Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

Abstract The Bayesian solution to a statistical inverse problem can be summarised by mode of the posterior distribution, i.e. maximum posteriori (MAP) estimator. MAP estimator essentially coincides with (regularised) variational problem, seen as minimisation Onsager–Machlup (OM) functional measure. An open in stability analysis problems is establish relationship between convergence properties solutions obtained approach and approach. To address this we propose general theory for modes that based on Γ-convergence OM functionals, apply Gaussian edge-preserving Besov priors. Part II paper considers more prior distributions.