Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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

]}

.