  In this article, the solution of a statistical inverse problem $M =
AU+\mathcal{E}$ by the Bayesian approach is studied where $U$ is a function on
the unit circle $\mathbb{T}$, i.e., a periodic signal. The mapping $A$ is a
smoothing linear operator and $\mathcal{E}$ a Gaussian noise. The connection to
the solution of a finite-dimensional computational model $M_{kn} = A_k U_n +
\mathcal{E}_k$ is discussed. Furthermore, a novel hierarchical prior model for
obtaining edge-preserving conditional mean estimates is introduced. The
convergence of the method with respect to finer discretization is studied and
the posterior distribution is shown to converge weakly. Finally, theoretical
findings are illustrated by a numerical example with simulated data.
