Convexity/concavity of renyi entropy and α-mutual information

Siu Wai Ho, Sergio Verdú

Research output: Chapter in Book/Report/Conference proceedingConference contribution

48 Scopus citations


Entropy is well known to be Schur concave on finite alphabets. Recently, the authors have strengthened the result by showing that for any pair of probability distributions P and Q with Q majorized by P, the entropy of Q is larger than the entropy of P by the amount of relative entropy D(P||Q). This result applies to P and Q defined on countable alphabets. This paper shows the counterpart of this result for the Rényi entropy and the Tsallis entropy. Lower bounds on the difference in the Rényi (or Tsallis) entropy are given in terms of a new divergence which is related to the Rényi (or Tsallis) divergence. This paper also considers a notion of generalized mutual information, namely α-mutual information, which is defined through the Rényi divergence. The convexity/concavity for different ranges of α is shown. A sufficient condition for the Schur concavity is discussed and upper bounds on α-mutual information are given in terms of the Rényi entropy.

Original languageEnglish (US)
Title of host publicationProceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages5
ISBN (Electronic)9781467377041
StatePublished - Sep 28 2015
EventIEEE International Symposium on Information Theory, ISIT 2015 - Hong Kong, Hong Kong
Duration: Jun 14 2015Jun 19 2015

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8095


OtherIEEE International Symposium on Information Theory, ISIT 2015
Country/TerritoryHong Kong
CityHong Kong

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Modeling and Simulation
  • Applied Mathematics


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