Bayesian methods for linear inverse problems are studied using hierarchical Gaussian models. The problems are considered with different discretizations, and the phenomena are analysed which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. It is shown 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. Inverse problems under different assumptions are studied on the asymptotic behaviour of the noise as the discretization becomes finer. It is shown that the maximum a posteriori and conditional mean estimates converge under different conditions.