Multilevel Bayesian Quadrature

Kaiyu Li, Daniel Giles, Toni Karvonen, Serge Guillas, Francois-Xavier Briol
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:1845-1868, 2023.

Abstract

Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-li23a, title = {Multilevel Bayesian Quadrature}, author = {Li, Kaiyu and Giles, Daniel and Karvonen, Toni and Guillas, Serge and Briol, Francois-Xavier}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {1845--1868}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/li23a/li23a.pdf}, url = {https://proceedings.mlr.press/v206/li23a.html}, abstract = {Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.} }
Endnote
%0 Conference Paper %T Multilevel Bayesian Quadrature %A Kaiyu Li %A Daniel Giles %A Toni Karvonen %A Serge Guillas %A Francois-Xavier Briol %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-li23a %I PMLR %P 1845--1868 %U https://proceedings.mlr.press/v206/li23a.html %V 206 %X Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.
APA
Li, K., Giles, D., Karvonen, T., Guillas, S. & Briol, F.. (2023). Multilevel Bayesian Quadrature. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:1845-1868 Available from https://proceedings.mlr.press/v206/li23a.html.

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