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Chained Kullback-Leibler Divergences.

Dmitri S Pavlichin1, Tsachy Weissman1

  • 1Stanford University.

Proceedings. IEEE International Symposium on Information Theory
|November 14, 2017
PubMed
Summary
This summary is machine-generated.

We introduce chained Kullback-Leibler (KL) divergences for analyzing Markov chains, like the Wright-Fisher model. These divergences offer new tools for genetic drift and information geometry, with code provided.

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Area of Science:

  • Information Theory
  • Statistical Mechanics
  • Population Genetics
  • Information Geometry

Background:

  • The Kullback-Leibler (KL) divergence is a fundamental measure of difference between probability distributions.
  • Analyzing complex systems like genetic drift often requires extensions of standard information-theoretic tools.
  • Large deviations analysis provides a framework for understanding rare events in stochastic processes.

Purpose of the Study:

  • To define and characterize novel 'chained' Kullback-Leibler divergences for multi-step processes.
  • To apply these chained divergences to the large deviations analysis of the Wright-Fisher model of genetic drift.
  • To explore connections between chained divergences, information geometry, and thermodynamic principles.

Main Methods:

  • Definition and mathematical characterization of k-fold chained KL divergence.
  • Analysis of the properties of chained divergences, including joint convexity.
  • Application to the Wright-Fisher model, identifying optimal intermediate distributions (paths).

Main Results:

  • The chained KL divergence is defined and its properties analogous to the standard KL divergence are established.
  • The optimal k-step path of distributions is characterized for the chained divergence.
  • A connection to geodesics in Fisher information metric is shown in the continuum limit.
  • A thermodynamic interpretation as Maxwell's demon operation rate is provided.

Conclusions:

  • Chained KL divergences provide a powerful new framework for analyzing sequential processes in probability and statistics.
  • These divergences offer novel insights into the dynamics of genetic drift and related stochastic models.
  • The study bridges information theory, statistical mechanics, and information geometry, with practical implications for inference.