On the posterior pointwise convergence rate of a Gaussian signal under a conjugate prior

We consider the problem of Bayes estimation of a linear functional of the signal in the Gaussian white noise mode, under the assumption that the unknown signal is from a Sobolev smoothness class. We propose a family of conjugate (Gaussian) priors and prove that the resulting Bayes estimators are rate minimax from both frequentist and Bayes perspectives. Finally, we show that the posterior distribution of the functional concentrates around the true value of the functional with the minimax rate uniformly over the Sobolev class.