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Conditioning in Tropical Probability Theory.

Entropy (Basel, Switzerland)·2023
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Arrow Contraction and Expansion in Tropical Diagrams.

Rostislav Matveev1, Jacobus W Portegies2

  • 1Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.

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|December 23, 2023
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Summary

Arrow contraction modifies probability space diagrams by replacing morphisms with isomorphisms. This technique, related to rate regions, helps reveal the entropic cone

Keywords:
entropic conetropical probability

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Area of Science:

  • Information theory
  • Category theory
  • Probability theory

Background:

  • Arrow contraction is a modification of tropical diagrams in probability spaces.
  • It involves replacing a morphism with an isomorphism while maintaining diagram integrity.
  • This concept is linked to rate regions established by Ahlswede and Körner.

Purpose of the Study:

  • To introduce and define the operation of arrow contraction on tropical diagrams.
  • To establish the relationship between arrow contraction and previously defined rate regions.
  • To utilize arrow contraction for deriving insights into the structure of the entropic cone.

Main Methods:

  • Modification of tropical diagrams by replacing specific morphisms.
  • Application of arrow contraction to analyze the properties of probability spaces.
  • Leveraging arrow contraction in a companion study to investigate the entropic cone's shape.

Main Results:

  • Arrow contraction is defined as a method to alter probability space diagrams.
  • The study establishes a connection between arrow contraction and rate regions.
  • Arrow contraction is shown to be a valuable tool for understanding the entropic cone.

Conclusions:

  • Arrow contraction provides a novel way to modify and analyze probability diagrams.
  • The operation has implications for understanding rate regions in information theory.
  • Further research using arrow contraction can elucidate the geometry of the entropic cone.