  We derive rates of contraction of posterior distributions on nonparametric
models resulting from sieve priors. The aim of the paper is to provide general
conditions to get posterior rates when the parameter space has a general
structure, and rate adaptation when the parameter space is, e.g., a Sobolev
class. The conditions employed, although standard in the literature, are
combined in a different way. The results are applied to density, regression,
nonlinear autoregression and Gaussian white noise models. In the latter we have
also considered a loss function which is different from the usual l2 norm,
namely the pointwise loss. In this case it is possible to prove that the
adaptive Bayesian approach for the l2 loss is strongly suboptimal and we
provide a lower bound on the rate.
