## Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces

## Abstract

In a variety of applications it is important to extract information from a probability measure μ 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 ν, from within a simple class of measures, which approximates μ. Here, 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 ν 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.

- Authors:

- Univ. of Cincinnati, OH (United States). Dept. of Physics
- Drexel Univ., Philadelphia, PA (United States). Dept. of Mathematics
- Warwick Univ., Coventry (United Kingdom). Mathematics Inst.

- Publication Date:

- Research Org.:
- Warwick Univ., Coventry (United Kingdom)

- Sponsoring Org.:
- USDOE Office of Science (SC); European Research Council (ERC); Engineering and Physical Sciences Research Council (EPSRC); US Department of the Navy, Office of Naval Research (ONR); National Science Foundation (NSF)

- OSTI Identifier:
- 1459163

- Grant/Contract Number:
- SC0002085; OISE-0967140

- Resource Type:
- Accepted Manuscript

- Journal Name:
- SIAM Journal of Mathematical Analysis

- Additional Journal Information:
- Journal Volume: 47; Journal Issue: 6; Journal ID: ISSN 0036-1410

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Kullback–Leibler divergence; relative entropy; Gaussian measures

### Citation Formats

```
Pinski, Frank J., Simpson, G., Stuart, A. M., and Weber, Hendrik. Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces. United States: N. p., 2015.
Web. doi:10.1137/140962802.
```

```
Pinski, Frank J., Simpson, G., Stuart, A. M., & Weber, Hendrik. Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces. United States. doi:10.1137/140962802.
```

```
Pinski, Frank J., Simpson, G., Stuart, A. M., and Weber, Hendrik. Thu .
"Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces". United States. doi:10.1137/140962802. https://www.osti.gov/servlets/purl/1459163.
```

```
@article{osti_1459163,
```

title = {Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces},

author = {Pinski, Frank J. and Simpson, G. and Stuart, A. M. and Weber, Hendrik},

abstractNote = {In a variety of applications it is important to extract information from a probability measure μ 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 ν, from within a simple class of measures, which approximates μ. Here, 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 ν 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.},

doi = {10.1137/140962802},

journal = {SIAM Journal of Mathematical Analysis},

number = 6,

volume = 47,

place = {United States},

year = {2015},

month = {11}

}

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