A statistical manifold modeled on Orlicz spaces using Kaniadakis κ-exponential models

Abstract

Given a probability measure μ, on the space of strictly positive densities Mμ, we construct a topological manifold on which the elements are connected by κ-exponential models in the form q=expκ⁡(u⊖κKp,κ(u))p, where expκ⁡(x)=(1+κ2x2+κx)1/κ, x⊖κy=x1+κ2y2−y1+κ2x2, p,q∈Mμ, and their local representations are elements of an Orlicz space, i.e. the manifold is modeled on Orlicz spaces. Parameter k is the G. Kaniadakis parameter for κ-deformed exponentials which is strongly relevant to relativity and statistical complex models in statistical mechanics. Functional Kp,κ is the deformed counterpart of the cumulant mapping and satisfies that, if κ→0, we obtain the usual cumulant functional of the exponential manifold; moreover in this limit case the exponential manifold constructed by Pistone and Sempi is recovered. In the context of deformed exponentials, we prove that the function ϕκ(⋅)=coshκ⁡(⋅)−1, where coshκ⁡(x) is the κ-deformed hyperbolic cosine, is a Young function and generates the Orlicz space on which the κ-exponential manifold is modeled, namely Lϕκ(p⋅μ). This construction differs from the one made by Pistone on the paper κ-exponential models from the geometrical viewpoint, since this last one is based on divergence functionals and modeled on Lebesgue spaces L1/κ(p⋅μ). The use of κ-deformed models is interesting since they generalize the exponential models and extend them to non-additive systems.