An infinite-dimensional statistical manifold modelled on Hilbert space |
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Authors: | Nigel J. Newton |
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Affiliation: | 1. School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom;2. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA |
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Abstract: | 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 -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. |
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