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May 2016 Infinite-dimensional statistical manifolds based on a balanced chart
Nigel J. Newton
Bernoulli 22(2): 711-731 (May 2016). DOI: 10.3150/14-BEJ673


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.


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Nigel J. Newton. "Infinite-dimensional statistical manifolds based on a balanced chart." Bernoulli 22 (2) 711 - 731, May 2016.


Received: 1 September 2013; Revised: 1 August 2014; Published: May 2016
First available in Project Euclid: 9 November 2015

zbMATH: 06562295
MathSciNet: MR3449798
Digital Object Identifier: 10.3150/14-BEJ673

Keywords: Banach manifold , Bayesian estimation , Fisher metric , information geometry , Non-parametric statistics

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 2 • May 2016
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