Biomedical Engineering Reference
In-Depth Information
The maximum a posteriori parameters estimation involves the determination
of
x
that maximizes
p
(
x
|
y
) with respect to
x
. By Bayes' rule,
p
(
y
|
x
)
p
(
x
)
p
(
y
)
p
(
x
|
y
)
=
(9.4)
.
Since the denominator of Eq. 9.4 does not affect the optimization, the MAP
parameters estimation can be obtained, equivalently, by maximizing the numer-
ator of Eq. 9.4 or its natural logarithm; that is, we need to find
x
which maximizes
the following criterion:
L
(
x
|
y
)
=
ln
p
(
y
|
x
)
+
ln
p
(
x
)
.
(9.5)
The first term in Eq. 9.5 is the likelihood due to the low-level process and the
second term is due to the high-level process. Based on the models of the high-
level and low-level processes, the MAP estimate can be obtained.
In order to carry out the MAP parameters estimation in Eq. 9.5, one needs to
specify the parameters of the two processes. A popular model for the high-level
process is the Gibbs Markov model. In the following sections we introduce a new
accurate model to model the low-level process. In this model we will assume
that each class consists of a mixture of normal distributions which follow the
following equation:
l
=
1
π
l
p
(
y
|
C
l
)
,
n
i
p
(
y
|
i
)
=
for
i
=
1
,
2
,...,
m
,
(9.6)
where
n
i
is the number of normal components that formed class
i
,
π
is the
n
i
corresponding mixing proportion, and
{
C
l
}
l
=
1
is the number of Gaussian com-
ponents that formed class
i
. So the overall model for the low-level process can
be expressed as follows:
m
p
es
(
y
)
=
p
(
i
)
p
(
y
|
i
)
.
(9.7)
i
=
1
In our proposed algorithm the priori probability
p
(
i
) is included in the mixing
proportion for each class.
9.2.5 Parameter Estimation for Low-Level Process
In order to estimate the parameters for low-level process, we need to esti-
mate the number of Gaussian components that formed the distribution for each
class, their means, the variances, and mixing proportions for each Gaussian
