Oracle Wiener filtering of a Gaussian signal

We study the problem of filtering a Gaussian process whose trajectories, in some sense, have an unknown smoothness $\beta_0$ from the white noise of small intensity $\epsilon$. If we knew the parameter $\beta_0$, we would use the Wiener filter which has the meaning of oracle. Our goal is now to mimic the oracle, i.e., construct such a filter without the knowledge of the smoothness parameter $\beta_0$ that has the same quality (at least with respect to the convergence rate) as the oracle. It is known that in the pointwise minimax estimation, the adaptive minimax rate is worse by a log factor as compared to the nonadaptive one. By constructing a filter which mimics the oracle Wiener filter, we show that there is no loss of quality in terms of rate for the Bayesian counterpart of this problem - adaptive filtering problem.