7.3.3
The following analysis examines how the minimum 
changes as the observation changes from d0 to 
where 
is a perturbation. In the beginning when 
is close to 0, 
should remain as the minimum for a range of
such small 
. This is simply because 
is
continuous with respect to d. When the perturbation becomes larger and
larger, the minimum has to give way to another configuration.
When should a change happens? To see the effect more clearly, assume the
perturbation is in observation components relating to only a particular
i so that the only changes are 
and
, 
. First,
assume that 
is a non- NULL
label (
) and consider
such a perturbation 
that incurs a larger likelihood
potential. Obviously, as the likelihood potential (conditioned on
)
increases, it will eventually become cheaper for 
to
change to 
. More accurately, this should happen when
where
is the number of non- NULL
labeled sites in 
under 
.
Next, assume 
and consider such a perturbation that incurs a
smaller likelihood potential. The perturbation has to be such a
coincidence that d and D become more resemblant to each other. As
the conditional likelihood potential
decreases, it will eventually become cheaper for 
to change
to one of the non- NULL
labels, 
. More accurately, this
should happen when
The above analysis shows how the minimal configuration 
adjusts as
d changes when 
is fixed. On the other hand, the 
can be
maintained unchanged by adjusting 
; this means a different
encoding of constraints into E.
