Biomedical Engineering Reference
In-Depth Information
the posterior distribution and the values of hyperparameters by repeating the E and
M steps. As a result of this recursive procedure, the marginal likelihood
p
(
|
ʦ
,
ʛ
)
y
is increased. A proof may be found, for example, in [1].
B.5.6 Hyperparameter Update Equations when
ʦ
=
ʱ
I
and
ʛ
=
ʲ
I
x
k
, becomes equal to the
¯
L
2
-norm regularized minimum-norm solution if we use
I
in the
Gaussian model. Let us derive update equations for the scalar hyperparameters
ʦ
=
ʱ
I
and
ʛ
=
ʲ
ʱ
and
in this case.
The average data likelihood in this case is given by
ʲ
ʱ
−
2
E
K
NK
2
x
k
x
k
ʘ(ʱ,ʲ)
=
log
k
=
1
ʲ
−
2
E
K
MK
2
T
+
log
1
(
y
k
−
Hx
k
)
(
y
k
−
Hx
k
)
.
k
=
Therefore, using
2
E
K
∂
∂ʱ
ʘ(ʱ,ʲ)
=
NK
2
1
x
k
x
k
−
=
0
ʱ
k
=
1
and
E
K
tr
K
x
k
x
k
x
k
x
k
)
=
E
(
k
=
1
k
=
1
tr
K
ʓ
−
1
x
k
=
1
¯
x
k
¯
+
K
k
=
K
x
k
¯
(
ʓ
−
1
=
1
¯
x
k
+
K
tr
),
k
=
we get
NK
E
K
1
K
K
1
1
N
ʱ
−
1
x
k
x
k
2
(
ʓ
−
1
=
=
1
¯
x
k
+
tr
)
(B.46)
k
=
1
k
=
ʱ
for the update equation of
.
