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An infinite-dimensional statistical manifold modelled on Hilbert space

Newton, Nigel J (2012) 'An infinite-dimensional statistical manifold modelled on Hilbert space.' Journal of Functional Analysis, 263 (6). pp. 1661-1681. ISSN 0022-1236

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We construct an infinite-dimensional Hilbert manifold of probability measures on an abstract measurable space. The manifold, M, retains the first- and second-order features of finite-dimensional information geometry: the α-divergences admit first derivatives and mixed second derivatives, enabling the definition of the Fisher metric as a pseudo-Riemannian metric. This is enough for many applications; for example, it justifies certain projections of Markov processes onto finite-dimensional submanifolds in recursive estimation problems. M was constructed with the Fenchel-Legendre transform between Kullback-Leibler divergences, and its role in Bayesian estimation, in mind. This transform retains, on M, the symmetry of the finite-dimensional case. Many of the manifolds of finite-dimensional information geometry are shown to be C ∞-embedded submanifolds of M. In establishing this, we provide a framework in which many of the formal results of the finite-dimensional subject can be proved with full rigour. © 2012 Elsevier Inc.

Item Type: Article
Uncontrolled Keywords: Bayesian estimation; Fenchel-Legendre transform; Fisher metric; Hilbert manifold; Information geometry; Information theory
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Q Science > QC Physics
Divisions: Faculty of Science and Health
Faculty of Science and Health > Computer Science and Electronic Engineering, School of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 05 Mar 2013 16:31
Last Modified: 18 Aug 2022 11:05

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