
Monte Carlo methods in posterior analysis 109
E(
ˆ
Φ
) = E
1
R
K
X
k=1
φ
k
R
k
!
=
1
R
K
X
k=1
φ
k
E(R
k
)
Equation (4.27)
=
K
X
k=1
φ
k
Z
∆
k
p(f)df
∆
k
→0
=
K
X
k=1
Z
∆
k
φ(f)p(f )df
∆
k
→0
=
Z
B
A
φ(f)p(f )df. (4.30)
The above demonstrates that the result of the Monte Carlo integration
using estimator
ˆ
Φ on average will yield the integral (it is unbiased). The
va riance of the estimator is calculated as follows
var(
ˆ
Φ) = var
1
R
K
X
k=1
φ
k
R
k
!
=
1
R
2
K
X
k=1
var(φ
k
R
k
) −
K
X
k=1
K
X
k
′
=1,k
′
6=k
cov(φ
k
R
k
, φ
k
′
R
k
′
)
Eq.Equation (4.28)
=
1
R
2
K
X
k=1
φ
2
k
Rp
k
(1 − p
k
) −
K
X
k=1
K
X
k
′
=1,k
′
6=k
φ
k
φ
k
′
Rp
k
p
k
′
=
1
R
K
X
k=1
φ
2
k
p
k
−
K
X
k=1
φ
k
p
k
K
X
k
′
=1
φ
k
′
p
k
′
!
.
Using p
k
=
R
∆
k
p(f)df and ta king ∆
k
s to 0 we obtain
var(
ˆ
Φ) =
1
R
Z
B
A
φ
2
(f)df −
Z
B
A
φ(f)df
!
2
=
σ
2