Biomedical Engineering Reference
In-Depth Information
5.5.3 VBEM Algorithm
5.5.3.1 E-Step
The posterior
p
(
z
|
y
)
is Gaussian, and assumed to be,
K
z
k
,
ʓ
−
1
p
(
z
|
y
)
=
p
(
z
k
|
y
k
),
and
p
(
z
k
|
y
k
)
=
N(
z
k
|¯
),
(5.141)
k
=
1
¯
ʓ
(
z
k
|
y
k
)
where
z
k
and
are, respectively, the mean and precision of
p
. Using similar
(
z
k
|
y
k
)
(
z
k
|
y
k
)
arguments as for the E-step of the PFA algorithm, the estimate of
p
,
p
,
is given by
log
p
(
z
k
|
y
k
)
=
E
A
[
log
p
(
y
k
|
z
k
,
A
)
]+
log
p
(
z
k
)
]
,
(5.142)
where we omit terms that do not contain
z
k
. The precision
ʓ
of the posterior is
∂
derived from the coefficient of
z
k
p
(
z
k
|
y
k
)
, and the mean
¯
z
k
is derived as the
z
k
that
∂
makes this derivative zero.
Defining
¯
A
s
=
,
¯
A
E
A
(
[
L
,
A
]
)
=[
L
]
, we obtain
∂
∂
A
s
ʛ
(
A
s
z
k
)
]−
ʦ
log
p
(
z
k
|
y
k
)
=
E
A
[
y
k
−
z
k
z
k
A
s
ʛ
A
s
ʛ
z
k
−
ʦ
=
E
A
[
y
k
]−
E
A
[
A
s
]
z
k
,
(5.143)
where
y
k
]=
¯
A
T
A
s
ʛ
E
A
[
s
ʛ
y
k
and
E
A
[
L
T
L
T
ʛ
L
]
E
A
[
ʛ
A
]
A
s
ʛ
E
A
[
A
s
]=
A
T
A
T
E
A
[
ʛ
L
]
E
A
[
ʛ
A
]
L
T
M
00
0
ʨ
−
1
L
T
ʛ
¯
A
ʛ
L
=
¯
A
T
s
ʛ
¯
A
s
+
=
.
¯
A
T
¯
A
T
ʛ
¯
A
ʨ
−
1
ʛ
L
+
M
(5.144)
Hence, we have
z
k
=
(
¯
A
s
ʛ
¯
A
s
+
A
s
ʛ
ʨ
−
1
s
E
A
[
A
s
]
)
z
k
,
M
(5.145)
where
00
0
ʨ
−
1
ʨ
−
1
s
=
.
(5.146)
