Adaptive inference over Besov spaces in the white noise model using $p$-exponential priors

TL;DR

This paper investigates adaptive Bayesian methods for estimating functions with varying smoothness in Besov spaces using $p$-exponential priors, demonstrating near-minimax rates in both homogeneous and inhomogeneous cases.

Contribution

It introduces empirical and hierarchical Bayes approaches for tuning hyper-parameters of $p$-exponential priors, achieving adaptive minimax rates over Besov spaces.

Findings

Laplace priors attain minimax or near-minimax rates in inhomogeneous Besov spaces.

Gaussian priors only attain minimax rates in homogeneous Besov spaces.

Empirical and hierarchical Bayes methods effectively adapt hyper-parameters for optimal rates.

Abstract

In many scientific applications the aim is to infer a function which is smooth in some areas, but rough or even discontinuous in other areas of its domain. Such spatially inhomogeneous functions can be modelled in Besov spaces with suitable integrability parameters. In this work we study adaptive Bayesian inference over Besov spaces, in the white noise model from the point of view of rates of contraction, using $p$-exponential priors, which range between Laplace and Gaussian and possess regularity and scaling hyper-parameters. To achieve adaptation, we employ empirical and hierarchical Bayes approaches for tuning these hyper-parameters. Our results show that, while it is known that Gaussian priors can attain the minimax rate only in Besov spaces of spatially homogeneous functions, Laplace priors attain the minimax or nearly the minimax rate in both Besov spaces of spatially homogeneous functions and Besov spaces permitting spatial inhomogeneities.