On adaptive posterior concentration rates

TL;DR

This paper studies how the Bayesian posterior concentrates around the true parameter in nonparametric models, providing bounds and constructing priors that adaptively achieve optimal rates under various loss functions.

Contribution

It establishes a lower bound on posterior coverage, constructs adaptive spike-and-slab priors in Gaussian models, and extends results to density estimation.

Findings

Constructed priors achieve adaptive contraction rates under $L^2$ and $L^{inity}$ metrics.

Derived a lower bound relating metric loss and frequentist separation rates.

Extended results to density estimation with an upper bound on contraction 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 H\"{o}lder 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.