Biomedical Engineering Reference
In-Depth Information
5.4.2 Probability Model
The prior probability distributions for the factor vectors are:
P
(
u
k
)
=
N(
u
k
|
0
,
I
),
(5.108)
P
(
v
k
)
=
N(
v
k
|
0
,
I
).
(5.109)
Thus, we have the prior probability distribution for the augmented factor vector
z
k
,
such that
p
(
z
k
)
=
p
(
u
k
)
p
(
v
k
)
=
N(
u
k
|
0
,
I
)N(
v
k
|
0
,
I
)
exp
2
u
k
u
k
exp
k
v
k
1
/
2
1
/
2
I
2
1
I
2
1
2
v
T
=
−
−
ˀ
ˀ
exp
z
k
1
/
2
I
2
1
2
z
k
=
−
=
N(
z
k
|
0
,
I
).
(5.110)
ˀ
The noise is assumed to be Gaussian,
,
ʛ
−
1
P
(
ʵ
)
=
N(
ʵ
|
0
).
Therefore, we have the conditional probability,
P
(
y
k
|
u
k
,
v
k
)
, such that
v
k
,
ʛ
−
1
A
c
z
k
,
ʛ
−
1
P
(
y
k
|
u
k
,
v
k
)
=
P
(
y
k
|
z
k
)
=
N(
y
k
|
Au
k
+
B
)
=
N(
y
k
|
).
(5.111)
We assume the same prior for
A
as in Eq. (
5.42
).
5.4.3 VBEM Algorithm for PFA
5.4.3.1 E-Step
The E-step estimates the posterior probability
p
(
z
k
|
y
k
)
. Using the same arguments
in Sect.
5.3.2.1
,wehave,
E
A
log
p
A
c
)
log
p
(
z
k
|
y
k
)
=
(
z
k
,
y
k
,
E
A
log
p
A
c
)
.
=
(
y
k
|
z
k
,
A
c
)
+
log
p
(
z
k
)
+
log
p
(
(5.112)
Note that, since
B
is fixed,
p
, which is expressed in
Eq. (
5.42
). Omitting the terms not containing
z
k
, and using Eqs. (
5.110
) and (
5.111
),
we obtain
(
A
c
)
is the same as
p
(
A
)
E
A
1
2
(
1
2
z
k
T
log
p
(
z
k
|
y
k
)
=
−
y
k
−
A
c
z
k
)
ʛ
(
y
k
−
A
c
z
k
)
−
z
k
.
(5.113)
