Open Access
2022 Optional Pólya trees: Posterior rates and uncertainty quantification
Ismaël Castillo, Thibault Randrianarisoa
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Electron. J. Statist. 16(2): 6267-6312 (2022). DOI: 10.1214/22-EJS2086


We consider statistical inference in the density estimation model using a tree–based Bayesian approach, with Optional Pólya trees as prior distribution. We derive near-optimal convergence rates for corresponding posterior distributions with respect to the supremum norm. For broad classes of Hölder–smooth densities, we show that the method automatically adapts to the unknown Hölder regularity parameter. We consider the question of uncertainty quantification by providing mathematical guarantees for credible sets from the obtained posterior distributions, leading to near–optimal uncertainty quantification for the density function, as well as related functionals such as the cumulative distribution function. The results are illustrated through a brief simulation study.

Funding Statement

This work was funded by the ANR, project ANR-17-CE40-0001 (BASICS).


The authors would like to thank Li Ma for insightful comments. This work has been supported by the grant ANR-17-CE40-0001 (BASICS).


Download Citation

Ismaël Castillo. Thibault Randrianarisoa. "Optional Pólya trees: Posterior rates and uncertainty quantification." Electron. J. Statist. 16 (2) 6267 - 6312, 2022.


Received: 1 December 2021; Published: 2022
First available in Project Euclid: 24 November 2022

MathSciNet: MR4515717
zbMATH: 07633938
Digital Object Identifier: 10.1214/22-EJS2086

Primary: 62G05 , 62G07 , 62G20

Keywords: Bayesian nonparametrics , frequentist coverage of credible sets , Pólya trees , posterior convergence rates , supremum norm , uncertainty quantification

Vol.16 • No. 2 • 2022
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