Biomedical Engineering Reference
In-Depth Information
K
log
p
(
y
|
z
,
A
s
)
=
log
p
(
y
k
|
z
k
,
A
s
)
k
=
1
K
K
2
1
2
T
=
log
|
ʛ
|−
1
(
y
k
−
Ls
k
−
Au
k
)
ʛ
(
y
k
−
Ls
k
−
Au
k
),
k
=
(5.161)
K
K
K
2
1
2
|
ʦ
|−
z
k
ʦ
log
p
(
z
)
=
log
p
(
z
k
)
=
log
z
k
k
=
1
k
=
1
K
K
K
2
1
2
1
2
s
k
ʦ
u
k
u
k
,
=
log
|
ʦ
|−
s
k
−
(5.162)
k
=
1
k
=
1
and
M
M
1
2
1
2
a
j
ʻ
j
ʱ
log
p
(
A
)
=
log
|
ʻ
j
ʱ
|−
a
j
.
(5.163)
j
=
1
j
=
1
Let us derive the update equation for
ʦ
.InEq.(
5.160
), the only term that contains
ʦ
is log
p
(
z
)
. Thus, we have
K
1
∂
ʦ
F(
ʛ
,
ʱ
,
ʦ
)
=
1
∂
ʦ
K
2
1
∂
ʦ
1
2
E
(
z
,
A
)
s
k
ʦ
log
|
ʦ
|−
s
k
k
=
1
K
K
2
ʦ
−
1
1
2
E
s
K
2
ʦ
−
1
1
2
R
ss
,
s
k
s
k
=
−
=
−
(5.164)
k
=
1
and setting the right-hand side to zero gives the update equation,
1
K
ʦ
−
1
=
R
ss
,
(5.165)
where
R
ss
is obtained in Eq. (
5.156
). We can derive the update equation for
ʱ
in
a similar manner. However, Since in Eq. (
5.160
), the only term that contains
ʱ
is
log
p
(
A
)
, the update equation for
ʱ
is exactly the same as that in Eq. (
5.79
). The
ʛ
update equation for
is given by
1
K
ʨ
¯
A
T
ʛ
−
1
R
ys
L
T
LR
ss
L
T
−
¯
A
=
diag
[
R
yy
−
−
LR
sy
+
]
.
(5.166)
Since the derivation of the equation above is lengthy, it is presented in Sect.
5.7.4
.
