November 2022 Smoothing and adaptation of shifted Pólya tree ensembles
Thibault Randrianarisoa
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Bernoulli 28(4): 2492-2517 (November 2022). DOI: 10.3150/21-BEJ1426

Abstract

Recently, S. Arlot and R. Genuer have shown that a random forest model outperforms its single-tree counterpart in estimating α-Hölder functions, 1α2. This backs up the idea that ensembles of tree estimators are smoother estimators than single trees. On the other hand, most positive optimality results on Bayesian tree-based methods assume that α1. Naturally, one wonders whether Bayesian counterparts of forest estimators are optimal on smoother classes, just as observed with frequentist estimators for α2. We focus on density estimation and introduce an ensemble estimator from the classical (truncated) Pólya tree construction in Bayesian nonparametrics. Inspired by the work mentioned above, the resulting Bayesian forest estimator is shown to lead to optimal posterior contraction rates, up to logarithmic terms, for the Hellinger and L1 distances on probability density functions on [0;1) for arbitrary Hölder regularity α>0. This improves upon previous results for constructions related to the Pólya tree prior, whose optimality was only proven when α1. Also, by adding a hyperprior on the trees’ depth, we obtain an adaptive version of the prior that does not require α to be specified to attain optimality.

Acknowledgements

I sincerely thank Pr. Ismaël Castillo (Sorbonne Université), whose guidance and insights into the study of Bayesian tree-based methods were of tremendous help in the making of this paper.

Citation

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Thibault Randrianarisoa. "Smoothing and adaptation of shifted Pólya tree ensembles." Bernoulli 28 (4) 2492 - 2517, November 2022. https://doi.org/10.3150/21-BEJ1426

Information

Received: 1 October 2020; Published: November 2022
First available in Project Euclid: 17 August 2022

zbMATH: 07594067
MathSciNet: MR4474551
Digital Object Identifier: 10.3150/21-BEJ1426

Keywords: Bayesian nonparametrics , ensemble , forest , Pólya tree

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Vol.28 • No. 4 • November 2022
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