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Revisiting Chernoff Information with Likelihood Ratio Exponential Families.

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Summary
This summary is machine-generated.

This study revisits Chernoff information, a statistical divergence, and explores its applications. New methods provide exact or approximate calculations for Gaussian distributions, enhancing its utility in various fields.

Keywords:
Bhattacharyya distanceBregman divergenceChernoff informationChernoff information distributionChernoff–Bregman divergenceChernoff–Jensen divergenceGaussian measuresKullback–Leibler divergenceL1 measurable spaceRényi α-divergencesaffine groupexponential arcinformation geometryregular/steep exponential family

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Area of Science:

  • Information Theory
  • Statistical Divergence
  • Probability Measures

Background:

  • Chernoff information, a statistical divergence, measures deviation between probability measures via Bhattacharyya distance.
  • Originally for Bayes error bounding, it's robust and used in information fusion and quantum information.
  • It can be viewed as a minmax symmetrization of Kullback-Leibler divergence.

Purpose of the Study:

  • To revisit Chernoff information within likelihood ratio exponential families.
  • To develop exact and approximate calculation methods for Chernoff information, particularly for Gaussian distributions.

Main Methods:

  • Utilizing geometric mixtures to define exponential families for Chernoff information analysis.
  • Employing symbolic computation for exact solutions and developing numerical schemes for approximations.

Main Results:

  • Exact solutions for Chernoff information between univariate Gaussian distributions.
  • Closed-form formulas for centered Gaussians with scaled covariance matrices.
  • A fast numerical scheme for approximating Chernoff information between multivariate Gaussians.

Conclusions:

  • The study provides enhanced analytical and computational tools for Chernoff information.
  • These advancements facilitate broader applications of Chernoff information in statistical analysis and machine learning.