Biomedical Engineering Reference
In-Depth Information
B.5.4
Update Equation for Noise Precision
ʛ
We next derive the update equation for
ʛ
. The derivative of the average data likelihood
with respect to
ʛ
is given by
2
E
K
T
∂
∂
ʛ
ʘ(
ʦ
,
ʛ
)
=
K
2
ʛ
−
1
1
−
1
(
y
k
−
Hx
k
)(
y
k
−
Hx
k
)
.
(B.43)
k
=
Setting this derivative to zero, we get
K
E
K
T
1
ʛ
−
1
=
1
(
y
k
−
Hx
k
)(
y
k
−
Hx
k
)
.
(B.44)
k
=
Using the mean and the precision of the posterior distribution, the above equation is
rewritten as
K
1
K
ʛ
−
1
T
ʓ
−
1
H
T
=
1
(
y
k
−
H
x
k
)(
¯
y
k
−
H
x
k
)
¯
+
H
.
(B.45)
k
=
B.5.5
Summary of the EM Algorithm
The EM algorithm is a recursive algorithm. We first set appropriate initial values to
the hyperparameters
, which are then used for computing the posterior dis-
tribution, i.e., for computing the mean and the precision of the posterior distribution
using
ʦ
and
ʛ
H
T
ʓ
=
ʦ
+
ʛ
H
,
x
k
=
ʓ
−
1
H
T
H
T
)
−
1
H
T
ʛ
y
k
=
(
ʦ
+
ʛ
H
ʛ
y
k
.
With these parameter values of the posterior distribution, the hyperparameters
ʦ
and
ʛ
are updated using
K
1
K
ʦ
−
1
x
k
+
ʓ
−
1
=
1
¯
x
k
¯
,
k
=
K
1
K
ʛ
−
1
ʓ
−
1
H
T
T
=
1
(
y
k
−
x
k
)(
¯
y
k
−
x
k
)
¯
+
.
H
H
H
k
=
The step that computes the posterior distribution is called the E step, and the step
that estimates the hyperparameters is called the M step. The EM algorithm updates
