Biomedical Engineering Reference
In-Depth Information
ʲ
Regarding the hyperparameter
, using a similar derivation, we have
MK
E
K
1
ʲ
−
1
T
=
1
(
y
k
−
Hx
k
)
(
y
k
−
Hx
k
)
.
(B.47)
k
=
Here, considering the relationships
E
K
E
K
x
k
H
T
Hx
k
y
k
y
k
−
T
x
k
H
T
y
k
−
y
k
Hx
k
+
1
(
y
k
−
Hx
k
)
(
y
k
−
Hx
k
)
=
k
=
k
=
1
y
k
y
k
−
K
x
k
]
H
T
y
k
−
y
k
H
E
x
k
H
T
Hx
k
]
=
E
[
[
x
k
]+
E
[
k
=
1
y
k
y
k
− ¯
K
x
k
H
T
y
k
−
y
k
H
x
k
H
T
Hx
k
]
=
x
k
+
¯
E
[
k
=
1
(B.48)
and
x
k
H
T
Hx
k
]=
H
T
Hx
k
x
k
)
]
E
[
E
[
tr
(
tr
H
T
H
E
x
k
x
k
=
tr
H
T
H
+
ʓ
−
1
x
k
=
x
k
¯
¯
tr
H
T
H
ʓ
−
1
x
k
H
T
H
= ¯
x
k
+
¯
,
(B.49)
we finally obtain
1
K
ʓ
−
1
tr
H
T
H
K
1
M
ʲ
−
1
2
=
1
y
k
−
H
x
k
¯
+
.
(B.50)
k
=
B.6
Variational Bayesian Inference
In the preceding sections, the unknown
x
is estimated based on the posterior distribu-
tion
p
, and the hyperparameters are estimated by using the EM algorithm. In this
section, we introduce a method called variational Bayesian inference, which makes
it possible to derive approximate posterior distributions for hyperparameters [2].
(
x
|
y
)
