A Locally Adaptive Bayesian Cubature Method

Matthew Fisher, Chris Oates, Catherine Powell, Aretha Teckentrup
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:1265-1275, 2020.

Abstract

Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods are locally adaptive in the sense that they focus computational resources on spatial regions in which local error estimates are largest. The main contributions of this work are twofold; first we establish that existing BC methods do not possess local adaptivity in the sense of many classical adaptive methods and secondly, we developed a novel BC method whose behaviour, demonstrated empirically, is analogous to such methods. Finally we present evidence that the novel method provides improved cubature performance, relative to standard BC, in a detailed empirical assessment.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-fisher20a, title = {A Locally Adaptive Bayesian Cubature Method}, author = {Fisher, Matthew and Oates, Chris and Powell, Catherine and Teckentrup, Aretha}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {1265--1275}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/fisher20a/fisher20a.pdf}, url = {https://proceedings.mlr.press/v108/fisher20a.html}, abstract = {Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods are locally adaptive in the sense that they focus computational resources on spatial regions in which local error estimates are largest. The main contributions of this work are twofold; first we establish that existing BC methods do not possess local adaptivity in the sense of many classical adaptive methods and secondly, we developed a novel BC method whose behaviour, demonstrated empirically, is analogous to such methods. Finally we present evidence that the novel method provides improved cubature performance, relative to standard BC, in a detailed empirical assessment. } }
Endnote
%0 Conference Paper %T A Locally Adaptive Bayesian Cubature Method %A Matthew Fisher %A Chris Oates %A Catherine Powell %A Aretha Teckentrup %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-fisher20a %I PMLR %P 1265--1275 %U https://proceedings.mlr.press/v108/fisher20a.html %V 108 %X Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods are locally adaptive in the sense that they focus computational resources on spatial regions in which local error estimates are largest. The main contributions of this work are twofold; first we establish that existing BC methods do not possess local adaptivity in the sense of many classical adaptive methods and secondly, we developed a novel BC method whose behaviour, demonstrated empirically, is analogous to such methods. Finally we present evidence that the novel method provides improved cubature performance, relative to standard BC, in a detailed empirical assessment.
APA
Fisher, M., Oates, C., Powell, C. & Teckentrup, A.. (2020). A Locally Adaptive Bayesian Cubature Method. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:1265-1275 Available from https://proceedings.mlr.press/v108/fisher20a.html.

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