Gamma-convergence of Onsager-Machlup functionals
- 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.
Author details: | Birzhan AyanbayevORCiD, Ilja KlebanovORCiD, Han Cheng LiORCiD, Tim J. SullivanORCiDGND |
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DOI: | https://doi.org/10.1088/1361-6420/ac3f81 |
ISSN: | 0266-5611 |
ISSN: | 1361-6420 |
Title of parent work (English): | Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data |
Subtitle (English): | I. With applications to maximum a posteriori estimation in Bayesian inverse problems |
Publisher: | IOP Publ. Ltd. |
Place of publishing: | Bristol |
Publication type: | Article |
Language: | English |
Date of first publication: | 2021/12/28 |
Publication year: | 2021 |
Release date: | 2024/01/29 |
Tag: | Bayesian inverse problems; Gamma-convergence; Onsager-Machlup functional; estimation; maximum a posteriori; small ball probabilities;; transition path theory |
Volume: | 38 |
Issue: | 2 |
Article number: | 025005 |
Number of pages: | 32 |
Funding institution: | Deutsche Forschungsgemeinschaft (DFG) [415980428]; DFG [TrU-2, EF1-10,; EXC-2046/1, 390685689, 318763901-SFB1294] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Hybrid Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |