Oracle convergence rate of posterior under projection prior and Bayesian model selection

Abstract

We apply the Bayes approach to the problem of projection estimation of a signal observed in the Gaussian white noise model and we study the rate at which the posterior distribution concentrates about the true signal from the space ℓ₂ as the information in observations tends to infinity. A benchmark is the rate of a so-called oracle projection risk, i.e., the smallest risk of an unknown true signal over all projection estimators. Under an appropriate hierarchical prior, we study the performance of the resulting (appropriately adjusted by the empirical Bayes approach) posterior distribution and establish that the posterior concentrates about the true signal with the oracle projection convergence rate. We also construct a Bayes estimator based on the posterior and show that it satisfies an oracle inequality. The results are nonasymptotic and uniform over ℓ₂. Another important feature of our approach is that our results on the oracle projection posterior rate are always stronger than any result about posterior convergence with the minimax rate over all nonparametric classes for which the corresponding projection oracle estimator is minimax over this class. We also study implications for the model selection problem, namely, we propose a Bayes model selector and assess its quality in terms of the so-called false selection probability.