On Bregman Distances and Divergences of Probability Measures

This paper introduces scaled Bregman distances of probability distributions which admit nonuniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes and signals. It is shown that the scaled Bregman distances extend not only the classical ones studied in the previous literature, but also the information divergence and the related wider class of convex divergences of probability measures. An information-processing theorem is established too, but only in the sense of invariance w.r.t. statistically sufficient transformations and not in the sense of universal monotonicity. Pathological situations where coding can increase the classical Bregman distance are illustrated by a concrete example. In addition to the classical areas of application of the Bregman distances and convex divergences such as recognition, classification, learning, and evaluation of proximity of various features and signals, the paper mentions a new application in 3-D exploratory data analysis. Explicit expressions for the scaled Bregman distances are obtained in general exponential families, with concrete applications in the binomial, Poisson, and Rayleigh families, and in the families of exponential processes such as the Poisson and diffusion processes including the classical examples of the Wiener process and geometric Brownian motion.