Biomedical Engineering Reference
In-Depth Information
ʱ
(
|
ʱ
)
(
|
ʱ
)
holds,
is obtained as the one that maximizes
p
y
.This
p
y
is referred to
as the data evidence or the marginal likelihood.
Let us summarize the procedure to estimate the source distribution
x
.First,we
estimate the hyperparameter
ʱ
by maximizing the marginal likelihood function,
ʱ
=
argmax
ʱ
p
(
y
|
ʱ
).
Next, this
ʱ
is substituted into the posterior distribution
p
(
x
|
y
,
ʱ
)
to obtain
. When this posterior is the Gaussian distribution in Eq. (
4.12
), the pre-
cision and mean are obtained by substituting
p
(
x
|
y
,
ʱ
)
into Eqs. (
4.13
) and
(
4.14
). The voxel time courses are reconstructed by computing
ʦ
=
diag
(
ʱ
)
x
k
in Eq. (
4.14
)for
¯
k
=
1
,...,
K
.
4.3 Cost Function for Marginal Likelihood Maximization
As described in the preceding section, the hyperparameter
ʱ
is estimated bymaximiz-
ing the marginal likelihood
p
. In this section, we describe the maximization
of the marginal likelihood, and to do so, let us derive an explicit form of the log
marginal likelihood, log
p
(
y
|
ʱ
)
. Substituting
3
(
y
|
ʱ
)
exp
|
ʦ
|
K
K
K
1
/
2
1
2
x
k
ʦ
p
(
x
|
ʱ
)
=
p
(
x
k
|
ʱ
)
=
−
x
k
,
(4.17)
N
/
2
(
2
ˀ)
k
=
1
k
=
1
and
2
2
K
exp
2
K
K
−
2
p
(
y
|
x
)
=
p
(
y
k
|
x
k
)
=
1
y
k
−
Hx
k
,
(4.18)
ˀ
k
=
1
k
=
into
p
p
p
(
y
|
ʱ
)
=
(
y
,
x
|
ʱ
)
d
x
=
(
y
|
x
)
p
(
x
|
ʱ
)
d
x
,
we obtain
M
/
2
K
2
exp [
1
/
2
|
ʦ
|
p
(
y
|
ʱ
)
=
−
D
] d
x
,
(4.19)
N
/
2
(
2
ˀ)
ˀ
where
K
K
=
2
1
2
2
x
k
ʦ
1
y
k
−
Hx
k
+
x
k
,
D
(4.20)
k
=
k
=
1
3
Note that
M
and
N
are respectively the sizes of
y
k
and
x
k
.
