  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
$1\le r\le2$, 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, $1\le r\le\infty$.
