Convergence rate for the Bayesian approach to linear inverse problems

Recently, the metrics of Ky Fan and Prokhorov were introduced as a tool for studying convergence of regularization methods for stochastic ill-posed problems. In this work, we show that the Bayesian approach to linear inverse problems can be examined in the new framework as well. We consider the finite-dimensional case where the measurements are disturbed by an additive normal noise, and the prior distribution is normal. Convergence and convergence rate results for the posterior distribution are obtained when the covariance matrices are proportional to the identity matrix.