7.3.2
Parameters involved in 
are the noise variances 
and the prior penalties 
(n=1,2). Let the parameters be
denoted uniformly by 
. For k=0,
and for 
,
Note all 
. Let the different energy components be
uniformly denoted by 
. For
k=0,
is the number of NULL labels in f and
is the number of label pairs at least one of which is NULL
. For 
, 
relates to the likelihood energy components; They
measure how much the observations 
deviate from the should-be
true values 
under f:
and
After some manipulation, the energy can be written as
where 
and 
are column vectors of 
components.
Given an instance 
, the 
is a known
vector of real numbers. The 
is the vector of unknown
weights to be determined. The stability follows immediately as
where 
.
Some remarks on 
, 
and E are in order. Obviously, all
and 
are non-negative. In the ideal case of
exact (possibly partial) matching, all 
(
) are zeros
because 
and 
are exactly the same and so are
and 
. In the general case of inexact
matching, the sum of the 
should be as small as possible for
the minimal solution. The following are some properties of E:
is linear in 
. Given
, it is linear in 
.
, 
and 
are equivalent,
as have been discussed.
relative to those of
(
) affect the rate of NULL
labels. The higher
the penalties 
are, the more sites in 
will be
assigned non- NULL
labels; and vice versa.
.
According to the second property, a larger 
relative to
the rest makes the constraint 
and 
play a more
important role. Useless and mis-leading constraints 
and
should be weighted by 0.
Using the third property, one can decrease 
values to
increase the number of the NULL
labels. This is because for the
minimum energy matching, lower cost for NULL
labels makes more sites be
labeled NULL
, which is equivalent to discarding more not-so-reliable
non- NULL
labels into the NULL
bin.
