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usually assumed to be Gaussian. Since there is no unique solution to this problem,
additional priors are needed to find solutions of interest. These algorithms can be
best presented from the standpoint of a general Bayesian framework that makes ex-
plicit the source prior assumptions using probability density functions (pdfs). Bayes
theorem
p
(
B
|
J
,H
)
p
(
J
|H
)
p
(
J
|
B
,H
)=
(8.6)
p
(
B
|H
)
states that the posterior probability of
J
given the measurements
B
and hypothesis or
Bayesian model
(consisting of all implicit assumptions and parameters) is equal
to the likelihood of
J
multiplied by the marginal prior probability of
J
, divided by
the normalizing constant of the posterior called the evidence for
H
H
which is defined
by
p
p
(
B
|H
)=
(
B
|
J
,H
)
p
(
J
|H
)
dJ
.
(8.7)
8.6.1 Bayesian Maximum a Posteriori (MAP) Estimates
A Gaussian likelihood model is usually assumed,
−
d
b
d
v
/
2
exp
B
LJ
,H
)=
2
1
2
F
2
ϒ
p
(
B
|
J
πσ
−
−
,
(8.8)
2
ϒ
2
σ
together with a prior pdf, which assigns a probability density to every possible esti-
mate before the measurement data has been taken into account. A very useful family
of prior models can be obtained with the generalized Gaussian marginal pdfs
exp
i
=
1
J
i
:
d
n
p
q
(
|H
)
∝
−
(
)
,
p
J
sgn
p
(8.9)
where
d
n
is the total number of source points,
p
specifies the shape of the pdf or
equivalently the
p
-
norm
-like measure to be minimized, which controls the sparsity
of the estimate, and
q
specifies the norm of
J
i
:
(the matrix containing the row vectors
of
J
associated with the
i
th source point as indexed by
i
), which here is assumed to
be the Frobenius norm. The signum function, sgn
(
p
)
, takes values of 1, 0, or
−
1for
=
positive, zero, or negative
p
, respectively. However, the special case of
p
0 (i.e.,
the so-called zero norm) rather implies minimizing the number of
J
i
:
's with nonzero
Frobenius norms. Other priors are also possible for MAP estimation.
Since the normalizing constant
p
does not affect the location of the poste-
rior mode, it can be ignored, and thus the MAP point estimate can be computed by
(
B
|H
)
2
(
map
)
J
(
map
)
,
J
J
J
σ
ˆ
=
arg max
J
log
p
(
|
B
)
∝
log
p
(
B
|
,H
)+
log
p
(
|H
)
.
(8.10)
ϒ
2
ϒ
,
σ
