Biomedical Engineering Reference
In-Depth Information
∂
∂
ʛ
log
p
(
y
|
z
,
A
)
K
2
K
∂
∂
ʛ
1
2
T
=
log
|
ʛ
|−
1
(
y
k
−
Ls
k
−
Au
k
)
ʛ
(
y
k
−
Ls
k
−
Au
k
)
k
=
K
K
2
ʛ
−
1
1
2
T
=
−
1
(
y
k
−
Ls
k
−
Au
k
)(
y
k
−
Ls
k
−
Au
k
)
.
(5.181)
k
=
According to Eq. (
5.81
), we have
2
diag
A
A
T
∂
∂
ʛ
L
2
ʛ
−
1
1
log
p
(
A
)
=
−
ʱ
.
(5.182)
We can thus compute the derivative of
F(
ʛ
,
ʱ
,
ʦ
)
with respect to
ʛ
, which is
K
2
ʛ
−
1
K
∂F
∂
ʛ
1
2
T
=
E
(
A
,
z
)
−
1
(
y
k
−
Ls
k
−
Au
k
)(
y
k
−
Ls
k
−
Au
k
)
k
=
A
T
L
1
2
A
2
ʛ
−
1
+
−
ʱ
K
2
ʛ
−
1
1
2
[
R
yu
¯
A
T
¯
AR
uy
−
R
ys
L
T
=
−
R
yy
−
−
−
LR
sy
LR
su
¯
A
T
¯
AR
us
L
T
LR
ss
L
T
+
+
+
]
1
2
E
A
[
L
2
ʛ
−
1
1
2
E
A
[
AR
uu
A
T
A
T
−
]+
−
A
ʱ
]
.
(5.183)
According to Eqs. (
5.83
) and (
5.84
), we get
AR
uu
A
T
A
T
A
T
E
A
[
+
ʱ
]=
E
A
[
(
R
uu
+
ʱ
)
]
A
A
ʨ
¯
A
T
A
T
]=
¯
A
ʛ
−
1
=
E
A
[
A
ʨ
+
L
.
(5.184)
¯
A
Using the equation above and the relationship
, and setting
the right-hand side of Eq. (
5.183
) equal to zero, we derive the update equation for
ʨ
=
(
R
yu
−
LR
su
)
ʛ
,
such that,
1
K
ʨ
¯
A
T
ʛ
−
1
¯
A
R
ys
L
T
LR
ss
L
T
=
diag
[
R
yy
−
−
LR
sy
+
−
]
.
The equation above is equal to Eq. (
5.166
).
