Biomedical Engineering Reference
In-Depth Information
where
u
1
,...,
ʵ
u
K
are collectively denoted
u
. The noise
is assumed to be Gaussian
with the mean of zero, i.e.,
,
ʛ
−
1
p
(
ʵ
)
=
N(
ʵ
|
0
),
(5.6)
where
ʛ
is a diagonal precision matrix. With this assumption, the conditional prob-
ability
p
(
y
k
|
u
k
)
is expressed as
Au
k
,
ʛ
−
1
p
(
y
k
|
u
k
)
=
N(
y
k
|
).
(5.7)
The noise
ʵ
is also assumed to be independent across time. Thus, we have
K
K
Au
k
,
ʛ
−
1
p
(
y
|
u
)
=
p
(
y
1
,...,
y
K
|
u
1
,...,
u
K
)
=
p
(
y
k
|
u
k
)
=
1
N(
y
k
|
),
k
=
1
k
=
(5.8)
where
y
1
,...,
y
K
are collectively denoted
y
. Using the probability distributions
defined above, the Bayesian factor analysis can factorize the sensor data into
L
independent factor activity and additive sensor noise. This factorization is achieved
using the EM algorithm [
4
,
5
]. Explanation of the basics of the EM algorithm is
provided in Sect. B.5 in the Appendix.
5.2.3 EM Algorithm
The E-step of the EM algorithm derives the posterior distribution
p
. Deriva-
tion of the posterior distribution in the Gaussian model is described in Sect. B.3 in the
Appendix. Since the posterior distribution is also Gaussian, we define the posterior
distribution
p
(
u
k
|
y
k
)
(
u
k
|
y
k
)
such that
u
k
,
ʓ
−
1
u
k
|
¯
p
(
u
k
|
y
k
)
=
N(
),
(5.9)
¯
ʓ
where
u
k
is the mean and
is the precision matrix. Using Eqs. (B.24) and (B.25)
with setting
ʦ
to
I
and
H
to
A
in these equations, we get
A
T
ʓ
=
(
ʛ
A
+
I
),
(5.10)
A
T
)
−
1
A
T
u
k
=
(
¯
ʛ
A
+
I
ʛ
y
k
.
(5.11)
The equations above are the E-step update equations in the Bayesian factor analysis.
Let us derive the M-step update equations for the mixing matrix
A
and the noise
precision matrix
ʛ
. To do so, we derive the average log likelihood,
Θ(
A
,
ʛ
)
, and
according to Eqs. (B.34), (
5.5
), and (
5.8
), it is expressed as
