Can one use total variation prior for edge-preserving Bayesian inversion?

Estimation of non-discrete physical quantities from indirect linear measurements is considered. Bayesian solution of such an inverse problem involves discretizing the problem and expressing available a priori information in the form of a prior distribution in a finite-dimensional space. Since a priori information is independent of the measurement, the discretization of the unknown quantity can be arbitrarily fine regardless of the number of measurements. The main result is that Bayesian conditional mean estimates for total variation prior distribution are not edge-preserving with very fine discretizations of the model space. Theoretical findings are illustrated by a numerical example with computer simulated data.