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Kullback--Leibler approximation for probability measures on infinite dimensional spaces
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Pinski, F. J., Simpson, G., Stuart, A. M. and Weber, Hendrik (2015) Kullback--Leibler approximation for probability measures on infinite dimensional spaces. SIAM Journal on Mathematical Analysis, 47 (6). pp. 4091-4122. doi:10.1137/140962802 ISSN 0036-1410.
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Official URL: http://dx.doi.org/10.1137/140962802
Abstract
In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Gaussian measures, Probability measures, Calculus of variations | ||||||||
| Journal or Publication Title: | SIAM Journal on Mathematical Analysis | ||||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||||
| ISSN: | 0036-1410 | ||||||||
| Official Date: | 5 November 2015 | ||||||||
| Dates: |
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| Volume: | 47 | ||||||||
| Number: | 6 | ||||||||
| Page Range: | pp. 4091-4122 | ||||||||
| DOI: | 10.1137/140962802 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 13 January 2016 | ||||||||
| Date of first compliant Open Access: | 18 February 2016 | ||||||||
| Adapted As: |
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