Biomedical Engineering Reference
In-Depth Information
(
|
)
The E-step of the VBEM algorithm is a step that estimates
p
, and according
to the arguments in Sect. B.6, the estimate of the posterior distribution,
u
y
(
|
)
p
u
y
,is
obtained as
E
A
log
p
)
,
log
p
(
u
|
y
)
=
(
u
,
y
,
A
(5.47)
where
E
A
[
·
] indicates the expectation with respect to
p
(
A
|
y
)
. To obtain
p
(
u
|
y
)
,we
substitute
p
(
u
,
y
,
A
)
=
p
(
y
|
u
,
A
)
p
(
u
,
A
)
=
p
(
y
|
u
,
A
)
p
(
u
)
p
(
A
),
(5.48)
into Eq. (
5.47
). Neglecting constant terms, we can derive,
E
A
log
p
)
log
p
(
u
|
y
)
=
(
y
|
u
,
A
)
+
log
p
(
u
)
+
log
p
(
A
E
A
log
p
)
=
(
y
|
u
,
A
)
+
log
p
(
u
K
E
A
log
p
u
k
)
.
=
(
y
k
|
u
k
,
A
)
+
log
p
(
(5.49)
k
=
1
In the equation above, the term log
p
is omitted because it does not contain
u
.
Since we have assumed that the prior and the noise probability distributions are
independent across time, we have the independence of the posterior with respect to
time, i.e.,
(
A
)
K
p
(
u
|
y
)
=
1
p
(
u
k
|
y
k
).
(5.50)
k
=
Using Eqs. (
5.49
) and (
5.50
), we obtain
E
A
log
p
log
p
(
u
k
|
y
k
)
=
(
y
k
|
u
k
,
A
)
+
log
p
(
u
k
)
.
(5.51)
Substituting Eqs. (
5.4
) and (
5.7
)into(
5.51
), we get
E
A
2
u
k
u
k
1
2
(
1
T
log
p
(
u
k
|
y
k
)
=
−
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
−
.
(5.52)
Since the posterior
p
(
u
k
|
y
k
)
is Gaussian, we assume
u
k
,
ʓ
−
1
p
(
u
k
|
y
k
)
=
N(
u
k
| ¯
),
(5.53)
where
u
k
and
¯
ʓ
are the mean and the precision of this posterior distribution.
