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Gaussian Approximations for Probability Measures on R^d

Lu, Yulong and Stuart, Andrew and Weber, Hendrik (2017) Gaussian Approximations for Probability Measures on R^d. SIAM/ASA Journal on Uncertainty Quantification, 5 (1). pp. 1136-1165. ISSN 2166-2525. doi:10.1137/16M1105384.

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This paper concerns the approximation of probability measures on R^d with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asympotic behavior is characterized using Γ-convergence. The theory developed is then applied to understand the frequentist consistency of Bayesian inverse problems in finite dimensions. For a fixed realization of additive observational noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure.

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Additional Information:© 2017 SIAM and ASA. Published by SIAM and ASA under the terms of the Creative Commons 4.0 license. Received by the editors November 28, 2016; accepted for publication (in revised form) June 22, 2017; published electronically November 16, 2017. The first author was supported by EPSRC as part of MASDOC DTC at the University of Warwick with grant EP/HO23364/1. The second author was supported by DARPA, EPSRC, and ONR. The third author was supported by the Royal Society through University Research Fellowship UF140187.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/HO23364/1
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Royal SocietyUF140187
Subject Keywords:Gaussian approximation, Kullback-Leibler divergence, Gamma-convergence, Bernstein-Von Mises theorem
Issue or Number:1
Classification Code:AMS subject classifications. 60B10, 60H07, 62F15
Record Number:CaltechAUTHORS:20161221-163341129
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73114
Deposited By: Linda Taddeo
Deposited On:22 Dec 2016 00:41
Last Modified:11 Nov 2021 05:11

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