On infinite-dimensional hierarchical probability models in statistical inverse problems

Abstract

In this article, the solution of a statistical inverse problem $M = AU+$ε by the Bayesian approach is studied where$U$ is a function on the unit circle $\T$, i.e., a periodic signal. The mapping $A$ is a smoothing linear operatorand ε a Gaussian noise. The connection to the solution ofa finite-dimensional computational model $M_{kn} = A_k U_n + $εk 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 andthe posterior distribution is shown to converge weakly.Finally, theoretical findings are illustrated by a numerical example with simulated data.