On φ-Families of Probability Distributions

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