On the \(f\)-divergence for discrete non-additive measures
Distances of probability measures, including f-divergences, Hellinger distance, Kullback-Leibler divergence, etc., play an important role in information theory and statistics. In several cases, the connection between these distances and Radon-Nikodym derivatives is exploited. The authors' aim in this paper is to introduce related distances considering monotone measures, hence replacing the sigma-additivity of classical measures by monotonicity. However, then the existence of a Radon-Nikodym derivative can be guaranteed only under quite restrictive constraints. These constraints (considering the discrete case only) are discussed. Also, when considering monotone measures, the Lebesgue integral cannot be exploited, in general, and therefore the integration in this paper is realized by means of the Choquet integral (observe that for sigma-additive measures, Lebesgue and Choquet integrals coincide). Restricting their considerations on a fixed finite space X, the authors have introduced and discussed f-divergence, Moebius transform-based f-divergence and Hellinger distance for monotone measures (called non-additive measures in this paper). Note that in some cases, e.g. in Definition 29, one can consider only belief functions as the considered monotone measures.
scientific article; zbMATH DE number 7325029
scientific article