Γ-convergence of Onsager-Machlup functionals. Part 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 a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup 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 Onsager-Machlup 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.

arXiv comment30 pages

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Cited by in corpus (7)

• Most probable transition paths in piecewise-smooth stochastic differential equations
• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
• Maximum a posteriori estimators in ${\ell }^p$ are well-defined for diagonal Gaussian priors
• An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems
• Are minimizers of the Onsager-Machlup functional strong posterior modes?
• Strong maximum a posteriori estimation in Banach spaces with Gaussian priors
• On Unifying Randomized Methods For Inverse Problems