IDEAS home Printed from https://ideas.repec.org/a/gam/jstats/v6y2023i3p49-772d1204044.html
   My bibliography  Save this article

Khinchin’s Fourth Axiom of Entropy Revisited

Author

Listed:
  • 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)

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.

Suggested Citation

  • Zhiyi Zhang & Hongwei Huang & Hao Xu, 2023. "Khinchin’s Fourth Axiom of Entropy Revisited," Stats, MDPI, vol. 6(3), pages 1-10, July.
  • Handle: RePEc:gam:jstats:v:6:y:2023:i:3:p:49-772:d:1204044
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2571-905X/6/3/49/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2571-905X/6/3/49/
    Download Restriction: no
    ---><---

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jstats:v:6:y:2023:i:3:p:49-772:d:1204044. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.