Bayesian Optimal Adaptive Estimation Using a Sieve Prior

摘要

We derive rates of contraction of posterior distributions on non-parametric models resulting from sieve priors. The aim of the study was to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter space is, for example, 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 auto-regression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l~2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the r loss is strongly suboptimal and we provide a lower bound on the rate.