On additive-combinatorial affine inequalities for Shannon entropy and differential entropy

To be considered for the 2016 IEEE Jack Keil Wolf ISIT Student Paper Award. This paper addresses the question of to what extent do discrete entropy inequalities for weighted sums of independent group-valued random variables continue to hold for differential entropies. We show that all balanced affine inequalities (with the sum of coefficients being zero) of Shannon entropy extend to differential entropy; conversely, any affine inequality for differential entropy must be balanced. In particular, this result recovers recently proved differential entropy inequalities by Kontoyiannis and Madiman [1] from their discrete counterparts due to Tao [2] in a unified manner. Our proof relies on a result of Renyi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum.