A proof of the Shepp–Olkin entropy monotonicity conjecture

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 $q$.