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Research on the Characteristics of Joint Distribution Based on Minimum Entropy

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ya-Jing Ma & Feng Wang & Xian-Yuan Wu & Kai-Yuan Cai, 2025. "Research on the Characteristics of Joint Distribution Based on Minimum Entropy," Mathematics, MDPI, vol. 13(6), pages 1-12, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:972-:d:1612852
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