  The Bayesian methods for linear inverse problems is studied using
hierarchical Gaussian models. The problems are considered with different
discretizations, and we analyze the phenomena which appear when the
discretization becomes finer. A hierarchical solution method for signal
restoration problems is introduced and studied with arbitrarily fine
discretization. We show that the maximum a posteriori estimate converges to a
minimizer of the Mumford--Shah functional, up to a subsequence. A new result
regarding the existence of a minimizer of the Mumford--Shah functional is
proved.
  Moreover, we study the inverse problem under different assumptions on the
asymptotic behavior of the noise as discretization becomes finer. We show that
the maximum a posteriori and conditional mean estimates converge under
different conditions.
