Necessary and Sufficient Conditions for Characterizing Finite Discrete Distributions with Generalized Shannon's Entropy

TL;DR

This paper identifies the exact conditions under which generalized Shannon's entropy uniquely characterizes finite discrete distributions, providing a foundation for inference and goodness-of-fit testing in unordered sample spaces.

Contribution

It establishes necessary and sufficient conditions for GSE to characterize finite distributions, extending the theoretical understanding and practical application of entropy-based methods.

Findings

K-1 GSE orders suffice for distribution identification

Results apply to distributions with known multiplicity structures

Enable new goodness-of-fit and model comparison methods

Abstract

This article establishes necessary and sufficient conditions under which a finite set of Generalized Shannon's Entropy (GSE) characterizes a finite discrete distribution up to permutation. For an alphabet of cardinality K, it is shown that K-1 distinct positive real orders of GSE are sufficient (and necessary if no multiplicity) to identify the distribution up to permutation. When the distribution has a known multiplicity structure with s distinct values, s-1 orders are sufficient and necessary. These results provide a label-invariant foundation for inference on unordered sample spaces and enable practical goodness-of-fit procedures across disparate alphabets. The findings also suggest new approaches for testing, estimation, and model comparison in settings where moment-based and link-based methods are inadequate.