Convexity/concavity of renyi entropy and α-mutual information

Abstract

Entropy is well known to be Schur concave on finite alphabets. Recently, the authors have strengthened the result by showing that for any pair of probability distributions P and Q with Q majorized by P, the entropy of Q is larger than the entropy of P by the amount of relative entropy D(P||Q). This result applies to P and Q defined on countable alphabets. This paper shows the counterpart of this result for the Rényi entropy and the Tsallis entropy. Lower bounds on the difference in the Rényi (or Tsallis) entropy are given in terms of a new divergence which is related to the Rényi (or Tsallis) divergence. This paper also considers a notion of generalized mutual information, namely α-mutual information, which is defined through the Rényi divergence. The convexity/concavity for different ranges of α is shown. A sufficient condition for the Schur concavity is discussed and upper bounds on α-mutual information are given in terms of the Rényi entropy.