Bayesian Quadrature for Multiple Related Integrals

Xiaoyue Xi, Francois-Xavier Briol, Mark Girolami
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:5373-5382, 2018.

Abstract

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-xi18a, title = {{B}ayesian Quadrature for Multiple Related Integrals}, author = {Xi, Xiaoyue and Briol, Francois-Xavier and Girolami, Mark}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {5373--5382}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/xi18a/xi18a.pdf}, url = {https://proceedings.mlr.press/v80/xi18a.html}, abstract = {Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.} }
Endnote
%0 Conference Paper %T Bayesian Quadrature for Multiple Related Integrals %A Xiaoyue Xi %A Francois-Xavier Briol %A Mark Girolami %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-xi18a %I PMLR %P 5373--5382 %U https://proceedings.mlr.press/v80/xi18a.html %V 80 %X Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.
APA
Xi, X., Briol, F. & Girolami, M.. (2018). Bayesian Quadrature for Multiple Related Integrals. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:5373-5382 Available from https://proceedings.mlr.press/v80/xi18a.html.

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