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Journal article
Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces
SIAM Journal on Mathematical Analysis, v 47(6), pp 4091-4122
29 Oct 2013
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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 minimising sequences. Furthermore,
parameterisation of the class of Gaussians through the mean and inverse
covariance is introduced, the need for regularisation is explained, and a
regularised minimisation is studied in detail. The calculus of variations
framework resulting from this work provides the appropriate underpinning for
computational algorithms.
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Details
- Title
- Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces
- Creators
- Frank Pinski - University of CincinnatiGideon Simpson - Drexel UniversityAndrew Stuart - University of WarwickHendrik Weber - University of WarwickWarwick Univ., Coventry (United Kingdom)
- Publication Details
- SIAM Journal on Mathematical Analysis, v 47(6), pp 4091-4122
- Publisher
- Society for Industrial and Applied Mathematics (SIAM)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000367020000001
- Scopus ID
- 2-s2.0-84953274306
- Other Identifier
- 991019168903704721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied