On Bayesian supremum norm contraction rates

TL;DR

This paper develops a method to analyze the convergence rates of Bayesian posterior distributions in the sup-norm, demonstrating that certain priors can achieve optimal minimax rates in density estimation and Gaussian white noise models.

Contribution

It introduces a novel approach for studying sup-norm convergence rates in Bayesian nonparametrics and proves optimal rates for common prior families.

Findings

Log-density priors achieve optimal sup-norm rates

Dyadic density histograms attain minimax convergence

Method applies to Gaussian white noise models

Abstract

Building on ideas from Castillo and Nickl [Ann. Statist. 41 (2013) 1999-2028], a method is provided to study nonparametric Bayesian posterior convergence rates when "strong" measures of distances, such as the sup-norm, are considered. In particular, we show that likelihood methods can achieve optimal minimax sup-norm rates in density estimation on the unit interval. The introduced methodology is used to prove that commonly used families of prior distributions on densities, namely log-density priors and dyadic random density histograms, can indeed achieve optimal sup-norm rates of convergence. New results are also derived in the Gaussian white noise model as a further illustration of the presented techniques.