Image Processing Reference
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where and are prior probabilities of the hypothesis and evidence,
correspondingly, and the conditional probability is known as a likeli-
hood function. Thus, Equation 8.19 can be viewed as a method of combining the
results of conventional likelihood analyses for multiple hypotheses into the pos-
terior probability of the hypotheses or some function of it, after being
exposed to the data. The derived posterior probability can be used to select the
most probable hypothesis, i.e., the one with the highest probability
p
()
p
()
pD H
(
|
)
pH D
(
|
)
ˆ
|
=
arg max
pH D
(
|
)
=
arg maxlog
pD H
(
|
)
+
log
pH
(
)
(8.19)
H
D
H
H
leading to the maximum a posteriori (MAP) estimate, where the prior data
probability (often called a partition function) is omitted because the data
does not depend on the choice of the hypothesis, and it does not influence the
maximization over
H
.
For the class of problems related to signal processing, hypothesis
H
generally
consists of a model
M
characterized by a set of nuisance parameters .
The primary goal usually is to find a MAP estimate of some quantity of interest or,
more generally, its posterior probability distribution . can be an
arbitrary function of the hypothesis or its components or often just a
specific nuisance parameter of the model . To obtain the posterior probability
of the nuisance parameter, its marginal probability has to be computed by integration
over the rest of the parameters of the model
p
()
Θ=
{
θ
12…
n
∆
,
}
,
,
pDM
(
∆| ,
,Θ ∆
)
∆=
f
()
∆≡θ
1
∫
(
pDM
(
θ
|
, =
)
p
(
θ θ
,
|
DMd
,
)
θ
1
1
2
…n
2
…n
(8.20)
∫
=
p
θθ
|
,
D M p
,
)
(
θ
|
D M d
,
)
θ
12
…n
2
…n
2
…n
Due to the integration operation involved in determination of any marginal prob-
ability, Bayesian analysis becomes very computationally intensive if an analytical
integral solution does not exist. Therefore, sampling techniques (e.g., MCMC, Gibbs
sampler) are often used to estimate full posterior probability ,
or some statistics such as an expected value
of the quantity of interest.
The Bayesian approach sounds very appealing for the development of multimodal
methods. It is inherently able to incorporate all available evidence, which is, in our
case, obtained from the fMRI and E/MEG data ( ) to support the hypothesis
on the location of neuronal activations, which in the case of DECD model
is However, detailed analysis of Equation 8.18 leads to necessary sim-
plifications and assumptions of the prior probabilities in order to derive a computa-
tionally tractable formulation. Therefore, it often loses its generality. Thus, to derive
a MAP estimator for , Trujilli-Barreto et al. [150] had to condition the com-
putation by a set of smplifying model assumptions such as: noise is formally distrib-
uted, no same parameters of forward models have inverse Gamma prior distributions,
and neuronal activation is described by a linear function of hemodynamic response.
pDM
(
∆| ,
)
ˆ
MAP
=
argmax
pDM
(
∆ |
,
),
∆
|,
DM
∆
EDM
[
∆| ,
]
D
=,
{}
XB
H
=,
{ .
Q
M
ˆ
Q
|,,
XB
M
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