Γ -convergence of Onsager–Machlup functionals: 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 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.

Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

“…The next result states this equivalence between OM minimisers and weak modes under the M-property and shows that the M-property is inherited by a posterior of the form (1.1) from the prior. This generalises proposition 4.1 and lemma B.8 of Ayanbayev et al (2022a) to potentials that are merely continuous rather than locally uniformly continuous. In the specific case that µ 0 is a Gaussian measure on a separable Banach space X, the claim (a) generalises theorem 3.2 of Dashti et al (2013), which requires that the potential is locally bounded and Lipschitz.…”

“…Strong modes were proposed by Dashti et al (2013), and weak modes were later suggested by Helin and Burger (2015) as a more convenient definition when connecting MAP estimators with variational solutions to inverse problems. Following Ayanbayev et al (2022a), we consider only global weak modes in this article.…”

“…As an OM functional need not be defined on the entire space X, it is not immediate that an OM minimiser is in any sense 'most likely' under the measure µ, and this is the motivation to study small-ball modes as in definition 2.1 instead. A weak mode is always a minimiser of any OM functional for µ, however, and the M-property of Ayanbayev et al (2022a) gives a sufficient condition to ensure that an OM minimiser is a weak mode.…”

Maximum a posteriori estimators in ℓp are well-defined for diagonal Gaussian priors

“…Agapiou et al (2018) studied the MAP estimator for Bayesian inversion with sparsity-promoting Besov priors. The connection between weak and strong modes was further explored in Lie and Sullivan (2018), and Ayanbayev et al (2021a, 2021b discussed stability and convergence of global weak modes using Γ-convergence. Recently, Lambley and Sullivan (2022) presented a perspective on modes via order theory.…”

On unifying randomized methods for inverse problems

“…Therefore, the novelty of theorem 1 is not in the proof that the cost functions converge, but that the randomized cost functions yield solutions that asymptotically converge to the solution of the non-randomized cost function. While we constructively prove Γ−convergence for (4), our result follows immediately under the much stronger assumption that the prior term already exhibits Γ−convergence and the likelihood term converges continuously based on the results presented in Ayanbayev et al (2021). In fact, the cost functions themselves do not converge as there is an additional bias term in expected value of the randomized cost function, J , compared to the deterministic cost function, J .…”