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In this paper, we provide some results related to the ∆ 2 -condition of Musielak-Orlicz functions and ϕ-families of probability distributions, which are modeled on Musielak-Orlicz spaces. We show that if two ϕ-families are modeled on Musielak-Orlicz spaces generated by Musielak-Orlicz functions satisfying the ∆ 2 -condition, then these ϕ-families are equal as sets. We also investigate the behavior of the normalizing function near the boundary of the set on which a ϕ-family is defined.

Abstract:In this paper, we propose a generalization of Rényi divergence, and then we investigate its induced geometry. This generalization is given in terms of a ϕ-function, the same function that is used in the definition of non-parametric ϕ-families. The properties of ϕ-functions proved to be crucial in the generalization of Rényi divergence. Assuming appropriate conditions, we verify that the generalized Rényi divergence reduces, in a limiting case, to the ϕ-divergence. In generalized statistical manifold, the ϕ-divergence induces a pair of dual connections D (−1) and D (1) . We show that the family of connections D

Abstract:In this paper, we investigate the mixture arc on generalized statistical manifolds. We ensure that the generalization of the mixture arc is well defined and we are able to provide a generalization of the open exponential arc and its properties. We consider the model of a ϕ-family of distributions to describe our general statistical model.

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