Biomedical Engineering Reference
In-Depth Information
x
1
,
x
2
,...,
x
K
and the whole sensor time series
y
1
,
y
2
,...,
y
K
. We first formulate
the Champagne algorithm omitting the source orientation estimation. That is, we
assume that the source orientation is predetermined at each voxel and use the mea-
is expressed as
y
k
=
Hx
k
+
ʵ
,
(4.3)
4.2 Probabilistic Model and Method Formulation
T
, the prior distribution is
Defining a column vector
ʱ
such that
ʱ
=[
ʱ
1
,...,ʱ
N
]
expressed as
K
K
,
ʦ
−
1
p
(
x
|
ʱ
)
=
p
(
x
k
|
ʱ
)
=
1
N(
x
k
|
0
),
(4.4)
k
=
1
k
=
where
, which indicates a diagonal matrix whose diagonal
elements are those of a vector in the parenthesis. Thus, we have
ʦ
is equal to
ʦ
=
diag
(
ʱ
)
N
,
ʦ
−
1
, ʱ
−
1
j
p
(
x
k
|
ʱ
)
=
N(
x
k
|
0
)
=
1
N(
x
j
|
0
).
(4.5)
j
=
ʵ
Since we assume that the noise
is independent across time, the conditional proba-
bility
p
(
y
|
x
)
is expressed as
K
K
Hx
k
, ʲ
−
1
I
p
(
y
|
x
)
=
p
(
y
k
|
x
k
)
=
1
N(
y
k
|
).
(4.6)
k
=
1
k
=
In this chapter, the noise precision matrix is assumed to be
is known.
Therefore, the parameters we must estimate is the voxel source distribution
x
and
the hyperparameter
ʲ
I
where
ʲ
.
In truly Bayesian formulation, we have to derive the joint posterior distribution
of all the unknown parameters,
p
ʱ
(
x
,
ʱ
|
y
)
,using
(
|
,
ʱ
)
(
,
ʱ
)
p
y
x
p
x
(
,
ʱ
|
)
=
.
p
x
y
(4.7)
p
(
y
)
However, we cannot compute
2
p
p
(
y
)
=
(
y
|
x
,
ʱ
)
p
(
x
,
ʱ
)
d
x
d
ʱ
,
(4.8)
2
In Eq. (
4.8
), the notation d
x
indicates d
x
1
d
x
2
...
d
x
K
.
