On the interplay between conditional entropy and error probability

Siu Wai Ho, Sergio Verdú

Research output: Contribution to journalArticlepeer-review

64 Scopus citations


Fano's inequality relates the error probability of guessing a finitely-valued random variable X given another random variable Y and the conditional entropy of X given Y. It is not necessarily tight when the marginal distribution of X is fixed. This paper gives a tight upper bound on the conditional entropy of X given Y in terms of the error probability and the marginal distribution of X. A new lower bound on the conditional entropy for countably infinite alphabets is also found. The relationship between the reliability criteria of vanishing error probability and vanishing conditional entropy is also discussed. A strengthened form of the Schur-concavity of entropy which holds for finite or countably infinite random variables is given.

Original languageEnglish (US)
Article number5625631
Pages (from-to)5930-5942
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2010

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Entropy
  • Fano's inequality
  • Schur-concavity
  • Shannon theory
  • equivocation
  • majorization theory


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