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Symmetries among Multivariate Information Measures Explored Using Möbius Operators.

David J Galas1, Nikita A Sakhanenko1

  • 1Pacific Northwest Research Institute, 720 Broadway, Seattle, WA 98122, USA.

Entropy (Basel, Switzerland)
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Summary
This summary is machine-generated.

This study introduces Möbius operators to unify and reveal new relationships between information measures. The formalism systematizes symmetries, offering a comprehensive framework for understanding diverse information functions.

Keywords:
MaxEntMöbius inversionentropyinformationinteraction-informationlatticesmulti-informationmultivariable dependencenetworkssymmetric group

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Area of Science:

  • Information Theory
  • Lattice Theory
  • Algebraic Structures

Background:

  • Information measures often exhibit duality relations derived from lattice symmetries.
  • Existing frameworks lack a systematic approach to explore the full spectrum of these relationships.

Purpose of the Study:

  • To develop a unifying formalism for information measures using lattice and functional symmetries.
  • To systematically examine and reveal new relationships among information functions.
  • To establish a novel operator algebra for information theory.

Main Methods:

  • Utilizing Möbius inversion on lattices of variable sets.
  • Defining and analyzing Möbius operators that map functions on lattices.
  • Investigating the algebraic structure formed by these operators (isomorphic to S3).

Main Results:

  • A systematic examination of information function relationships, revealing new connections.
  • Demonstration that Möbius operators form a group isomorphic to the symmetric group S3.
  • Expression of lattice function relations through a transparent operator algebra.

Conclusions:

  • The developed formalism provides a unified approach to understanding diverse information measures.
  • The Möbius operator algebra offers a powerful tool for deriving a wide range of new relationships.
  • Generalization of the algebra opens avenues for discovering even more complex information theoretic connections.