On $\phi$-families of probability distributions

Abstract

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.

References in corpus (1)

• $\kappa$-exponential models from the geometrical viewpoint

Cited by in corpus (4)

• Infinite-dimensional statistical manifolds based on a balanced chart
• Conditions for the existence of a generalization of Rényi divergence
• A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space
• New Metric and Connections in Statistical Manifolds