Biomedical Engineering Reference
In-Depth Information
(
u
k
|
y
k
)
¯
The E-step of the VBEM algorithm estimates
p
, i.e., it estimates
u
k
, and
∂
ʓ
(
u
k
|
y
k
)
. For this estimation, we compute the derivative
log
p
.UsingEq.(
5.52
),
∂
u
k
the derivative is given by
E
A
A
T
∂
log
p
(
u
k
|
y
k
)
=
ʛ
(
y
k
−
Au
k
)
−
u
k
∂
u
k
E
A
A
T
A
u
k
−
¯
A
T
=
ʛ
y
k
−
ʛ
u
k
,
(5.54)
)
=
¯
A
, i.e.,
¯
A
(defined in Eq. (
5.45
)) is the mean of the posterior
p
where
E
A
(
A
(
A
|
y
)
.
Let us compute
E
A
A
T
A
contained in the right-hand side of Eq. (
5.54
). Noting
that the diagonal elements of
ʛ
ʻ
1
,...,ʻ
M
,
A
T
ʛ
are denoted
ʛ
A
is rewritten as
⊡
⊤
⊡
⊤
a
1
.
a
T
M
ʻ
1
...
0
M
⊣
⊦
⊣
⊦
=
.
.
.
A
T
1
ʻ
j
a
j
a
j
,
ʛ
=[
a
1
,...,
a
M
]
A
j
=
0
... ʻ
M
giving
E
A
A
T
A
M
1
ʻ
j
E
A
a
j
a
j
ʛ
=
.
(5.55)
j
=
Since the precision matrix of the posterior distribution of
a
j
is
ʻ
j
ʨ
,wehave
E
A
a
j
a
j
1
ʻ
j
ʨ
−
1
a
j
= ¯
a
j
¯
+
.
Therefore, the relationship
1
ʻ
j
ʻ
j
ʨ
−
1
E
A
A
T
A
M
1
a
j
ʛ
=
a
j
¯
¯
+
j
=
M
¯
A
T
1
ʻ
j
a
j
a
j
ʨ
−
1
ʛ
¯
A
ʨ
−
1
=
+
M
=
+
M
(5.56)
j
=
holds.
Substituting Eq. (
5.56
)into(
5.54
), we get
I
u
k
.
∂
y
k
)
=
¯
A
T
¯
A
T
ʛ
¯
A
ʨ
−
1
log
p
(
u
k
|
ʛ
y
k
−
+
M
+
(5.57)
∂
u
k
