Concavity of entropy under thinning

Abstract

Building on the recent work of Johnson (2007) and Yu (2008), we prove that entropy is a concave function with respect to the thinning operation T_a. That is, if X and Y are independent random variables on Z_+ with ultra-log-concave probability mass functions, then H(T_a X+T_{1-a} Y)>= a H(X)+(1-a)H(Y), 0 <= a <= 1, where H denotes the discrete entropy. This is a discrete analogue of the inequality (h denotes the differential entropy) h(sqrt(a) X + sqrt{1-a} Y)>= a h(X)+(1-a) h(Y), 0 <= a <= 1, which holds for continuous X and Y with finite variances and is equivalent to Shannon's entropy power inequality. As a consequence we establish a special case of a conjecture of Shepp and Olkin (1981).