Inequalities between entropy and index of coincidence derived from information diagrams

To any discrete probability distribution P we can associate its entropy H(P)=-Σp_i ln p_i and its index of coincidence IC(P)=Σp_i². The main result of the paper is the determination of the precise range of the map P→(IC(P), H(P)). The range looks much like that of the map P→(P_max, H(P)) where P_max is the maximal point probability, cf. research from 1965 (Kovalevskij (1965)) to 1994 (Feder and Merhav (1994)). The earlier results, which actually focus on the probability of error 1-P_max rather than P_max, can be conceived as limiting cases of results obtained by methods presented here. Ranges of maps as those indicated are called information diagrams. The main result gives rise to precise lower as well as upper bounds for the entropy function. Some of these bounds are essential for the exact solution of certain problems of universal coding and prediction for Bernoulli sources. Other applications concern Shannon theory (relations between various measures of divergence), statistical decision theory, and rate distortion theory. Two methods are developed. One is topological; the other involves convex analysis and is based on a “lemma of replacement” which is of independent interest in relation to problems of optimization of mixed type (concave/convex optimization)