RATES OF CONTRACTION FOR POSTERIOR DISTRIBUTIONS IN L r -METRICS, 1 . r . .

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 minimaxoptimal 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 . ..