Biomedical Engineering Reference
In-Depth Information
log
p
(
y
|
ʸ
)
=
d
u
p
(
u
|
y
)
[
log
p
(
u
,
y
|
ʸ
)
−
log
p
(
u
|
y
)
]
E
u
log
p
|
ʸ
)
+
H
p
)
,
=
(
u
,
y
(
u
|
y
(5.31)
where the second term on the right-hand side is the entropy regarding
p
(
u
|
y
)
.Using
Eqs. (
5.9
) and (C.9) in the Appendix, we get
K
H
p
)
=
1
H
p
y
k
)
=−
K
2
(
u
|
y
(
u
k
|
log
|
ʓ
|
.
(5.32)
k
=
Using Eqs. (
5.5
) and (
5.8
), the first term in Eq. (
5.31
) can be rewritten as
E
u
log
p
|
ʸ
)
=
E
u
log
p
)
(
u
,
y
(
y
|
u
,
ʸ
)
+
log
p
(
u
E
u
u
k
u
k
K
1
2
T
=−
(
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
+
k
=
1
K
2
+
log
|
ʛ
|
,
(5.33)
where constant terms are omitted. On the right-hand side of the equation above,
E
u
u
k
u
k
u
k
E
u
u
k
A
T
I
u
k
E
u
u
k
ʓ
u
k
A
T
ʛ
Au
k
+
=
ʛ
A
+
=
E
u
tr
u
k
u
k
ʓ
tr
E
u
u
k
u
k
=
=
ʓ
tr
u
k
+
ʓ
−
1
u
k
ʓ
¯
=
(
¯
u
k
¯
)
ʓ
= ¯
u
k
,
(5.34)
where a constant term is again omitted. Also, we can show that
E
u
y
k
ʛ
y
k
u
k
A
T
y
k
ʛ
u
k
A
T
Au
k
+
ʛ
=
A
u
k
+ ¯
¯
ʛ
y
k
y
k
ʛ
ʓ
−
1
u
k
ʓʓ
−
1
A
T
u
k
ʓ
¯
=
A
ʓ
¯
u
k
+ ¯
ʛ
y
k
=
2
¯
u
k
,
(5.35)
where Eq. (
5.11
) is used. Substituting the above four equations into Eq. (
5.31
), we
can get Eq. (
5.30
).
5.2.5 Summary of the BFA Algorithm
Let us summarize the BFA algorithm. The algorithm is based on the factor analysis
model presented in Eq. (
5.2
). The algorithm estimates the factor activity
u
k
,the
mixing matrix
A
, and the sensor-noise precision matrix
ʛ
. In the estimation, first
