Biomedical Engineering Reference
In-Depth Information
5.4.3.2 M-Step
The M-step estimates the posterior probability
p
(
A
c
|
y
k
)
. Using the same arguments
as in Sect.
5.3.2.2
,wehave
log
p
(
A
c
|
y
)
E
z
log
p
A
c
)
=
E
z
log
p
A
c
)
=
(
y
,
z
,
(
y
|
z
,
A
c
)
+
log
p
(
z
)
+
log
p
(
K
=
E
z
log
p
(
y
k
|
z
k
,
A
c
)
+
log
p
(
A
c
)
k
=
1
⊡
⊤
K
M
1
2
1
2
⊣
−
T
a
j
ʻ
j
ʱ
⊦
,
=
E
z
1
(
y
k
−
Au
k
−
B
v
k
)
ʛ
(
y
k
−
Au
k
−
B
v
k
)
−
a
j
k
=
j
=
1
(5.121)
where
z
collectively expresses
z
1
,...,
z
K
, and terms not containing
A
are omitted.
The form of the posterior distribution in Eq. (
5.44
) is also assumed:
M
a
j
,(ʻ
j
ʨ
)
−
1
a
j
|
¯
p
(
A
c
|
y
)
=
p
(
A
|
y
)
=
1
N(
).
j
=
¯
A
is obtained as the
A
that makes
∂
The mean
∂
A
log
p
(
A
|
y
)
equal to zero, and the
precision
as the coefficient of
a
j
in that derivative. Using Eqs. (
5.65
) and (
5.66
)
the derivative is computed as
ʻ
j
ʨ
K
∂
∂
u
k
(
|
y
k
)
=
ʛ
1
(
y
k
−
Au
k
−
v
k
)
−
ʛ
ʱ
A
log
p
A
E
z
B
A
k
=
=
ʛ
R
yu
−
ʛ
BR
v
u
−
ʛ
A
(
R
uu
+
ʱ
).
(5.122)
In Eq. (
5.122
), the coefficient of
a
j
is
ʻ
j
(
R
uu
+
ʱ
)
, and thus the matrix
ʨ
is equal
to
ʨ
=
R
uu
+
ʱ
,
(5.123)
¯
A
is obtained as
and
¯
A
)
ʨ
−
1
=
(
R
yu
−
BR
,
(5.124)
v
u
where
R
yu
is obtained as
K
u
k
.
R
yu
=
y
k
¯
(5.125)
k
=
1
