7.3.4
Assuming the functional form of the noise distribution is known, then we
can take the advantage of (partial) parametric modeling for the
estimation. When the noise is additive white Gaussian with unknown
, the estimate can be obtained in closed-form. The
closed-form estimation is performed in two steps: (1) to estimate the
noise variances 
(
); then to compute the
weights 
(
) using the relationship in
Eq.(7.36); and (2) to compute allowable
, relative to 
(
), to
satisfy the correctness in (7.7). Optimization as
(7.16) derived using a non-parametric
principle may not be applicable in this case.
Given d, D and 
, the Gaussian noise variances can be
estimated by maximizing the joint likelihood function 
(ML estimation). The ML estimates are simply
and
where
is the number of non- NULL labels in f and
is the number of label pairs neither of which is the NULL
. The
optimal weights for 
can be obtained immediately by
So far, only the exemplary configurations 
, not others, are
used in computing the 
.
Now the remaining problem is to determine 
to meet the
correctness (7.7). Because of
, this is to estimate the MRF parameters 
in the prior distributions implied in the given exemplar. There may be
a range of 
under which each 
is
correctly encoded. The range is determined by the lower and upper
bounds.
In doing so, only those configurations in 
, which reflect
transitions from a non- NULL
to the NULL
label and the other way
around, are needed; the other configurations, which reflect transitions
from one non- NULL
to another non- NULL
label, are not. This subset is
obtained by changing each of the non- NULL
labels in 
to the
NULL
label or changing each of the NULL
labels to a non- NULL
label.
First, consider label changes from a non- NULL
label to the NULL
label.
Assume a configuration change from 
to f is due to the change
from 
to 
for just one 
. The
corresponding energy change is given by
The above change must be positive, 
.
Suppose there are N non- NULL
labeled sites under 
and
therefore 
has N such neighboring configurations. Then N
such inequalities of 
can be obtained. The
two unknowns, 
and 
, can be solved for and
used as the lower bounds 
and
.
Similarly, the upper bounds can be computed by considering label changes
from the NULL
label to a non- NULL
label. The corresponding energy
change due to a change from 
to 
is given by
The above change must also be positive, 
.
Suppose there are N NULL
labeled sites under 
and recall that
there are M possible non- NULL
labels in 
. Then 
inequalities can be obtained. The two unknowns, 
and
, can be solved and used as the upper bounds
and 
. If the exemplary
configurations 
are minimal for the corresponding 
and
, then the solution must be consistent, that is
, for each of the instances.
Now, the space of all correct parameters is given by
A correct 
makes 
for all 
. The hyperplane
partitions 
into two parts, with all 
in the
positive side of it. The value for 
may be simply set to
the average 
.
When there are L>1 instances, 
(
) are obtained
from the data set computed from all the instances. Given the common
, L correct ranges can be computed. The correct range
for the L instances as a whole is the intersection of the L ranges.
As the result, the overall 
is the maximum of all
the lower bounds and 
the minimum of all the upper
bounds. Although each range can often be consistent, i.e.
for each n, there is less
chance to guarantee that they as a whole are consistent for all
: The intersection may be empty when L>1. This
inconsistency means a correct estimate does not exist for all the
instances as a whole. There are several reasons for this. First of all,
the model assumptions such as Gaussian is not verified by the data set,
especially when the data set is small. In this case, the noise in
different instances has different variances; when the ranges are
computed under the assumption that the ML estimate is common to all the
instances, they may not be consistent to each other. This is the most
direct reason for the inconsistency. Second, 
in some exemplary
instances can not be embedded as the minimal energy configuration to
satisfy the given constraints. Such instances are mis-leading and also
causes the inconsistency.
