On Bayesian supremum norm contraction rates
The Annals of Statistics · 2014 · Volume 42 · Issue 5
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.
Keywords
62G05, 62G07, 62G20, Bayesian nonparametrics, contraction rates, supremum norm
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