On the $φ$-family of probability distributions

Details On the $φ$-family of probability distributions On the $φ$-family of probability distributions

2011-05-05

arxiv.org/abs/1105.1118v2

Mathematics

Probability

16 pages

Scientific paper

We generalize the exponential family of probability distributions $\mathcal{E}_{p}$. In our approach, the exponential function is replaced by the $\phi$-function, resulting in the $\phi$-family of probability distributions $\mathcal{F}_{c}^{\phi}$. We provide how $\phi$-families are constructed. In the $\phi$-family, the analogous of the cumulant-generating functional 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. We found that the Kaniadakis' $\kappa$-exponential function satisfies the definition of $\phi$-functions. A formula for the $\phi$-divergence where the $\phi$-function is the $\kappa$-exponential function is derived.

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