May 2022 Adaptive Bayesian density estimation in sup-norm
Zacharie Naulet
Author Affiliations +
Bernoulli 28(2): 1284-1308 (May 2022). DOI: 10.3150/21-BEJ1387


We investigate the problem of deriving adaptive posterior rates of contraction on L balls in density estimation. Although it is known that log-density priors can achieve optimal rates when the true density is sufficiently smooth, adaptive rates were still to be proven. Here we establish that the so-called spike-and-slab prior can achieve adaptive and optimal posterior contraction rates. Along the way, we prove a generic L contraction result for log-density priors with independent wavelet coefficients. Interestingly, our approach is different from previous works on L contraction and is reminiscent of the classical test-based approach used in Bayesian nonparametrics. Moreover, we require no lower bound on the smoothness of the true density, albeit the rates are deteriorated by an extra log(n) factor in the case of low smoothness.

Funding Statement

This work was supported by U.S. Air Force Office of Scientific Research grant #FA9550-15-1-0074.


The author thanks Daniel M. Roy for helpful discussions and the opportunity to work on this project.


Download Citation

Zacharie Naulet. "Adaptive Bayesian density estimation in sup-norm." Bernoulli 28 (2) 1284 - 1308, May 2022.


Received: 1 April 2020; Revised: 1 February 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388939
zbMATH: 07526585
Digital Object Identifier: 10.3150/21-BEJ1387

Keywords: Adaptation , Bayesian density estimation , supremum norm

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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