Biomedical Engineering Reference
In-Depth Information
Therefore, setting the right-hand side of the equation above to zero, we have the
update equation for
ʱ
, such that,
diag
1
+
ʨ
−
1
M
¯
A
T
ʱ
−
1
ʛ
¯
A
=
.
(5.79)
5.3.2.4 Update Equation for the Noise Precision
ʛ
We next derive the update equation for
ʛ
. To do so, we maximize the free energy in
Eq. (
5.71
) with respect to
ʛ
. On the right-hand-side of Eq. (
5.71
), the terms containing
ʛ
are log
p
(
y
|
u
,
A
)
and log
p
(
A
)
. Thus, omitting the terms not containing
ʛ
,the
free energy is expressed as
log
p
)
F
[
ʱ
,
ʛ
]=
E
(
y
|
u
,
A
)
+
log
p
(
A
(
A
,
u
)
K
2
K
1
2
T
=
E
log
|
ʛ
|−
1
(
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
(
A
,
u
)
k
=
M
L
M
L
1
2
1
2
1
ʻ
j
ʱ
A
j
,
+
(ʻ
j
ʱ
)
−
.
log
(5.80)
j
=
1
=
1
j
=
1
=
To maximize
F
[
ʱ
,
ʛ
]
with respect to
ʛ
, we consider the relationship
1
2
log
M
L
1
∂ʻ
j
1
2
ʻ
j
ʱ
A
j
,
(ʻ
j
ʱ
)
−
j
=
1
=
1
1
2
L
L
1
ʻ
j
−
1
2
ʱ
A
j
,
L
2
1
ʻ
j
−
1
2
1
ʱ
A
j
,
,
=
=
=
1
=
and obtain
1
2
log
M
L
1
∂
ʛ
1
2
ʻ
j
ʱ
A
j
,
(ʻ
j
ʱ
)
−
j
=
1
=
1
⊡
⊤
1
/ʻ
1
0
...
0
⊣
⊦
01
/ʻ
2
...
0
L
2
=
.
.
.
.
0
0
00
...
1
/ʻ
M
⊡
⊣
⊤
⊦
L
=
1
ʱ
A
1
,
...
0
0
L
=
1
ʱ
A
2
,
...
0
0
1
2
−
.
.
.
.
0
0
...
L
=
A
2
M
,
0
0
1
ʱ
