Bayesian inverse problems with heterogeneous variance

We consider inverse problems in Hilbert spaces under correlated Gaussian noise, and use a Bayesian approach to find their regularized solution. focus on mildly ill-posed with fractional using novel wavelet-based vaguelette–vaguelette approach. It allows us apply sequence space methods without assuming that all operators are simultaneously diagonalizable. The results proved for more general bases covariance operators. Our primary aim is study posterior contraction rate such over Sobolev classes compare it the derived minimax rate. Secondly, we effect of plugging consistent estimator variances This result applied problem error forward operator. Thirdly, show empirical Bayes distribution plugged-in maximum marginal likelihood prior scale contracts at optimal rate, adaptively, sense.