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Two Measures of Dependence.

Amos Lapidoth1, Christoph Pfister1

  • 1Signal and Information Processing Laboratory, ETH Zurich, 8092 Zurich, Switzerland.

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Summary
This summary is machine-generated.

Two new dependence measures for random variables are introduced, generalizing Shannon's mutual information. These measures, based on Rényi divergence and relative α-entropy, offer new insights into information theory and hypothesis testing.

Keywords:
Rényi divergenceRényi entropydata processingdependence measurerelative α-entropy

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Area of Science:

  • Information Theory
  • Statistical Inference
  • Random Variables

Background:

  • Dependence measures quantify relationships between random variables.
  • Shannon's mutual information is a fundamental measure of dependence.
  • Generalizations of mutual information are needed for advanced applications.

Purpose of the Study:

  • Introduce two novel families of dependence measures.
  • Generalize Shannon's mutual information using Rényi divergence and relative α-entropy.
  • Explore properties and applications of these new measures.

Main Methods:

  • Defined dependence measures based on Rényi divergence of order α.
  • Defined dependence measures based on relative α-entropy.
  • Analyzed properties such as the data-processing inequality.

Main Results:

  • Both measures generalize Shannon's mutual information for α=1.
  • The first measure satisfies the data-processing inequality and relates to hypothesis testing.
  • The second measure does not satisfy the data-processing inequality but is relevant for distributed encoding.

Conclusions:

  • The introduced dependence measures extend the scope of information-theoretic analysis.
  • These measures provide valuable tools for understanding complex dependencies in random variables.
  • Potential applications include advanced statistical inference and distributed information processing.