Integrals over Gaussians under Linear Domain Constraints

Alexandra Gessner, Oindrila Kanjilal, Philipp Hennig
; Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:2764-2774, 2020.

Abstract

Integrals of linearly constrained multivariate Gaussian densities are a frequent problem in machine learning and statistics, arising in tasks like generalized linear models and Bayesian optimization. Yet they are notoriously hard to compute, and to further complicate matters, the numerical values of such integrals may be very small. We present an efficient black-box algorithm that exploits geometry for the estimation of integrals over a small, truncated Gaussian volume, and to simulate therefrom. Our algorithm uses the Holmes-Diaconis-Ross (HDR) method combined with an analytic version of elliptical slice sampling (ESS). Adapted to the linear setting, ESS allows for rejection-free sampling, because intersections of ellipses and domain boundaries have closed-form solutions. The key idea of HDR is to decompose the integral into easier-to-compute conditional probabilities by using a sequence of nested domains. Remarkably, it allows for direct computation of the logarithm of the integral value and thus enables the computation of extremely small probability masses. We demonstrate the effectiveness of our tailored combination of HDR and ESS on high-dimensional integrals and on entropy search for Bayesian optimization.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-gessner20a, title = {Integrals over Gaussians under Linear Domain Constraints}, author = {Gessner, Alexandra and Kanjilal, Oindrila and Hennig, Philipp}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {2764--2774}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, address = {Online}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/gessner20a/gessner20a.pdf}, url = {http://proceedings.mlr.press/v108/gessner20a.html}, abstract = {Integrals of linearly constrained multivariate Gaussian densities are a frequent problem in machine learning and statistics, arising in tasks like generalized linear models and Bayesian optimization. Yet they are notoriously hard to compute, and to further complicate matters, the numerical values of such integrals may be very small. We present an efficient black-box algorithm that exploits geometry for the estimation of integrals over a small, truncated Gaussian volume, and to simulate therefrom. Our algorithm uses the Holmes-Diaconis-Ross (HDR) method combined with an analytic version of elliptical slice sampling (ESS). Adapted to the linear setting, ESS allows for rejection-free sampling, because intersections of ellipses and domain boundaries have closed-form solutions. The key idea of HDR is to decompose the integral into easier-to-compute conditional probabilities by using a sequence of nested domains. Remarkably, it allows for direct computation of the logarithm of the integral value and thus enables the computation of extremely small probability masses. We demonstrate the effectiveness of our tailored combination of HDR and ESS on high-dimensional integrals and on entropy search for Bayesian optimization.} }
Endnote
%0 Conference Paper %T Integrals over Gaussians under Linear Domain Constraints %A Alexandra Gessner %A Oindrila Kanjilal %A Philipp Hennig %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-gessner20a %I PMLR %J Proceedings of Machine Learning Research %P 2764--2774 %U http://proceedings.mlr.press %V 108 %W PMLR %X Integrals of linearly constrained multivariate Gaussian densities are a frequent problem in machine learning and statistics, arising in tasks like generalized linear models and Bayesian optimization. Yet they are notoriously hard to compute, and to further complicate matters, the numerical values of such integrals may be very small. We present an efficient black-box algorithm that exploits geometry for the estimation of integrals over a small, truncated Gaussian volume, and to simulate therefrom. Our algorithm uses the Holmes-Diaconis-Ross (HDR) method combined with an analytic version of elliptical slice sampling (ESS). Adapted to the linear setting, ESS allows for rejection-free sampling, because intersections of ellipses and domain boundaries have closed-form solutions. The key idea of HDR is to decompose the integral into easier-to-compute conditional probabilities by using a sequence of nested domains. Remarkably, it allows for direct computation of the logarithm of the integral value and thus enables the computation of extremely small probability masses. We demonstrate the effectiveness of our tailored combination of HDR and ESS on high-dimensional integrals and on entropy search for Bayesian optimization.
APA
Gessner, A., Kanjilal, O. & Hennig, P.. (2020). Integrals over Gaussians under Linear Domain Constraints. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in PMLR 108:2764-2774

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