Heavy-tailed Bayesian nonparametric adaptation

TL;DR

This paper introduces a Bayesian approach using heavy-tailed priors for adaptive nonparametric estimation, achieving near-optimal rates without hyperparameter tuning across various models.

Contribution

It develops a novel heavy-tailed prior method for adaptive Bayesian nonparametrics that attains minimax contraction rates without hyperparameter sampling.

Findings

Achieves adaptive rates in Gaussian regression.

Performs well in linear inverse problems and Besov classes.

Numerical simulations support theoretical results.

Abstract

We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the $L^2$- and $L^\infty$-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.