A proof of the Shepp-Olkin entropy concavity conjecture

TL;DR

This paper proves the Shepp-Olkin conjecture, demonstrating that the entropy of sums of independent Bernoulli variables is concave in their parameters, and extends the conjecture to Renyi and Tsallis entropies.

Contribution

The authors provide a complete proof of the Shepp-Olkin entropy concavity conjecture and generalize it to other entropy measures, refining previous partial results.

Findings

Proved the Shepp-Olkin conjecture for Bernoulli sums.

Identified the monotonic case as the worst case for entropy concavity.

Extended the conjecture to Renyi and Tsallis entropies.

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 is a refinement of 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 Renyi and Tsallis entropies.