
Regional Fertility Data Analysis: A Small Area Bayesian Approach 213
into the Metropolis-Hasting ratio in (10.14) which becomes
g(̟|̟
∗
)
g(̟
∗
|̟)
×
π(∆|̟
∗
, µ
∗
, δ
2∗
) π(µ
∗
, δ
2∗
|̟
∗
) π(µ, δ
2
|̟, ∆)
π(∆|̟, µ, δ
2
) π(µ, δ
2
|̟) π(µ
∗
, δ
2∗
|̟
∗
, ∆)
×
π(̟
∗
)
π(̟)
(10.17)
Using the fa c t that
π(∆|̟) =
π(∆|̟, µ, δ
2
) π(µ, δ
2
|̟)
π(µ, δ
2
|̟, ∆)
,
the ratio (10.17) r e duces to
g(̟|̟
∗
)
g(̟
∗
|̟)
×
π(∆|̟
∗
)
π(∆|̟)
×
π(̟
∗
)
π(̟)
(10.18)
When the prior and proposal density of ̟ are non-informative, the acceptance
rate is dominated by the marginal likelihood ratio r = π(∆|̟
∗
)/π(∆|̟) with
π(∆|̟) =
Z
d
Y
r=1
Z
π(∆
r
|δ
2
, µ
r
) π
0
(µ
r
|̟) π
2
(δ
2
) dµ
r
!
dδ
2
(10.19)
Letting n
∗
= (N −d)J/2, we can simplify it into
log(π(∆