When the conditions for MaxEnt PDF projection
exist, then (3.5) holds.
All reference hypotheses in class
(3.5), determined by function
produce exactly the same projected PDF,
which will be the maximum entropy projected PDF.
This follows from the fact that if (3.5)
holds, then the J-function (2.3)
is independent of when is fixed.
This means that in the generation process
(for generating samples of
explained in Section 3.3,
the manifold distribution (2.5)
is in fact the uniform distribution.
This is true regardless of which reference hypothesis
is chosen within class (3.5).
Discussion on the choice of
It may seem difficult to choose a reference
hypothesis by selecting the function
Although any reference hypothesis in class (3.5)
will result in the same projected PDF,
there are some that are easier to analyze.
After all, it is necessary to derive the
, preferably in closed form.
Since we must also assume that the function
in (3.4) exists,
then we can restrict ourselves to
the exponetial class (3.7).
This class is determined by the type of norm in effect.
The feature may also contain more than one norm.
For example, the feature
has two different norms.
In this case, both the Laplacian and Gaussian reference hypotheses
(See Table 3.1) could be used and would in fact produce the same
For simple norms, there are canonical reference hypotheses
In all practical problems we have encountered,
either the canonical Exponential or Gaussian
So, in summary, we recommend that you insure that the
feature contains a simple energy statistic,
such as found in Table 3.1, and use the
corresponding canonical reference hypothesis.
For additional discussion of the reference hypothesis, see Section 2.2.3.