-convergence of Onsager-Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems
INVERSE PROBLEMS(2022)
Abstract
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.
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Key words
Bayesian inverse problems,Gamma-convergence,maximum a posteriori estimation,Onsager-Machlup functional,small ball probabilities,transition path theory
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