Necessary and Sufficient Conditions for Characterizing Finite Discrete Distributions with Generalized Shannon's Entropy
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915598689107968 |
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| author | Zhang, Jialin |
| author_facet | Zhang, Jialin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Necessary and Sufficient Conditions for Characterizing Finite Discrete Distributions with Generalized Shannon's Entropy Zhang, Jialin Probability 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. |
| title | Necessary and Sufficient Conditions for Characterizing Finite Discrete Distributions with Generalized Shannon's Entropy |
| topic | Probability |
| url | https://arxiv.org/abs/2511.02955 |