Biomedical Engineering Reference
In-Depth Information
First, we have
N
∂
∂ʱ
j
∂
∂ʱ
j
=
ʱ
−
1
j
log
|
ʦ
|=
log
ʱ
j
.
(4.34)
j
=
1
Defining
th component is equal to 1
and all other components are zero, we can compute the second term of Eq. (
4.33
),
such that
ʠ
j
,
j
as an
(
N
×
N
)
matrix in which the
(
j
,
j
)
tr
tr
H
T
H
ʦ
+
ʲ
∂
∂ʱ
j
∂
∂ʱ
j
ʓ
∂
∂ʱ
j
ʓ
−
1
ʓ
−
1
log
|
ʓ
|=
=
tr
tr
ʠ
j
,
j
∂
∂ʱ
j
ʦ
ʓ
−
1
ʓ
−
1
=[
ʓ
−
1
=
=
]
j
,
j
,
(4.35)
where
[·]
j
,
j
indicates the
(
j
,
j
)
th element of a matrix in the squared brackets. Note
[
ʓ
−
1
that
th element of the posterior covariance matrix.
Next, we compute the second term of the right-hand side of Eq. (
4.32
). Using
Eqs. (
4.98
) and (
4.99
), we can derive,
]
j
,
j
above is the
(
j
,
j
)
ʲ
x
k
K
K
∂
∂ʱ
j
1
K
∂
∂ʱ
j
1
K
y
k
ʣ
−
1
2
x
k
ʦ
¯
y
k
=
y
k
−
H
x
k
¯
+ ¯
y
k
=
1
k
=
1
x
k
∂
K
K
1
K
1
K
x
k
ʠ
j
,
j
¯
=
1
¯
∂ʱ
j
ʦ
x
k
=
¯
1
¯
x
k
k
=
k
=
K
1
K
x
j
(
=
1
¯
t
k
).
(4.36)
k
=
[
ʓ
−
1
Therefore, denoting
]
j
,
j
as
j
,
j
, the relationship
K
∂F(
ʱ
)
∂ʱ
j
1
K
=
j
,
j
−
ʱ
−
1
x
j
(
+
1
¯
t
k
)
=
0
(4.37)
j
k
=
holds, and we get
K
1
K
ʱ
−
1
j
x
j
(
=
j
,
j
+
1
¯
t
k
).
(4.38)
k
=
This is equal to the update equation in the EM algorithm derived in Eq. (B.42).
On the other hand, we can derive a different update equation. To do so, we rewrite
the expression for the posterior precision matrix in Eq. (
4.13
)to
I
−
ʓ
−
1
ʦ
=
ʲ
ʓ
−
1
H
T
H
, where the
(
j
,
j
)
th element of this equation is
ʲ
ʓ
−
1
H
T
H
−
ʱ
j
[
ʓ
−
1
1
]
j
,
j
=
j
.
(4.39)
j
,
