Transversely Hessian foliations and information geometry

Abstract

A family of probability distributions parametrized by an open domain [Formula: see text] in [Formula: see text] defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry, the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannian metric on [Formula: see text]. It seems to be quite a strong condition. In general, not much can be said about the Fisher information matrix tensor. To develop a more general theory, we weaken the assumption and replace “positive definite” by the existence of a suitable torsion-free connection. It permits us to define naturally a foliation with a transversely Hessian structure. We develop the theory of transversely Hessian foliations along the lines of the classical theory.