Research on the Characteristics of Joint Distribution Based on Minimum Entropy

Ya-Jing Ma, Feng Wang, Xian-Yuan Wu () and Kai-Yuan Cai
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Ya-Jing Ma: School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Feng Wang: School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Xian-Yuan Wu: School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Kai-Yuan Cai: School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Mathematics, 2025, vol. 13, issue 6, 1-12

Abstract: This paper focuses on the extreme-value issue of Shannon entropy for joint distributions with specified marginals, a subject of growing interest. It introduces a theorem showing that the coupling with minimal entropy must be essentially order-preserving, whereas the coupling with maximal entropy aligns with independence. This means that the minimum-entropy coupling in a two-dimensional system forms an upper triangular discrete joint distribution by exchanging the rows and columns of the joint distribution matrix. Consequently, entropy is interpreted as a measure of system disorder. This manuscript’s key academic contribution is in clarifying the physical meaning behind optimal-entropy coupling, where a special ordinal relationship is pinpointed and methodically outlined. Furthermore, it offers a computational approach for order-preserving coupling as a practical illustration.

Keywords: Shannon entropy; minimum-entropy coupling; essentially order-preserving coupling; local optimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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