重新审视钦钦的熵的第四公理

IF 0.9 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Stats Pub Date : 2023-07-27 DOI:10.3390/stats6030049
Zhiyi Zhang, Hongwei Huang, Hao Xu
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引用次数: 0

摘要

Boltzmann–Gibbs–Shannon(BGS)熵是唯一满足四个条件的熵形式,即Khinchin公理。BGS熵的唯一性定理,加上Shannon的互信息完全表征了两个潜在随机元素之间的独立性,使BGS熵在许多研究领域中处于特殊地位。在本文中,第四个公理被一个稍微削弱的条件所取代:一个熵,其相关的互信息为零,当且仅当两个潜在的随机元素是独立的。在较弱的第四公理下,通过护送变换来寻求熵的其他形式。本文报告了两个主要结果。首先,除了BGS熵之外,还有许多熵满足较弱的条件,但保留了BGS熵的所有期望效用。其次,通过护送变换,新确定的熵是唯一满足较弱公理的熵。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Khinchin’s Fourth Axiom of Entropy Revisited
The Boltzmann–Gibbs–Shannon (BGS) entropy is the only entropy form satisfying four conditions known as Khinchin’s axioms. The uniqueness theorem of the BGS entropy, plus the fact that Shannon’s mutual information completely characterizes independence between the two underlying random elements, puts the BGS entropy in a special place in many fields of study. In this article, the fourth axiom is replaced by a slightly weakened condition: an entropy whose associated mutual information is zero if and only if the two underlying random elements are independent. Under the weaker fourth axiom, other forms of entropy are sought by way of escort transformations. Two main results are reported in this article. First, there are many entropies other than the BGS entropy satisfying the weaker condition, yet retaining all the desirable utilities of the BGS entropy. Second, by way of escort transformations, the newly identified entropies are the only ones satisfying the weaker axioms.
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CiteScore
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