Information Technology Reference
In-Depth Information
d
v
log
Σ
B
+
tr
B
T
ˆ
B
,
2
(
ml
)
α
(
ml
)
,
2
ϒ
)=
ˆ
Σ
−
1
B
ˆ
σ
ˆ
=
arg min
α
,
σ
−
log
p
(
B
|
α
,
σ
(8.31)
ϒ
2
ϒ
where
ˆ
Σ
J
L
T
(
α
)
−
1
Σ
B
=
L
+
Σ
ϒ
is the model data covariance, and
Σ
J
=
diag
is the
prior source covariance matrix. The noise covariance matrix,
Σ
ϒ
, can be assumed
2
ϒ
2
ϒ
to be a multiple of the identity matrix (e.g.,
is the noise variance,
a hyperparameter that can also be learned from the data), or can be empirically
obtained from the measurements.
In the case of Gamma hyperpriors (i.e.,
a
,
b
,
c
, and
d
are nonzero), the poste-
rior probability of the log hyperparameters given the data, that is, the product of the
marginal likelihood and the hyperprior,
p
σ
I
, where
σ
2
2
(
|
α
,
ϒ
)
(
α
,
ϒ
)
B
log
log
σ
p
log
log
σ
, is maxi-
mized, or equivalently the negative log posterior is minimized,
(
)
α
(
map
)
,
2
map
2
2
log
ˆ
log ˆ
σ
=
arg
min
−
log
p
(
B
|
log
α
,
log
σ
ϒ
)
p
(
log
α
,
log
σ
ϒ
)
ϒ
2
ϒ
log
α
,
log
σ
d
v
log
Σ
B
+
tr
B
T
ˆ
B
+
ˆ
Σ
−
1
B
=
arg
min
2
ϒ
log
α
,
log
σ
d
α
i
=
1
(
a
logα
i
−
b
α
i
)+
c
logσ
−
2
σ
−
2
ϒ
−
d
ϒ
.
(8.32)
Evidence maximization is usually achieved by using Expectation-Maximization
update rules
1
d
v
d
r
2
b
−
1
Ω
(
k
)
i
:
B
d
r
tr
I
L
:
i
1
2
F
+
α
(
k
+
1
)
i
−
Ω
(
k
)
i
:
α
−
1
(
k
)
i
=(
1
+
2
a
)
+
,
(8.33)
1
d
v
2
d
LJ
(
k
)
2
F
+
σ
2
(
k
+
1
)
ϒ
2
(
k
)
ϒ
R
(
k
)
)+
σ
=
B
−
tr
(
/
(
d
b
+
2
c
)
,
(8.34)
or alternatively using the MacKay gradient update rules
1
2
a
1
d
v
d
r
2
b
d
r
tr
I
i
:
L
:
i
Ω
i
:
B
2
F
+
(
k
+
1
)
i
(
k
)
−
1
(
k
)
i
(
k
)
α
=
−
Ω
α
+
/
,
(8.35)
1
d
v
2
d
LJ
(
k
)
d
b
−
2
c
2
F
+
2
(
k
+
1
)
R
(
k
)
)+
σ
=
B
−
/
tr
(
,
(8.36)
ϒ
where
L
:
i
is a matrix with column vectors from
L
that are controlled by the
same
i
th hyperparameter,
d
r
is the rank of
L
:
i
L
:
i
,
L
:
i
ˆ
Σ
(
k
B
−
1
Ω
(
k
)
i
:
=
α
−
1
(
k
)
i
, and
R
(
k
)
=
Σ
(
k
J
L
T
ˆ
Σ
(
k
B
−
1
L
is the
k
th resolution matrix. With fixed dipole orienta-
tions
L
:
i
is a vector, but with loose orientations
L
:
i
is a
d
b
by three matrix. For
patch source models involving dipoles within a region,
L
:
i
is a matrix containing
all gain vectors associated with the local patch of cortex. The gradient update rule
is much faster than the EM rule and is similar to the update rules used in several
