Lower bound for the oracle projection posterior convergence rate

In Babenko and Belitser (2010), a new notion for the posterior concentration rate is proposed, the so-called oracle risk rate, the best possible rate over an appropriately chosen estimators family, which is a local quantity (as compared, e.g., with global minimax rates). The program of oracle estimation and Bayes oracle posterior optimality is fully implemented in the above paper for the Gaussian white noise model and the projection estimators family. In this note, we complement the upper bound results of Babenko and Belitser (2010) on the posterior concentration rate by a lower bound result, namely that the concentration rate of the posterior distribution around the ‘true’ value cannot be faster than the oracle projection rate.