Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

680 Bayesian Statistics

ith step of the algorithm is

Θi|x,γi− 1 ∼ g(θ|x,γi− 1 )

Γi|x,θi ∼ g(γ|x,θi).

Recall from our discussion in Section 11.3 that

Θi

D

→ k(θ|x)

Γi

D

→ g(γ|x),

asi→∞. Furthermore, the arithmetic average

1

m

∑m

i=1

W(Θi)

P

→E[W(Θ)|x]=δW(x)asm→∞. (11.4.4)

In practice, to obtain the Bayes estimate ofW(θ) by the Gibbs sampler, we
generate by Monte Carlo the stream of values (θ 1 ,γ 1 ),(θ 2 ,γ 2 ).... Then choosing
large values ofmandn∗>m, our estimate ofW(θ) is the average,

1

n∗−m

∑n∗

i=m+1

W(θi). (11.4.5)

Because of the Monte Carlo generation these procedures are often calledMCMC,
forMarkov Chain Monte Carloprocedures. We next provide two examples.

Example 11.4.1.Reconsider the conjugate family of normal distributions dis-

cussed in Example 11.1.3, withθ 0 =0. Hereweusethemodel

X|Θ ∼ N

(

θ,

σ^2

n

)

,σ^2 is known

Θ|τ^2 ∼ N(0,τ^2 )

1

τ^2

∼ Γ(a, b),aandbare known. (11.4.6)

To set up the Gibbs sampler for this hierarchical Bayes model, we need the condi-
tional pdfsg(θ|x, τ^2 )andg(τ^2 |x, θ). For the first, we have

g(θ|x, τ^2 )∝f(x|θ)h(θ|τ^2 )ψ(τ−^2 ).

As we have been doing, we can ignore standardizing constants; hence, we need
only consider the productf(x|θ)h(θ|τ^2 ). But this is a product of two normal pdfs
which we obtained in Example 11.1.3. Based on those results,g(θ|x, τ^2 )isthepdf
of aN({τ^2 /[(σ^2 /n)+τ^2 ]}x,(τ^2 σ^2 )/[σ^2 +nτ^2 ]). For the second pdf, by ignoring
standardizing constants and simplifying, we obtain

g

(

1

τ^2

|x, θ

)

∝ f(x|θ)g(θ|τ^2 )ψ(1/τ^2 )

∝

1

τ

exp

{

−

1

2

θ^2

τ^2

}(

1

τ^2

)a− 1

exp

{

−

1

τ^2

1

b

}

∝

(

1

τ^2

)a+(1/2)− 1

exp

{

−

1

τ^2

[

θ^2

2

+

1

b

]}

, (11.4.7)