Open Access
August 2019 Bayesian mode and maximum estimation and accelerated rates of contraction
William Weimin Yoo, Subhashis Ghosal
Bernoulli 25(3): 2330-2358 (August 2019). DOI: 10.3150/18-BEJ1056


We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor-product B-splines and endowing the coefficients with Gaussian priors. In the usual fixed-in-advanced sampling plan, we establish posterior contraction rates for mode and maximum and show that they coincide with the minimax rates for this problem. To quantify estimation uncertainty, we construct credible sets for these two quantities that have high coverage probabilities with optimal sizes. If one is allowed to collect data sequentially, we further propose a Bayesian two-stage estimation procedure, where a second stage posterior is built based on samples collected within a credible set constructed from a first stage posterior. Under appropriate conditions on the radius of this credible set, we can accelerate optimal contraction rates from the fixed-in-advanced setting to the minimax sequential rates. A simulation experiment shows that our Bayesian two-stage procedure outperforms single-stage procedure and also slightly improves upon a non-Bayesian two-stage procedure.


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William Weimin Yoo. Subhashis Ghosal. "Bayesian mode and maximum estimation and accelerated rates of contraction." Bernoulli 25 (3) 2330 - 2358, August 2019.


Received: 1 August 2017; Revised: 1 March 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066259
MathSciNet: MR3961250
Digital Object Identifier: 10.3150/18-BEJ1056

Keywords: anisotropic Hölder space , Credible set , maximum value , Mode , Nonparametric regression , posterior contraction , sequential , tensor-product B-splines , two-stage

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
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