λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

This paper systematically presents the [Formula: see text]-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the [Formula: see text]-deformed case: [Formula: see text]-convexity, [Formula: see text]-conjugation, [Formula: see text]-biorthogonality, [Formula: see text]-logarithmic divergence, [Formula: see text]-exponential and [Formula: see text]-mixture families, etc. In particular, [Formula: see text]-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical [Formula: see text]-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the [Formula: see text]-exponential family, in turn, coincides with the [Formula: see text]-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, [Formula: see text]-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.