Γ-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems. (28th December 2021)

• Record Type: Journal Article
• Authors: Ayanbayev, Birzhan; Klebanov, Ilja; Lie, Han Cheng; Sullivan, T J
• Is Part Of: Inverse problems. Volume 38:Number 2(2022)
• Publication Date: 2021-12-28
• Subjects: Bayesian inverse problems -- Γ-convergence -- maximum a posteriori estimation -- Onsager–Machlup functional -- small ball probabilities -- transition path theory; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac3f81
• Languages: English
• ISSNs: 0266-5611

Abstract

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 Γ-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.