Bayesian mode and maximum estimation and accelerated rates of contraction

TL;DR

This paper develops Bayesian methods using tensor-product B-splines for estimating the mode and maximum of an unknown function, achieving optimal contraction rates and proposing a sequential approach that accelerates convergence.

Contribution

It introduces a Bayesian two-stage estimation procedure that accelerates contraction rates for mode and maximum estimation in noisy settings.

Findings

Posterior contraction rates match minimax bounds.

Credible sets achieve high coverage with optimal size.

Sequential procedure improves convergence speed.

Abstract

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.