Non-asymptotic Bayesian Minimax Adaptation

TL;DR

This paper introduces a Bayesian method for nonparametric estimation that adaptively achieves minimax optimality without prior knowledge of smoothness or radius, applicable in Gaussian white-noise and regression models.

Contribution

It presents a Bayesian approach that attains rate-exact minimax adaptation in non-asymptotic settings without requiring prior smoothness or radius information.

Findings

Achieves minimax rates adaptively in non-asymptotic regimes.

Provides risk bounds quantifying effects of smoothness and radius.

Applicable to Gaussian white-noise and non-parametric regression models.

Abstract

This paper studies a Bayesian approach to non-asymptotic minimax adaptation in nonparametric estimation. Estimating an input function on the basis of output functions in a Gaussian white-noise model is discussed. The input function is assumed to be in a Sobolev ellipsoid with an unknown smoothness and an unknown radius. Our purpose in this paper is to present a Bayesian approach attaining minimaxity up to a universal constant without any knowledge regarding the smoothness and the radius. Our Bayesian approach provides not only a rate-exact minimax adaptive estimator in large sample asymptotics but also a risk bound for the Bayes estimator quantifying the effects of both the smoothness and the ratio of the squared radius to the noise variance, where the smoothness and the ratio are the key parameters to describe the minimax risk in this model. Application to non-parametric regression models is also discussed.