A -exponential statistical Banach manifold

Let μ be a given probability measure and M μ the set of μ -equivalent strictly positive probability densities. In this paper we construct a Banach manifold on M μ , modeled on the space L ∞ ( p ⋅ μ ) where p is a reference density, for the non-parametric q -exponential statistical models (Tsallis’s deformed exponential), where 0 < q < 1 is any real number. This family is characterized by the fact that when q → 1 , then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space. The coordinate mappings of the manifold are given in terms of Csiszár’s Φ -divergences; the tangent vectors are identified with the one-dimensional q -exponential models and q -deformations of the score function.