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Statistical Computing in Nuclear Imaging
book

Statistical Computing in Nuclear Imaging

by Arkadiusz Sitek
December 2014
Intermediate to advanced content levelIntermediate to advanced
275 pages
9h 12m
English
CRC Press
Content preview from Statistical Computing in Nuclear Imaging
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
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Publisher Resources

ISBN: 9781439849347