Biomedical Engineering Reference
In-Depth Information
(
z
k
|
y
k
)
Since
p
is Gaussian, we assume that
z
k
,
ʓ
−
1
p
(
z
k
|
y
k
)
=
N(
z
k
|¯
).
∂
The mean
¯
z
k
is obtained as
z
k
that makes
log
p
(
z
k
|
y
k
)
equal to zero, and the
∂
z
k
precision
is obtained from the coefficient of
z
k
in this derivative. Using Eq. (
5.113
),
the derivative is given by
ʓ
A
c
ʛ
y
k
−
A
c
z
k
]−
∂
∂
log
p
(
z
k
|
y
k
)
=
E
A
[
z
k
z
k
=
¯
A
T
A
c
ʛ
c
ʛ
y
k
−
E
A
[
A
c
]
z
k
−
z
k
.
(5.114)
A
c
ʛ
In the expression above, using Eq. (
5.56
),
E
A
[
A
c
]
is expressed as
E
A
[
E
A
A
T
B
T
]
¯
A
T
A
T
ʛ
A
ʛ
B
A
c
ʛ
A
T
B
T
E
A
[
A
c
]=
ʛ
[
]
=
ʛ
¯
AB
T
B
T
ʛ
B
¯
A
T
¯
A
T
ʛ
¯
A
ʨ
−
1
+
ʛ
M
B
¯
A
c
ʛ
¯
A
c
+
ʨ
−
1
c
=
=
,
M
(5.115)
B
T
ʛ
¯
A
T
ʛ
B
where
ʨ
−
1
0
00
ʨ
−
1
c
=
,
(5.116)
and
¯
A
c
=
¯
A
B
.
,
(5.117)
Therefore, we finally have
∂
y
k
)
=
¯
A
c
ʛ
y
k
−
(
¯
A
c
ʛ
¯
A
c
z
k
+
ʨ
−
1
c
log
p
(
z
k
|
M
+
I
)
z
k
.
(5.118)
∂
z
k
The precision matrix of the posterior distribution is obtained as
¯
A
T
c
ʛ
¯
A
c
+
ʨ
−
1
c
ʓ
=
M
+
I
,
(5.119)
and the mean of the posterior as
=
ʓ
−
1
¯
¯
A
T
B
T
u
k
v
k
¯
z
k
=
ʛ
y
k
.
(5.120)
Equations (
5.119
) and (
5.120
) are the E-step update equations in the PFA algorithm.
