Biomedical Engineering Reference
In-Depth Information
K
K
2
log
1
2
x
k
ʦ
log
p
(
y
,
x
|
ʦ
,
ʛ
)
=
|
ʦ
|−
x
k
k
=
1
K
K
2
log
1
2
T
+
|
ʛ
|−
1
(
y
k
−
Hx
k
)
ʛ
(
y
k
−
Hx
k
).
(B.37)
k
=
Thus, the average data likelihood is obtained as
2
E
K
K
2
log
1
x
k
ʦ
ʘ(
ʦ
,
ʛ
)
=
|
ʦ
|−
x
k
k
=
1
2
E
K
K
2
log
1
T
+
|
ʛ
|−
1
(
y
k
−
Hx
k
)
ʛ
(
y
k
−
Hx
k
)
.
(B.38)
k
=
B.5.3
Update Equation for Prior Precision
ʦ
,
ʦ
Let us first derive the update equation for
ʦ
. The updated value of
ʦ
, is the one
that maximizes the average data likelihood,
ʘ(
ʦ
,
ʛ
)
. The derivative of
ʘ(
ʦ
,
ʛ
)
with respect to
ʦ
is given by
2
E
K
∂
∂
ʦ
ʘ(
ʦ
,
ʛ
)
=
K
2
ʦ
−
1
1
x
k
x
k
−
.
(B.39)
k
=
1
Here we use Eqs. (
C.89
) and (
C.90
). Setting this derivative to zero, we have
K
E
K
1
ʦ
−
1
x
k
x
k
=
.
(B.40)
k
=
1
Using the posterior precision
ʓ
, the relationship
E
K
K
x
k
x
k
x
k
ʓ
−
1
=
1
¯
x
k
¯
+
K
(B.41)
k
=
1
k
=
holds. Therefore, the update equation for
ʦ
is expressed as
k
=
1
¯
K
1
K
ʦ
−
1
x
k
+
ʓ
−
1
=
x
k
¯
.
(B.42)
