Biomedical Engineering Reference
In-Depth Information
ʓ
The posterior precision matrix
is obtained as the coefficient of
u
k
in the right-hand
ʓ
side of Eq. (
5.57
), i.e.,
is given by
¯
A
T
ʛ
¯
A
ʨ
−
1
ʓ
=
+
M
+
I
.
(5.58)
¯
The mean
u
k
is obtained as the value of
u
k
that makes the right-hand side of Eq. (
5.57
)
equal to zero. That is, we have
I
−
1
¯
A
T
¯
A
T
ʛ
¯
A
ʨ
−
1
u
k
=
¯
+
M
+
ʛ
y
k
.
(5.59)
5.3.2.2 M-Step
The M-step of the VBEM algorithm estimates
p
. According to the arguments
in Sect. B.6 in the Appendix, the estimate of the posterior distribution,
(
A
|
y
)
p
(
A
|
y
)
,is
obtained as
E
u
log
p
)
,
log
p
(
A
|
y
)
=
(
u
,
y
,
A
(5.60)
where
E
u
[
·
] indicates the expectation with respect to the posterior distribution,
p
(
u
|
y
)
. To obtain
p
(
A
|
y
)
, we substitute Eq. (
5.48
)into(
5.60
). Neglecting terms
(such as log
p
(
u
)
) that do not contain
A
, we derive,
E
u
log
p
)
.
log
p
(
A
|
y
)
=
(
y
|
u
,
A
)
+
log
p
(
A
(5.61)
Substituting Eqs. (
5.8
) and (
5.42
)into(
5.61
), omitting terms not containing
A
, we get
E
u
K
M
1
2
(
1
2
T
1
ʻ
j
a
j
ʱ
(
|
)
=
−
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
−
a
j
.
log
p
A
y
k
=
1
j
=
(5.62)
As in Eq. (
5.44
), the posterior
p
(
a
j
|
y
)
is Gaussian with the mean
a
j
and the
¯
precision
ʻ
j
ʨ
. The precision
ʻ
j
ʨ
can be obtained by the coefficient of
a
j
in the
∂
∂
(
|
)
¯
derivative
a
j
can be obtained as
a
j
that makes this derivative
equal to zero. To compute the derivative, we first rewrite:
A
log
p
A
y
, and
M
M
L
1
ʻ
j
a
j
ʱ
A
j
,
,
a
j
=
1
ʻ
j
ʱ
(5.63)
j
=
j
=
1
=
and we can find
M
L
∂
1
2
1
ʻ
j
ʱ
A
j
,
=
ʻ
j
ʱ
A
j
,
=
ʛ
ʱ
]
j
,
.
[
A
(5.64)
∂
A
j
,
j
=
1
=
