On adaptive posterior concentration rates

Abstract

We investigate the problem of deriving posterior concentration rates under different loss functions in nonparametric Bayes. We first provide a lower bound on posterior coverages of shrinking neighbourhoods that relates the metric or loss under which the shrinking neighbourhood is considered, and an intrinsic pre-metric linked to frequentist separation rates. In the Gaussian white noise model, we construct feasible priors based on a spike and slab procedure reminiscent of wavelet thresholding that achieve adaptive rates of contraction under $L^2$ or $L^{\infty }$ metrics when the underlying parameter belongs to a collection of Hölder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of posterior credible balls as well as frequentist minimax adaptive estimation. Our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment. The upper bound is attained for certain sieve priors and enables to extend our results to density estimation.

Published at http://dx.doi.org/10.1214/15-AOS1341 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Convergence rates of posterior distributions for noniid observations
• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• On adaptive inference and confidence bands
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$

Cited by in corpus (7)

• Adaptive Bernstein-von Mises theorems in Gaussian white noise
• Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
• Conditions for Posterior Contraction in the Sparse Normal Means Problem
• Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms
• Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators
• Rate exact Bayesian adaptation with modified block priors
• On frequentist coverage errors of Bayesian credible sets in moderately high dimensions