We apply the Bayes approach to the problem of projection estimation of the signal observed in Gaussian white noise model and we study the rate at which the posterior distribution concentrates around the true signal. A benchmark is the (local) rate of the so called oracle projection risk, i.e. the smallest risk of the unknown true signal over all projection estimators. 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.