Biomedical Engineering Reference
In-Depth Information
For convenience in the following arguments, we define the
j
th row of the mixing
matrix
A
as the column vector
a
j
, i.e., the
M
×
L
matrix
A
is expressed as
⊡
⊣
⊤
⊦
=
⊡
⊣
⊤
⊦
,
a
1
a
2
a
T
M
A
1
,
1
...
A
1
,
L
A
2
,
1
...
A
2
,
L
A
=
(5.38)
.
.
.
.
.
A
M
,
1
...
A
M
,
L
where
=
A
j
,
1
,...,
A
j
,
L
T
.
a
j
(5.39)
The prior distribution of
a
j
is assumed to be
2
,(ʻ
j
ʱ
)
−
1
p
(
a
j
)
=
N(
a
j
|
0
),
(5.40)
ʛ
ʻ
j
, and
where the
j
th diagonal element of the noise precision matrix
is denoted
ʱ
is a diagonal matrix given by
⊡
⊤
ʱ
1
0
...
0
⊣
⊦
.
0
ʱ
2
...
0
ʱ
=
(5.41)
.
.
.
.
.
.
00
... ʱ
L
Here, note that we assume that elements of
a
j
are statistically independent because
the precision matrix in Eq. (
5.40
) is diagonal. We further assume that
a
i
and
a
j
are
statistically independent when
i
=
j
. Thus, the prior probability distribution for the
whole mixing matrix
A
is expressed as
M
,(ʻ
j
ʱ
)
−
1
p
(
A
)
=
1
N(
a
j
|
0
).
(5.42)
j
=
This equation is equivalent to
M
L
,(ʻ
j
ʱ
)
−
1
p
(
A
)
=
1
N(
A
j
,
|
0
).
(5.43)
j
=
1
=
2
In the prior distribution in Eq. (
5.40
), the precision matrix has a form of a diagonal matrix
ʱ
multiplied by a scalar
ʻ
j
. The inclusion of this scalar,
ʻ
j
, is just for convenience in the mathematical
expressions for the update equations of
ʨ
and
a
j
. The inclusion of
¯
ʻ
j
actually makes these update
equations significantly simpler.
