Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$

Abstract

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le \infty$, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for $1\le r\le 2$, but deteriorate as $r$ increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all $L^r$-norms, $1\le r\le \infty$.

Published in at http://dx.doi.org/10.1214/11-AOS924 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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