A proof of the Shepp-Olkin entropy concavity conjecture

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.

References in corpus (1)

• Concavity of entropy under thinning

Cited by in corpus (11)

• Target space entanglement in quantum mechanics of fermions and matrices
• Symmetry-Resolved Entanglement: General considerations, calculation from correlation functions, and bounds for symmetry-protected topological phases
• Bernoulli sums and Rényi entropy inequalities
• The Poisson binomial distribution -- Old & New
• A proof of the Shepp-Olkin entropy monotonicity conjecture
• Breaking global symmetries with locality-preserving operations
• Entropy and thinning of discrete random variables
• On the Maximum Entropy of a Sum of Independent Discrete Random Variables
• Entanglement, loss, and quantumness: When balanced beam splitters are best
• Maximum Entropy of Sums of Independent Ternary Random Variables
• Non-uniform Bounds in the Poisson Approximation with Applications to Informational Distances. I