We consider statistical linear inverse problems in Hilbert spaces of the type ˆ Y = Kx + U where we want to estimate the function x from indirect noisy functional observations ˆY . In several applications the operator K has an inverse that is not continuous on the whole space of reference; this phenomenon is known as ill-posedness of the inverse problem. We use a Bayesian approach and a conjugate-Gaussian model. For a very general specification of the probability model the posterior distribution of x is known to be inconsistent in a frequentist sense. Our first contribution consists in constructing a class of Gaussian prior distributions on x that are shrinking with the measurement error U and we show that, under mild conditions, the corresponding posterior distribution is consistent in a frequentist sense and converges at the optimal rate of contraction. Then, a class ^ of posterior mean estimators for x is given. We propose an empirical Bayes procedure for selecting an estimator in this class that mimics the posterior mean that has the smallest risk on the true x.
TSE Working Paper, n. 10-175, May 2010