A deformed exponential statistical manifold

Abstract

Consider μ a probability measure and Pμ the set of μ-equivalent strictly positive probability densities. To endow Pμ with a structure of a C∞-Banach manifold we use the j-connection by an open arc, where ϕ is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and k-exponential functions. Moreover, we find the tangent space of Pμ at a point p, and as a consequence the tangent bundle of Pμ. We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and k-exponential functions can be used to generalize of Rényi divergence.