On the ϕ-family of probability distributions

摘要: We generalize the exponential family of probability distributions Ep. In our approach, the exponential function is replaced by the ϕ-function, resulting in the ϕ-family of probability distributions Fϕ c. We provide how ϕ-families are constructed. In the ϕ-family, the analogous of the cumulant-generating functional is a normalizing function. We define the ϕ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. We found that the Kaniadakis’ κ-exponential function satisfies the definition of ϕ-functions. A formula for the ϕ-divergence where the ϕ-function is the κ-exponential function is derived.