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Arrow Contraction and Expansion in Tropical Diagrams.

Entropy (Basel, Switzerland)·2023
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Conditioning in Tropical Probability Theory.

Rostislav Matveev1, Jacobus W Portegies2

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

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Summary

We introduce a new conditioning operation for tropical diagrams in probability spaces. This operation is proven to be Lipschitz continuous, ensuring stability with respect to asymptotic entropy distance.

Keywords:
tropical probability

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Area of Science:

  • Probability theory
  • Information theory
  • Tropical geometry

Background:

  • Tropical diagrams offer a geometric representation of probability spaces.
  • Understanding operations on these diagrams is crucial for analyzing probabilistic models.
  • The concept of conditioning is fundamental in probability theory.

Purpose of the Study:

  • To define a natural conditioning operation for tropical diagrams.
  • To analyze the mathematical properties of this new operation.
  • To establish its relationship with existing distance metrics in information theory.

Main Methods:

  • Definition of a novel conditioning operation on tropical diagrams.
  • Application of concepts from tropical geometry and probability theory.
  • Analysis of Lipschitz continuity using asymptotic entropy distance.

Main Results:

  • A well-defined conditioning operation for tropical diagrams has been established.
  • The operation is demonstrated to be Lipschitz continuous.
  • This continuity is shown with respect to the asymptotic entropy distance.

Conclusions:

  • The introduced conditioning operation is robust and well-behaved.
  • The Lipschitz continuity provides a theoretical foundation for its use in analyzing probability spaces.
  • This work bridges tropical geometry and information theory with applications in probabilistic modeling.