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Γ -convergence of Onsager–Machlup functionals : I. With applications to maximum a posteriori estimation in Bayesian inverse problems
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Ayanbayev, Birzhan, Klebanov, Ilja, Lie, Han Cheng and Sullivan, T. J. (2021) Γ -convergence of Onsager–Machlup functionals : I. With applications to maximum a posteriori estimation in Bayesian inverse problems. Inverse Problems, 38 (2). 025005. doi:10.1088/1361-6420/ac3f81 ISSN 1361-6420.
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Official URL: https://doi.org/10.1088/1361-6420/ac3f81
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.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| SWORD Depositor: | Library Publications Router | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Bayesian statistical decision theory, Inverse problems (Differential equations), Functionals, Gaussian measures, Besov spaces, Convergence | ||||||||||||
| Journal or Publication Title: | Inverse Problems | ||||||||||||
| Publisher: | IOP Publishing | ||||||||||||
| ISSN: | 1361-6420 | ||||||||||||
| Official Date: | 28 December 2021 | ||||||||||||
| Dates: |
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| Volume: | 38 | ||||||||||||
| Number: | 2 | ||||||||||||
| Article Number: | 025005 | ||||||||||||
| DOI: | 10.1088/1361-6420/ac3f81 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
| Date of first compliant deposit: | 1 February 2022 | ||||||||||||
| Date of first compliant Open Access: | 1 February 2022 | ||||||||||||
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