On Bayesian supremum norm contraction rates

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.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1253 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org