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The frequentist behavior of nonparametric Bayes estimates, more specifically,rates of contraction of the posterior distributions to shrinking $L^r$-normneighborhoods, $1\le r\le\infty$, of the unknown parameter, are studied. Atheorem for nonparametric density estimation is proved under generalapproximation-theoretic assumptions on the prior. The result is applied to avariety of common examples, including Gaussian process, wavelet series, normalmixture and histogram priors. The rates of contraction are minimax-optimal for$1\le r\le2$, but deteriorate as $r$ increases beyond 2. In the case ofGaussian nonparametric regression a Gaussian prior is devised for which theposterior contracts at the optimal rate in all $L^r$-norms, $1\le r\le\infty$.

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