Biomedical Engineering Reference
In-Depth Information
6.3 Bayesian Modeling Using General Gaussian Scale
Mixtures and Arbitrary Covariance Components
In this section, we present a general-purpose Bayesian framework for source
localization and discuss a central distinction between fixed-prior MAP estimation
schemes and empirical Bayesian approaches that adopt a flexible, parameterized
prior. While often derived using different assumptions and methodology, they can
be related via a simple hierarchical structure based on general Gaussian scale mix-
ture distributions with arbitrary covariance components. Numerous special cases of
this model have been considered previously in the context of MEG and EEG source
localization and related problems as will discussed in subsequent sections.
6.3.1 The Generative Model
To begin we invoke the noise model from (
6.1
), which fully defines the assumed data
likelihood
⊛
⊞
2
d
s
1
2
⊝
−
⊠
,
p
(
y
|
s
)
∝
exp
y
−
L
i
s
i
(6.4)
i
=
1
Σ
−
1
W
denotes the weighted matrix norm
trace
2
where
.
While the unknown noise covariance can also be parameterized and seamlessly
estimated from the data via the proposed paradigm, for simplicity we assume that
Σ
X
[
X
T
WX
]
is known, estimated from the data using a variational Bayesian factor analy-
sis (VBFA) model as discussed in Sect.
5.3
and that it is fixed. Next we adopt the
following source prior for
s
:
exp
d
ʳ
1
2
trace
s
T
Σ
−
1
s
p
(
s
|
ʳ)
∝
−
[
s
]
,Σ
s
=
1
ʳ
i
C
i
.
(6.5)
i
=
This is equivalent to applying independently, at each time point, a zero-mean
Gaussian distributionwith covariance
Σ
s
to each column of
s
.Here
ʳ
[
ʳ
1
,...,ʳ
d
ʳ
]
is a vector of
d
nonnegative hyperparameters that control the relative contribution of
each covariance basis matrix
C
i
. While the hyperparameters are unknown, the set of
components
C
ʳ
is assumed to fixed and known. Such a formu-
lation is extremely flexible however, because a rich variety of candidate covariance
bases can be proposed as will be discussed in more detail later. Moreover, this struc-
ture has been advocated by a number of others in the context of neuroelectromagnetic
source imaging [
7
,
11
]. Finally, we assume a hyperprior on
{
C
i
:
i
=
1
,...,
d
ʳ
}
ʳ
of the form
d
ʳ
1
2
exp
p
(ʳ)
=
[−
f
i
(ʳ
i
)
]
(6.6)
i
=
1
