## Energy Statistic (ES)

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 [5]. 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 scalability ( ), triangle inequality ( ), and zero property . Examples of norms are generalized sample moments

where and . The corresponding ES is, for example

 (3.6)

By insuring that contains an energy statistic, then for any fixed finite-valued feature value , is fixed, therefore the manifold , in (2.4), is compact. This is necessary to insure the MaxEnt property [5]. Also necessary for the maxEnt property is satisfying (3.5) for some function , which means that depends on only through . Given that exists, then it is easy to find a reference hypotheses that meets (3.5). One example can be written

 (3.7)

for , and is the apropriate scale factor.

Example 6   Let be the positive quadrant of , such that . The statistic

 (3.8)

leads to the 1-norm on and can be paired with the exponential reference hypothesis

 (3.9)

where and .

Example 7   The statistic

 (3.10)

leads to the 2-norm on and can be paired with the Gaussian

 (3.11)

where and .

Further examples of energy statistics and associated canonical reference hypotheses are provided in Table 3.1.

Interestingly, no matter which reference hypothesis meets (3.5) the resulting projected PDF is the same. So, if , and , then

We will explain this in Section 3.2.3.

Table: Reference PDFs and their energy statistics. The reference PDFs depend on the data only through the indicated energy statistics. Note that is the positive quadrant of where all elements of are positive and is the hypercube where all elements of are in .
 Name Data range Ref. Hyp. Energy Statistic Gaussian Laplacian Exponential Uniform n/a

Baggenstoss 2017-05-19