Dual differential geometry associated with the Kullbaek-Leibler information on the Gaussian distributions and its 2-parameter deformations

Abstract

Amari showed that the geometry of a family of probability distributions is characterized by a dual differential geometry determined by a couple of affine connections and a divergence associated with a couple of dual potential functions. In this paper, a 2-parameter class of dual differential geometries is constructed on the manifold of the family of multivariate Gaussian distributions with nonzero means, as well as a new class of divergences. This class of geometry includes the Riemannian geometry studied by Skovgaard and the geometry associated with the Kullbaek-Leibler information. The specific dually flat charts for the latter geometry is given in conjunction with a detailed analysis of the associated connections. In order to facilitate the various calculations of differential geometric quantities in the analysis of our geometries, we introduce a new coordinate free differential calculus of a function of a symmetric matrix argument, based on a specific bilinear form defined on the domain of the function and its dual space. This calculus enables us to obtain a parallel formalism of the Legendre transformation in convex analysis even for a function of a matrix argument.