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hybrid sLORETA/FOCUSS algorithms [81]. In practice, the discrepancy principle
is often used to avoid under-regularizing. Once the optimal maximum likelihood or
maximum a posteriori hyperparameters have been learned (i.e., they have stopped
changing), the SBL inverse operator can be expressed as
L
T
ˆ
−
1
Ω
(
sbl
)
=
Σ
(
sbl
)
J
Σ
(
sbl
)
B
,
(8.37)
and the posterior mean is given by
E
J
(
sbl
)
J
J
=
Ω
(
sbl
)
B
=
|
B
;
Σ
.
(8.38)
It is important to note that many useful SBL variants can be obtained by the
reparametrization of the source covariance matrix
1
C
i
α
−
i
. In fact, if only
a few hyperparameters are used, and each controls many source points, then the
parametrization cannot support sparse estimates. For example, in the restricted max-
imum likelihood (ReML) algorithm one of the source covariance components is the
identity matrix, which is controlled by a single hyperparameter [18, 68, 50, 99]. In
standard SBL,
C
i
d
α
i
Σ
J
=
∑
=
e
(
i
)
e
T
=
, where
e
(
i
)
is a vector with zeros everywhere except at
the
i
th element, where it is one. This delta function parametrization can be extended
to box car functions in which
e
(
i
)
takes a value of 1 for all three dipole components
or for a patch of cortex. Alternatively, each
e
(
i
)
can be substituted by a geodesic ba-
sis function
(
i
)
ψ
(
i
)
(e.g., a 2D Gaussian current density basis function) centered at the
i
th source point and with some spatial standard deviation [79, 70]. This approach
can be extended to a multiscale algorithm, in which the source covariance matrix
is composed of components across many possible spatial scales, by using multiple
ψ
(
i
)
vectors located at the
i
th source point but with different spatial standard devi-
ations [70, 73, 71, 72]. This approach can be used to estimate the spatial extent of
distributed sources by using a mixture model of geodesic Gaussian distributions at
different spatial scales. Such multiscale approach can also be used with parameter
MAP estimation [47, 72].
The problem of finding optimal hyperpriors to handle multimodal posteriors and
to eliminate the use of improper priors has been dealt with by using flat hyperpriors
or by introducing MCMC strategies [59, 60]. In practice, the noninformative hyper-
prior works well and helps avoid the problem of determining the optimal hyperprior.
Finally, as explained for parameter MAP estimation, to simultaneously localize the
generators of a very long time series of any length very quickly, instead of localizing
the times series matrix
B
, one can use the matrix
US
1
/
2
, where
U
and
S
are the left
singular vectors and singular values matrices of
BB
T
.
8.7 Spatial Scanning and Beamforming
An alternative approach to the
ill-posed
bioelectromagnetic inverse problem is to
independently scan for dipoles within a grid containing candidate locations (i.e.,
