Biomedical Engineering Reference
In-Depth Information
assess the relative strengths of many Bayesian strategies with respect to particular
applications; it will also help ensure that different methods are used to their full
potential and not underutilized.
6.2 Bayesian Modeling Framework
In a Bayesian framework all prior assumptions are embedded in the distribution
p
(
s
)
.
If under a given experimental or clinical paradigm this
p
(
s
)
were somehow known
exactly, then the posterior distribution
p
(
s
|
y
)
can be computed via Bayes rule:
p
(
y
|
s
)
p
(
s
)
p
(
s
|
y
)
=
.
(6.2)
p
(
y
)
This distribution contains all possible information about the unknown
s
conditioned
on the observed data
y
. Two fundamental problems prevent using
p
(
s
|
y
)
for source
localization. First, for most priors
p
(
s
)
, the distribution
p
(
y
)
given by:
p
(
y
)
=
p
(
y
|
s
)
p
(
s
)
d
s
.
(6.3)
cannot be computed. Because this quantity, which is sometimes referred to as the
model evidence or marginal likelihood, is required to compute posterior moments
and is also sometimes used to facilitate model selection, this deficiency can be very
problematic. Of course if only a point estimate for
s
is desired, then this normalizing
distribution may not be needed. For example, a popular estimator involves finding
the value of
s
that maximizes the posterior distribution, often called the maximum a
posteriori orMAP estimate, and is invariant to
p
. HoweverMAP estimates may be
unrepresentative of posterior mass and are unfortunately intractable to compute for
most
p
(
y
)
given reasonable computational resources. Secondly, we do not actually
know the prior
p
(
s
)
and so some appropriate distribution must be assumed, perhaps
based on neurophysiological constraints or computational considerations. In fact, it
is this choice, whether implicitly or explicitly, that differentiates a wide variety of
localization methods at a very high level.
Such a prior is often considered to be fixed and known, as in the case of minimum
(
s
)
2-norm approaches [
2
], minimum current estimation (MCE), FOCUSS, sLORETA,
and minimum variance beamformers. Alternatively, a number of empirical Bayesian
approaches have been proposed that attempt a form of model selection by using
the data to guide the search for an appropriate
p
. In this scenario, candidate
priors are distinguished by a set of flexible hyperparameters
(
s
)
that must be estimated
via a variety of data-driven iterative procedures including hierarchical covariance
component models, automatic relevance determination (ARD), and several related
variational Bayesian methods (for a more comprehensive list of citations see Wipf
et al. [
10
]).
Ęł
