Biomedical Engineering Reference
In-Depth Information
E
(
A
,
u
)
log
p
)
F
[
ʱ
,
ʛ
]=
(
,
,
)
−
(
|
)
−
(
|
u
y
A
log
p
u
y
log
p
A
y
=
E
(
A
,
u
)
[
(
|
,
)
+
(
)
log
p
y
u
A
log
p
u
+
log
p
(
A
)
−
log
p
(
u
|
y
)
−
log
p
(
A
|
y
)
]
,
(5.71)
where
E
)
[·]
indicates the average regarding both
p
(
A
|
y
)
and
p
(
u
|
y
)
.Onthe
(
A
,
u
right-most-side of the equation above, the terms containing
ʱ
are only log
p
(
A
)
.
Thus, omitting the terms not containing
ʱ
, we can rewrite the free energy as
log
p
)
.
F
[
ʱ
,
ʛ
]=
E
(
A
(5.72)
(
A
,
u
)
Considering
M
M
M
1
2
1
2
a
j
ʻ
j
ʱ
log
p
(
A
)
=
log
1
N(
a
j
|
0
, ʻ
j
ʱ
)
=
log
|
ʻ
j
ʱ
|−
a
j
j
=
j
=
1
j
=
1
M
L
M
L
1
2
1
2
1
ʻ
j
ʱ
A
j
,
,
=
log
(ʻ
j
ʱ
)
−
(5.73)
j
=
1
=
1
j
=
1
=
The free energy in Eq. (
5.72
) is rewritten as
⊡
⊤
M
L
M
L
1
2
1
2
⊣
1
ʻ
j
ʱ
A
j
,
⊦
.
F
[
ʱ
,
ʛ
]=
E
(
A
,
u
)
log
(ʻ
j
ʱ
)
−
(5.74)
j
=
1
=
1
j
=
1
=
To derive the estimate of
ʱ
, we differentiate the free energy in (
5.74
) with respect
to
ʱ
, resulting in
⊡
⊣
⊡
⊣
⊤
⊦
⊤
⊦
,
M
L
M
L
∂
∂
ʱ
F
[
ʱ
,
ʛ
]=
∂
∂
ʱ
1
2
1
2
1
ʻ
j
ʱ
A
j
,
(ʻ
j
ʱ
)
−
E
A
log
j
=
1
=
1
j
=
1
=
(5.75)
where the expectation
E
u
[·]
is omitted because the terms on the right-hand side of
the equation above do not contain
u
. To compute the equation above, we consider
the relationship
⊡
⊤
M
L
M
=
1
ʻ
j
ʱ
L
M
∂
∂ʱ
1
2
1
2
M
2
ʱ
−
1
1
2
⊣
A
j
,
⊦
=
1
ʻ
j
A
j
,
,
(5.76)
log
(ʻ
j
ʱ
)
−
−
j
=
=
1
j
=
j
=
1
1
