A proof of the Shepp-Olkin entropy monotonicity conjecture

Erwan Hillion, Oliver T Johnson

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

3 Citations (Scopus)
154 Downloads (Pure)


Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to q-R´enyi and q-Tsallis entropies, for a range of values of q. MSC 2010
Original languageEnglish
Article number126
Number of pages14
JournalElectronic Journal of Probability
Publication statusPublished - 9 Nov 2019


  • entropy
  • functional inequalities
  • mixing coefficients
  • Poisson–binomial distribution


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