Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

In this paper, we present a review of recent developments on the [Formula: see text]-deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the [Formula: see text]-formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the [Formula: see text]-deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the [Formula: see text]-Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the [Formula: see text] limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the [Formula: see text]-escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of [Formula: see text]-thermodynamics in the picture of the information geometry.