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This summary is machine-generated.

This study uses majorization theory to establish bounds on Rényi and guessing entropies, offering new ways to measure randomness in computer science.

Keywords:
Fano inequalityPinsker inequalityRényi entropySchur concavitydata processing inequalityentropyerror probabilityguessing entropyguessing momentsmajorizationtotal variation distance

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Area of Science:

  • Information Theory
  • Computer Science Theory
  • Probability and Statistics

Background:

  • Rényi and guessing entropies are key measures of uncertainty and information.
  • Existing bounds on these entropies have limitations in certain applications.
  • Majorization theory provides a framework for comparing probability distributions.

Purpose of the Study:

  • To derive optimal lower and upper bounds for Rényi and guessing entropies.
  • To connect these bounds to error probability and total variation distance.
  • To provide a unified framework for understanding randomness measurement.

Main Methods:

  • Application of majorization theory through "Robin Hood" elementary operations.
  • Derivation of bounds with respect to error probability.
  • Derivation of bounds with respect to total variation distance to the uniform distribution.

Main Results:

  • Established optimal lower and upper bounds on Rényi and guessing entropies.
  • Derived reverse-Fano and Fano inequalities using error probability.
  • Derived reverse-Pinsker and Pinsker inequalities using total variation distance.

Conclusions:

  • The derived bounds offer a comprehensive understanding of entropy measures.
  • This work unifies the study of randomness measurement across various computer science domains.
  • The "Robin Hood" approach provides a powerful tool for information-theoretic analysis.