# Multilevel Sequential Monte Carlo Samplers for Normalizing Constants

## Abstract

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for two Bayesian inverse problems arising from elliptic partial differential equations (PDEs). The examples involve the inversion of observations of the solution of (i) a 1-dimensional Poisson equation to infer the diffusion coefficient, and (ii) a 2-dimensional Poisson equation to infer the external forcing.

- Authors:

- Univ. Bordeaux (France). Center INRIA Bordeaux Sud-Ouest & Institut de Mathematiques de Bordeaux
- National Univ. of Singapore (Singapore). Dept. of Statistics & Applied Probability
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Div.

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1393885

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- ACM Transactions on Modeling and Computer Simulation

- Additional Journal Information:
- Journal Volume: 27; Journal Issue: 3; Journal ID: ISSN 1049-3301

- Publisher:
- Association for Computing Machinery

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Moral, Pierre Del, Jasra, Ajay, Law, Kody J. H., and Zhou, Yan.
```*Multilevel Sequential Monte Carlo Samplers for Normalizing Constants*. United States: N. p., 2017.
Web. doi:10.1145/3092841.

```
Moral, Pierre Del, Jasra, Ajay, Law, Kody J. H., & Zhou, Yan.
```*Multilevel Sequential Monte Carlo Samplers for Normalizing Constants*. United States. doi:10.1145/3092841.

```
Moral, Pierre Del, Jasra, Ajay, Law, Kody J. H., and Zhou, Yan. Thu .
"Multilevel Sequential Monte Carlo Samplers for Normalizing Constants". United States.
doi:10.1145/3092841.
```

```
@article{osti_1393885,
```

title = {Multilevel Sequential Monte Carlo Samplers for Normalizing Constants},

author = {Moral, Pierre Del and Jasra, Ajay and Law, Kody J. H. and Zhou, Yan},

abstractNote = {This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for two Bayesian inverse problems arising from elliptic partial differential equations (PDEs). The examples involve the inversion of observations of the solution of (i) a 1-dimensional Poisson equation to infer the diffusion coefficient, and (ii) a 2-dimensional Poisson equation to infer the external forcing.},

doi = {10.1145/3092841},

journal = {ACM Transactions on Modeling and Computer Simulation},

number = 3,

volume = 27,

place = {United States},

year = {Thu Aug 24 00:00:00 EDT 2017},

month = {Thu Aug 24 00:00:00 EDT 2017}

}