Infinite-dimensional statistical manifolds based on a balanced chart

Abstract

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are balanced between the density and log-density functions. The manifolds, (Mλ,λ e [2,∞)), retain many of the features of finite-dimensional information geometry; in particular, the α-divergences are of class C[λ]-1, enabling the definition of the Fisher metric and α-derivatives of particular classes of vector fields. Manifolds of probability measures, (Mλ,λ e [2,∞)), based on centred versions of the charts are shown to be C[λ]-1-embedded submanifolds of the. Mλ. The Fisher metric is a pseudo-Riemannian metric on. Mλ. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of α-covariant derivatives. Mλ and Mλ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.