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, $({\tilde{M}}_{\lambda },\lambda \in [2,\infty ))$, 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.

Published at http://dx.doi.org/10.3150/14-BEJ673 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Generalized Gibbs Ensembles in Discrete Quantum Gravity
• On $\phi$-families of probability distributions

Cited by in corpus (2)

• On some information geometric structures concerning Mercator projections
• A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space