variable. This is
called thehierarchical Bayesmodel, and it is of the formX|θ ∼ f(x|θ)
Θ|γ ∼ h(θ|γ)
Γ ∼ ψ(γ). (11.4.1)With this model we can exert control over the priorh(θ|γ) by modifying the pdf
of the random variable Γ. A second methodology,empirical Bayes, obtains an
estimate ofγand plugs it into the posterior pdf. We offer the reader a brief introduc-
tion of these procedures in this section. There are several good books on Bayesian
methods. In particular, Chapter 4 of Lehmann and Casella (1998) discusses these
procedures in some detail.
Consider first the hierarchical Bayes model given by (11.4.1). The parameterγ
can be thought of a nuisance parameter. It is often called ahyperparameter.As
with regular Bayes, the inference focuses on the parameterθ; hence, the posterior
pdf of interest remains the conditional pdfk(θ|x).
These discussions often involve several pdfs; hence, we frequently use gas a
generic pdf. It will always be clear from its arguments what distribution it repre-
sents. Keep in mind that the conditional pdff(x|θ) does not depend onγ; hence,
g(θ, γ|x)=
g(x,θ,γ)
g(x)=g(x|θ, γ)g(θ, γ)
g(x)=
f(x|θ)h(θ|γ)ψ(γ)
g(x).Therefore, the posterior pdf is given by
k(θ|x)=∫∞
∫ −∞f(x|θ)h(θ|γ)ψ(γ)dγ
∞
−∞∫∞
−∞f(x|θ)h(θ|γ)ψ(γ)dγdθ. (11.4.2)
Furthermore, assuming squared-error loss, the Bayes estimate ofW(θ)is
δW(x)=∫∞
−∞∫∞
∫ −∞W(θ)f(x|θ)h(θ|γ)ψ(γ)dγdθ
∞
−∞∫∞
−∞f(x|θ)h(θ|γ)ψ(γ)dγdθ. (11.4.3)
Recall that we defined the Gibbs sampler in Section 11.3. Here we describe it
to obtain the Bayes estimate ofW(θ). Fori=1, 2 ,...,m,wheremis specified, the