Biomedical Engineering Reference
In-Depth Information
(
)
is a so-called conjugate prior, the posterior probability distribution
has the form of the Gaussian distribution:
Since
p
A
a
j
,(ʻ
j
ʨ
)
−
1
p
(
a
j
|
y
)
=
N(
a
j
| ¯
),
and
M
M
a
j
,(ʻ
j
ʨ
)
−
1
p
(
A
|
y
)
=
p
(
a
1
,...,
a
M
|
y
)
=
p
(
a
j
|
y
)
=
1
N(
a
j
| ¯
),
(5.44)
j
=
j
=
1
where
are the mean and precision matrix of the posterior distribution.
Namely, the posterior distribution
p
¯
a
j
and
ʻ
j
ʨ
(
A
|
y
)
has a form identical to the prior distribution
p
(
A
)
with the diagonal
ʱ
replaced by the non-diagonal
ʨ
. We define for later use
¯
A
, such that
the matrix
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
.
a
1
¯
¯
A
1
,
1
...
A
1
,
L
A
2
,
1
...
A
2
,
L
.
.
.
.
.
A
M
,
1
...
A
M
,
L
a
2
.
¯
A
=
=
(5.45)
a
T
M
¯
In the VBFA algorithm, an overspecified model order
L
is used, i.e., the value of
L
is set greater than the true model order
L
0
. The posterior mean of
a
j
,
¯
a
j
:
=
A
j
,
1
,...,
A
j
,
L
0
,
A
j
,
L
0
+
1
,...,
A
j
,
L
T
a
j
is the Bayes estimate of the
j
th row of the mixing matrix. In this estimated mix-
ing matrix, the matrix elements
A
j
,
L
0
+
1
,...,
A
j
,
L
are those corresponding to non-
existing factors. In the VBFA algorithm, those elements are estimated to be sig-
nificantly small, and the influence of the overspecified factors are automatically
eliminated in the final estimation results.
5.3.2 Variational Bayes EM Algorithm (VBEM)
5.3.2.1 E-Step
We follow the arguments in Sect. B.6 in the Appendix, and derive the variational
Bayes EM algorithm. The posterior distribution
p
(
u
,
A
|
y
)
is approximated by
p
(
u
,
A
|
y
)
=
p
(
u
|
y
)
p
(
A
|
y
),
(5.46)
