On $\varphi$-families of probability distributions

Rui F. Vigelis, Charles C. Cavalcante

Published 2011-05-05, updated 2013-09-11Version 3

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a $\varphi$-function, resulting in a $\varphi$-family of probability distributions. We show how $\varphi$-families are constructed. In a $\varphi$-family, the analogue of the cumulant-generating function is a normalizing function. We define the $\varphi$-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback-Leibler divergence. A formula for the $\varphi$-divergence where the $\varphi$-function is the Kaniadakis' $\kappa$-exponential function is derived.