Calibrating Black-Box Probabilistic Numerical Methods

Juntao Chen, Chris J. Oates, Markus Michael Rau
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:9-24, 2026.

Abstract

Black-box probabilistic numerical methods offer the tantalising prospect of endowing any (scalar) output from any (consistent) numerical code with a Bayesian credible interval. The idea is to construct a dataset containing simulations of increasing precision, and to cast extrapolation of these data to the infinite precision limit as a prediction task. However, calibrating probabilistic predictions is challenging when working with a small dataset. For codes with multiple outputs, treating each (scalar) output independently can result in large variation in credible intervals between outputs whose numerical accuracy ought to be similar. A principled solution is to formulate a multivariate prediction task, but this requires a statistical model capable of describing the (possibly complex) multivariate phenomenon being simulated, which goes against the spirit of the black-box framework. This paper proposes and analyses a simple approach to couple together related prediction tasks while preserving the simplicity of the black-box framework.

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-chen26a, title = {Calibrating Black-Box Probabilistic Numerical Methods}, author = {Chen, Juntao and Oates, Chris J. and Rau, Markus Michael}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {9--24}, year = {2026}, editor = {Karvonen, Toni and Bosch, Nathanael and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp and Kouw, Wouter}, volume = {341}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v341/main/assets/chen26a/chen26a.pdf}, url = {https://proceedings.mlr.press/v341/chen26a.html}, abstract = {Black-box probabilistic numerical methods offer the tantalising prospect of endowing any (scalar) output from any (consistent) numerical code with a Bayesian credible interval. The idea is to construct a dataset containing simulations of increasing precision, and to cast extrapolation of these data to the infinite precision limit as a prediction task. However, calibrating probabilistic predictions is challenging when working with a small dataset. For codes with multiple outputs, treating each (scalar) output independently can result in large variation in credible intervals between outputs whose numerical accuracy ought to be similar. A principled solution is to formulate a multivariate prediction task, but this requires a statistical model capable of describing the (possibly complex) multivariate phenomenon being simulated, which goes against the spirit of the black-box framework. This paper proposes and analyses a simple approach to couple together related prediction tasks while preserving the simplicity of the black-box framework.} }
Endnote
%0 Conference Paper %T Calibrating Black-Box Probabilistic Numerical Methods %A Juntao Chen %A Chris J. Oates %A Markus Michael Rau %B Proceedings of the 2nd International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2026 %E Toni Karvonen %E Nathanael Bosch %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %E Wouter Kouw %F pmlr-v341-chen26a %I PMLR %P 9--24 %U https://proceedings.mlr.press/v341/chen26a.html %V 341 %X Black-box probabilistic numerical methods offer the tantalising prospect of endowing any (scalar) output from any (consistent) numerical code with a Bayesian credible interval. The idea is to construct a dataset containing simulations of increasing precision, and to cast extrapolation of these data to the infinite precision limit as a prediction task. However, calibrating probabilistic predictions is challenging when working with a small dataset. For codes with multiple outputs, treating each (scalar) output independently can result in large variation in credible intervals between outputs whose numerical accuracy ought to be similar. A principled solution is to formulate a multivariate prediction task, but this requires a statistical model capable of describing the (possibly complex) multivariate phenomenon being simulated, which goes against the spirit of the black-box framework. This paper proposes and analyses a simple approach to couple together related prediction tasks while preserving the simplicity of the black-box framework.
APA
Chen, J., Oates, C.J. & Rau, M.M.. (2026). Calibrating Black-Box Probabilistic Numerical Methods. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:9-24 Available from https://proceedings.mlr.press/v341/chen26a.html.

Related Material