Active Uncertainty Calibration in Bayesian ODE Solvers

Hans Kersting MPI for Intelligent Systems, Philipp Hennig MPI for Intelligent Systems
Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, PMLR R14:455-464, 2016.

Abstract

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are ’well-calibrated’ in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR14-systems16a, title = {Active Uncertainty Calibration in {B}ayesian {ODE} Solvers}, author = {Systems, Hans Kersting MPI for Intelligent and Systems, Philipp Hennig MPI for Intelligent}, booktitle = {Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence}, pages = {455--464}, year = {2016}, editor = {Ihler, Alexander and Janzing, Dominik}, volume = {R14}, series = {Proceedings of Machine Learning Research}, month = {25--29 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r14/main/assets/systems16a/systems16a.pdf}, url = {https://proceedings.mlr.press/r14/systems16a.html}, abstract = {There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are ’well-calibrated’ in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Active Uncertainty Calibration in Bayesian ODE Solvers %A Hans Kersting MPI for Intelligent Systems %A Philipp Hennig MPI for Intelligent Systems %B Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2016 %E Alexander Ihler %E Dominik Janzing %F pmlr-vR14-systems16a %I PMLR %P 455--464 %U https://proceedings.mlr.press/r14/systems16a.html %V R14 %X There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are ’well-calibrated’ in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations. %Z Reissued by PMLR on 04 October 2026.
APA
Systems, H.K.M.f.I. & Systems, P.H.M.f.I.. (2016). Active Uncertainty Calibration in Bayesian ODE Solvers. Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R14:455-464 Available from https://proceedings.mlr.press/r14/systems16a.html. Reissued by PMLR on 04 October 2026.

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