Dualistic Structure

Abstract

Statistical manifolds are abstract generalizations of statistical models. Even if a statistical manifold is treated as a purely geometric object, however, the motivation for the definitions is inspired from statistical models. In this new framework, the manifold of density functions is replaced by an arbitrary Riemannian manifold M, and the Fisher information matrix is replaced by the Riemannian metric g of the manifold M. The dual connections ∇⁽⁻¹⁾ and ∇⁽¹⁾ are replaced by a pair of dual connections ∇ and ∇^∗. The skewness tensor, which measures the cummulants of the third order on a statistical model, is replaced by a 3-covariant skewness tensor.