Khinchin’s Fourth Axiom of Entropy Revisited

Zhiyi Zhang (), Hongwei Huang and Hao Xu
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Zhiyi Zhang: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Hongwei Huang: Wells Fargo Bank, Charlotte, NC 28282, USA
Hao Xu: Wells Fargo Bank, Charlotte, NC 28282, USA

Stats, 2023, vol. 6, issue 3, 1-10

Abstract: 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.

Keywords: Khinchin’s axioms; escort distributions; independence-dependence preserving; power function; entropy uniqueness (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2023
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