Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞

Abstract

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking L^r-norm neighborhoods, 1 ≤ r ≤ ∞, 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 ≤ r ≤ 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 ≤ r ≤ ∞.