Constrained Weighted Bayesian Bootstrap

Sam Rosen, Jason Xu
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:5740-5763, 2026.

Abstract

We prove the weighted {Bayesian} bootstrap, a method for approximate sampling of a posterior distribution, can be extended to sample from general constrained posterior distributions under mild assumptions. The method entails a simple algorithm that can take advantage of fast tools from convex optimization. Under regularity conditions, we show the asymptotic distribution of samples from the constrained weighted {Bayesian} bootstrap has a covariance matching the restricted maximum likelihood estimator, an efficient estimator. We assess the method empirically on a variety of constrained {Bayesian} problems, demonstrating broad applicability of the method as well as advantages over existing peer methods. The constrained weighted {Bayesian} bootstrap quickly samples from constrained posteriors, providing adequate uncertainty quantification for problems typically solved via optimization methods designed to deliver only a point estimate. As a case study, using constraints required in European-style option prices, uncertainty estimates of an option pricing surface are derived with constrained weighted {Bayesian} bootstrap.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-rosen26a, title = {Constrained Weighted {Bayesian} Bootstrap}, author = {Rosen, Sam and Xu, Jason}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {5740--5763}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/rosen26a/rosen26a.pdf}, url = {https://proceedings.mlr.press/v337/rosen26a.html}, abstract = {We prove the weighted {Bayesian} bootstrap, a method for approximate sampling of a posterior distribution, can be extended to sample from general constrained posterior distributions under mild assumptions. The method entails a simple algorithm that can take advantage of fast tools from convex optimization. Under regularity conditions, we show the asymptotic distribution of samples from the constrained weighted {Bayesian} bootstrap has a covariance matching the restricted maximum likelihood estimator, an efficient estimator. We assess the method empirically on a variety of constrained {Bayesian} problems, demonstrating broad applicability of the method as well as advantages over existing peer methods. The constrained weighted {Bayesian} bootstrap quickly samples from constrained posteriors, providing adequate uncertainty quantification for problems typically solved via optimization methods designed to deliver only a point estimate. As a case study, using constraints required in European-style option prices, uncertainty estimates of an option pricing surface are derived with constrained weighted {Bayesian} bootstrap.} }
Endnote
%0 Conference Paper %T Constrained Weighted Bayesian Bootstrap %A Sam Rosen %A Jason Xu %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-rosen26a %I PMLR %P 5740--5763 %U https://proceedings.mlr.press/v337/rosen26a.html %V 337 %X We prove the weighted {Bayesian} bootstrap, a method for approximate sampling of a posterior distribution, can be extended to sample from general constrained posterior distributions under mild assumptions. The method entails a simple algorithm that can take advantage of fast tools from convex optimization. Under regularity conditions, we show the asymptotic distribution of samples from the constrained weighted {Bayesian} bootstrap has a covariance matching the restricted maximum likelihood estimator, an efficient estimator. We assess the method empirically on a variety of constrained {Bayesian} problems, demonstrating broad applicability of the method as well as advantages over existing peer methods. The constrained weighted {Bayesian} bootstrap quickly samples from constrained posteriors, providing adequate uncertainty quantification for problems typically solved via optimization methods designed to deliver only a point estimate. As a case study, using constraints required in European-style option prices, uncertainty estimates of an option pricing surface are derived with constrained weighted {Bayesian} bootstrap.
APA
Rosen, S. & Xu, J.. (2026). Constrained Weighted Bayesian Bootstrap. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:5740-5763 Available from https://proceedings.mlr.press/v337/rosen26a.html.

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