When is itself a compact set, such as the unit hypercube for , we can make the uniform distribution. Then, so long as the manifold is compact for all ^{3.1}, then will be a proper uniform distribution for all , which has maximum entropy. Alternatively, when is infinite in extent, the manifold can be forced to be compact by the inclusion of an energy statistic in (first proposed in [5]). The solution for compact and the solution for unbounded are formalized by the following two theorems.
The second case considers when is not compact.
The highest-entropy distribution, which is found by maximizing the entropy of over , is denoted by . Intuitively, has the MaxEnt property because all samples generated by for a given are equally likely (the uniform distribution, which has maximum entropy on a compact set). This is a result of the fact that for a given , all samples on the manifold have constant value on , making also constant on the manifold, implying that the manifold distribution is in fact the uniform distribution. The reader is referred to a previous article for additional details of the proof [5].
Baggenstoss 2017-05-19