Abstract

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 $q$-Rényi and $q$-Tsallis entropies, for a range of values of $q$.

Citation

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Erwan Hillion. Oliver Johnson. "A proof of the Shepp–Olkin entropy monotonicity conjecture." Electron. J. Probab. 24 1 - 14, 2019. https://doi.org/10.1214/19-EJP380

Information

Received: 10 December 2018; Accepted: 24 October 2019; Published: 2019

First available in Project Euclid: 9 November 2019

Digital Object Identifier: 10.1214/19-EJP380

Subjects:

Primary: 94A17

Secondary: 60E15

Keywords: Entropy , functional inequalities , mixing coefficients , Poisson–binomial distribution

JOURNAL ARTICLE
14 PAGES

Vol.24 • 2019

Electronic Journal of Probability cover image