Biomedical Engineering Reference
In-Depth Information
ciple of importance sampling [
15
,
16
], which usually assumes dependence on
x
and
N
only, the
k
-1
weights can be defined by (
6.9
).
i
p
(
x
x
|
N
)
w
i
µ
0:
k
1:
k
(6.9)
k
i
q
(
|
N
)
0:
k
1:
k
Here, we assume the importance density obeys the properties of Markov chain such that
q
(
x
|
N
)
=
q
(
x
|
x
,
N
) (
q
x
|
N
)
=
q
(
x
|
x
,
D
N
) (
q
x
|
N
)
(6.10)
0:
k
1:
k
k
0:
k
-
1
1:
k
0:
k
-
1
1:
k
-
1
k
k
-
1
k
0:
k
-
1
1:
k
-
1
p
x
(
|
N
)
At each iteration, the posterior density
can be estimated from the previous iteration
0
k
1
k
as (
6.11
):
p
(
D
N
|
x
,
N
) (
p
x
|
N
)
k
0:
k
1:
k
-
1
0:
k
1:
k
-
1
p
(
x
|
N
)
=
0:
k
1:
k
p
(
D
N
|
N
)
k
1:
k
-
1
p
(
D
N
|
x
,
N
) (
p
x
|
x
,
N
)
k
0:
k
1:
k
-
1
k
0:
k
-
1
1:
k
-
1
=
´
p
(
x
|
N
)
0:
k
-
1
1:
k
-
1
p
(
D
N
|
N
)
k
1:
k
-
1
p
(
D
N
|
x
) (
p
x
|
x
)
k
k
k
k
-
1
=
´
p
(
x
|
N
)
0:
k
-
1
1:
k
-
1
p
(
D
N
|
N
)
k
1:
k
-
1
µ D
p
(
N
|
x
) (
p
x
|
x
) (
p
x
|
N
)
(6.11)
k
k
k
k
-
1
0:
k
-
1
1:
k
-
1
Replacing (
6.10
) and (
6.11
) into (
6.9
), the weight can be updated recursively as (
6.12
):
i
i
i
i
i
i
i
p
(
D
N
|
x
) (
p
x
|
x
) (
p
x
|
N
)
p
(
D
N
|
x
) (
p
x
|
x
)
i
w
µ
k
k
k
k
-
1
0:
k
-
1
1:
k
-
1
=
k
k
k
k
-
1
w
i
(6.12)
k
k
-
1
q
(
x
i
|
x
i
,
D
N
) (
q
x
i
|
N
)
q
(
x
i
|
x
i
,
D
N
)
k
k
-
1
k
0:
k
-
1
1:
k
-
1
k
k
-
1
k
i
k
i
k
i
k
i
k
q
(
x
|
x
,
Δ
N
)
p
(
x
|
x
)
Usually, the importance density
is chosen to be the prior density
,
−
1
k
−
1
i
k
i
k
xx
by (6.6) as a prediction stage.
Sequential Importance Resampling [
17
] should also be applied at every time index to avoid
degeneration, so that the sample is i.i.d. from the discrete uniform density with weights
p
(
|
)
requiring the generation of new samples from
−
1
w
i
-
=
1
.
N
k
1
S
The weights then change proportionally given by
w
i
µ D
p
(
N
|
x
i
)
(6.13)
k
k
k
