A proof of the Shepp-Olkin entropy concavity conjecture

Erwan Hillion, Oliver T Johnson

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
316 Downloads (Pure)

Abstract

We prove the Shepp–Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin’s original conjecture, to consider Rényi and Tsallis entropies.
Original languageEnglish
Pages (from-to)3638-3649
Number of pages12
JournalBernoulli
Volume23
Issue number4B
Early online date23 May 2017
DOIs
Publication statusPublished - Nov 2017

Keywords

  • Bernoulli sums
  • concavity
  • entropy
  • transportation of measure
  • Poisson binomial distribution

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