2013/04/30 by Ismaël Castillo
Computer Science · Mathematics · #Applied mathematics #Bayesian Methods and Mixture Models #Bayesian probability #Combinatorics #Computer science #Density estimation #Gaussian #Gaussian Processes and Bayesian Inference #Infimum and supremum #Mathematical optimization #Mathematics #Minimax #Nonparametric statistics #Norm (philosophy) #Prior probability #Rate of convergence #Statistical Methods and Inference #Statistics #Uniform norm #math.ST #stat.TH
paper · pdf · doi:10.1214/14-aos1253
published as Annals of Statistics 2014, Vol. 42, No. 5, 2058-2091 · 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)
openalex publication_date 2014/09/11 · arxiv created 2014/10/14 · arxiv updated 2014/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.