Hessian structures on deformed exponential families and their conformal structures

Abstract

An exponential family is an important class of statistical models in statistical sciences. In information geometry, it is known that an exponential family naturally has dualistic Hessian structures. A deformed exponential family is a statistical model which is a generalization of exponential families. A deformed exponential family naturally has two kinds of dualistic Hessian structures. In this paper, such Hessian geometries are summarized.In addition, a deformed exponential family has a generalized conformal structure of statistical manifolds. In the case of q-exponential family, which is a special class of deformed exponential families, the family naturally has two kinds of different Riemannian metrics which are obtained from conformal transformations of Hessian metrics. Then it is showed that a q-exponential family is a Riemannian manifold of constant curvature.