Abstract
We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function 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. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.
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Acknowledgements
We are indebted to the anonymous referees for significant comments and suggestions leading to the current version. This work received financial support from CAPES—Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.
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Vigelis, R.F., Cavalcante, C.C. On φ-Families of Probability Distributions. J Theor Probab 26, 870–884 (2013). https://doi.org/10.1007/s10959-011-0400-5
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DOI: https://doi.org/10.1007/s10959-011-0400-5