11.4. Modern Bayesian Methods 681
which is the pdf of a Γ{a+(1/2),[(θ^2 /2) + (1/b)]−^1 }distribution. Thus the Gibbs
sampler for this model is given by:
Θi|x, τi^2 − 1 ∼ N
(
τi^2 − 1
(σ^2 /n)+τi^2 − 1
x,
τi^2 − 1 σ^2
σ^2 +nτi^2 − 1
)
1
τi^2
|x,Θi ∼ Γ
(
a+
1
2
,
(
θ^2 i
2
+
1
b
)− 1 )
, (11.4.8)
fori=1, 2 ,...,m. As discussed above, for a specified values of largemandn∗>
m, we collect the chain’s values ((Θm,τm),(Θm+1,τm+1),...,(Θn∗,τn∗)) and then
obtain the Bayes estimate ofθ(assuming squared-error loss):
̂θ=^1
n∗−m
∑n∗
i=m+1
Θi. (11.4.9)
The conditional distribution of Θ givenxandτi− 1 , though, suggests the second
estimate given by
θ̂∗=^1
n∗−m
∑n∗
i=m+1
τi^2
τi^2 +(σ^2 /n)
x. (11.4.10)
Example 11.4.2. Lehmann and Casella (1998, p. 257) presented the following
hierarchical Bayes model:
X|λ ∼ Poisson(λ)
Λ|b ∼ Γ(1,b)
B ∼ g(b)=τ−^1 b−^2 exp{− 1 /bτ},b> 0 ,τ > 0.
For the Gibbs sampler, we need the two conditional pdfs,g(λ|x, b)andg(b|x, λ).
The joint pdf is
g(x, λ, b)=f(x|λ)h(λ|b)ψ(b). (11.4.11)
Based on the pdfs of the model, (11.4.11), for the first conditional pdf we have
g(λ|x, b) ∝ e−λ
λx
x!
1
b
e−λ/b
∝ λx+1−^1 e−λ[1+(1/b)], (11.4.12)
which is the pdf of a Γ(x+1,b/[b+ 1]) distribution.
For the second conditional pdf, we have
g(b|x, λ) ∝
1
b
e−λ/bτ−^1 b−^2 e−^1 /(bτ)
∝ b−^3 exp
{
−
1
b
[
1
τ
+λ
]}
.