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Lu, Yulong, Stuart, A. M. and Weber, Hendrik (2017) Gaussian approximations for probability measures on Rd*. SIAM/ASA Journal on Uncertainty Quantification, 5 (1). pp. 1136-1165. doi:10.1137/16M1105384 ISSN 2166-2525.
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Official URL: https://doi.org/10.1137/16M1105384
Abstract
This paper concerns the approximation of probability measures on Rd with respect to the KullbackLeibler 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 behaviour 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
| Item Type: | Journal Article | ||||||||||||||||||
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| Alternative Title: | Gaussian Approximations for Probability Measures on $R^d$ | ||||||||||||||||||
| Subjects: | Q Science > QA Mathematics | ||||||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||||||||
| Library of Congress Subject Headings (LCSH): | Gaussian measures, Bayesian statistical decision theory | ||||||||||||||||||
| Journal or Publication Title: | SIAM/ASA Journal on Uncertainty Quantification | ||||||||||||||||||
| Publisher: | SIAM | ||||||||||||||||||
| ISSN: | 2166-2525 | ||||||||||||||||||
| Official Date: | 16 November 2017 | ||||||||||||||||||
| Dates: |
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| Volume: | 5 | ||||||||||||||||||
| Number: | 1 | ||||||||||||||||||
| Page Range: | pp. 1136-1165 | ||||||||||||||||||
| DOI: | 10.1137/16M1105384 | ||||||||||||||||||
| Status: | Peer Reviewed | ||||||||||||||||||
| Publication Status: | Published | ||||||||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||||||||
| Date of first compliant deposit: | 27 June 2017 | ||||||||||||||||||
| Date of first compliant Open Access: | 22 January 2018 | ||||||||||||||||||
| RIOXX Funder/Project Grant: |
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