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

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.