A Riemannian geometry in the q-exponential Banach manifold induced by q-divergences

Abstract

For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built. In this paper, the geometry induced on this manifold is characterized by q-divergence functionals. This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Moreover, the classical Amari's α-connections appears as special case of the q-connections ∇(q). The main result is the expected one, namely the zero curvature of the manifold. © 2013 Springer-Verlag.