Infinite-dimensional statistical manifolds based on a balanced chart

TL;DR

This paper constructs infinite-dimensional Banach manifolds of measures using balanced charts, extending finite-dimensional information geometry concepts to analyze divergence, Fisher metrics, and Bayesian approximations.

Contribution

It introduces a new family of infinite-dimensional manifolds with balanced charts, enabling the extension of geometric tools like Fisher metrics and $oldsymbol{ extalpha}$-divergences to measure spaces.

Findings

Manifolds retain finite-dimensional geometric features

Fisher metric becomes pseudo-Riemannian on the larger manifold

Embedded submanifolds have Riemannian metrics for finite dimensions

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, $(\tilde{M}_{\lambda},\lambda\in [2,\infty))$, retain many of the features of finite-dimensional information geometry; in particular, the $\alpha$-divergences are of class $C^{\lceil\lambda\rceil-1}$, enabling the definition of the Fisher metric and $\alpha$-derivatives of particular classes of vector fields. Manifolds of probability measures, $(M_{\lambda},\lambda\in [2,\infty))$, based on centred versions of the charts are shown to be $C^{\lceil\lambda \rceil-1}$-embedded submanifolds of the $\tilde{M}_{\lambda}$. The Fisher metric is a pseudo-Riemannian metric on $\tilde{M}_{\lambda}$. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of $\alpha$-covariant derivatives. $\tilde{M}_{\lambda}$ and $M_{\lambda}$ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.