In the second case above, when is not constrained to a compact set,
the condition (3.4) must be satisfied.
To do this, we need to insure that
contains an energy statistic .
An energy statistic is a statistic, usually scalar, that
contains information about the norm (size) of .
The energy statistic can be explicitly included as a
component of , such as a sample variance,
but does not need to be known explicitly.
The important thing is that if
contains an energy
statistic, then (3.4) is satisfied
for some norm. A norm
must meet the properties of
triangle inequality (
and zero property
Examples of norms are generalized sample moments
Energy Statistic (ES)
where and .
The corresponding ES is, for example
By insuring that contains an energy statistic,
then for any fixed finite-valued feature value ,
is fixed, therefore the manifold
(2.4), is compact.
This is necessary to insure the MaxEnt property .
Also necessary for the maxEnt property is satisfying (3.5)
for some function , which means that
on only through
then it is easy to find a reference hypotheses that meets
(3.5). One example can be written
for , and is the apropriate scale factor.
Further examples of energy statistics and associated canonical
reference hypotheses are provided in Table 3.1.
leads to the 2-norm on
and can be paired with the Gaussian
Interestingly, no matter which reference hypothesis meets
(3.5) the resulting projected PDF is the same.
We will explain this in Section 3.2.3.
Reference PDFs and their energy statistics.
The reference PDFs depend on the data only through the
indicated energy statistics. Note that
is the positive
where all elements of are positive
is the hypercube
where all elements of are in .