A Rényi-type quasimetric with random interference detection

Abstract

This paper introduces a new dissimilarity measure between two discrete and finite probability distributions. The followed approach is grounded jointly on mixtures of probability distributions and an optimization procedure. We discuss the clear interpretation of the constitutive elements of the measure under an information-theoretical perspective by also highlighting its connections with the Rényi divergence of infinite order. Moreover, we show how the measure describes the inefficiency in assuming that a given probability distribution coincides with a benchmark one by giving formal writing of the random interference between the considered probability distributions. We explore the properties of the considered tool, which are in line with those defining the concept of quasimetric—i.e. a divergence for which the triangular inequality is satisfied. As a possible usage of the introduced device, an application to rare events is illustrated. This application shows that our measure may be suitable in cases where the accuracy of the small probabilities is a relevant matter.