### #map-estimators

### Γ-convergence of Onsager-Machlup functionals in Inverse Problems

The articles “Γ-convergence of Onsager–Machlup functionals” (“I. With applications to maximum a posteriori estimation in Bayesian inverse problems” and “II. Infinite product measures on Banach spaces”) by Birzhan Ayanbayev, Ilja Klebanov, Han Cheng Lie, and myself have just appeared in their final form in the journal *Inverse Problems*.

The purpose of this work is to address a long-standing issue in the Bayesian approach to inverse problems, namely the joint stability of a Bayesian posterior and its modes (MAP estimators) when the prior, likelihood, and data are perturbed or approximated. We show that the correct way to approach this problem is to interpret MAP estimators as global weak modes in the sense of Helin and Burger (2015), which can be identified as the global minimisers of the Onsager–Machlup functional of the posterior distribution, and hence to provide a convergence theory for MAP estimators in terms of Γ-convergence of these Onsager–Machlup functionals. It turns out that posterior Γ-convergence can be assessed in a relatively straightforward manner in terms of prior Γ-convergence and continuous convergence of potentials (negative log-likelihoods). Over the two parts of the paper, we carry out this programme both in generality and for specific priors that are commonly used in Bayesian inverse problems, namely Gaussian and Besov priors (Lassas et al., 2009; Dashti et al., 2012).

B. Ayanbayev, I. Klebanov, H. C. Lie, and T. J. Sullivan. “Γ-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems.” *Inverse Problems* 38(2):025005, 32pp., 2022.

**Abstract (Part I).**
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 Γ-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.

B. Ayanbayev, I. Klebanov, H. C. Lie, and T. J. Sullivan. “Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces.” *Inverse Problems* 38(2):025006, 35pp., 2022.

**Abstract (Part II).**
We derive Onsager–Machlup functionals for countable product measures on weighted \(\ell^{p}\) subspaces of the sequence space \(\mathbb{R}^\mathbb{N}\). Each measure in the product is a shifted and scaled copy of a reference probability measure on \(\mathbb{R}\) that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter \( 1 \leq p \leq 2 \). Together with Part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

Published on Wednesday 5 January 2022 at 12:00 UTC #publication #inverse-problems #modes #map-estimators #ayanbayev #klebanov #lie

### Γ-convergence of Onsager-Machlup functionals

Birzhan Ayanbayev, Ilja Klebanov, Han Cheng Lie, and I have just uploaded preprints of our work “Γ-convergence of Onsager–Machlup functionals” to the arXiv; this work consists of two parts, “Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems” and “Part II: Infinite product measures on Banach spaces”.

The purpose of this work is to address a long-standing issue in the Bayesian approach to inverse problems, namely the joint stability of a Bayesian posterior and its modes (MAP estimators) when the prior, likelihood, and data are perturbed or approximated. We show that the correct way to approach this problem is to interpret MAP estimators as global weak modes in the sense of Helin and Burger (2015), which can be identified as the global minimisers of the Onsager–Machlup functional of the posterior distribution, and hence to provide a convergence theory for MAP estimators in terms of Γ-convergence of these Onsager–Machlup functionals. It turns out that posterior Γ-convergence can be assessed in a relatively straightforward manner in terms of prior Γ-convergence and continuous convergence of potentials (negative log-likelihoods). Over the two parts of the paper, we carry out this programme both in generality and for specific priors that are commonly used in Bayesian inverse problems, namely Gaussian and Besov priors (Lassas et al., 2009; Dashti et al., 2012).

**Abstract (Part I).**
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 Γ-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.

**Abstract (Part II).**
We derive Onsager–Machlup functionals for countable product measures on weighted \(\ell^{p}\) subspaces of the sequence space \(\mathbb{R}^\mathbb{N}\). Each measure in the product is a shifted and scaled copy of a reference probability measure on \(\mathbb{R}\) that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter \( 1 \leq p \leq 2 \). Together with Part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

Published on Wednesday 11 August 2021 at 12:00 UTC #preprint #inverse-problems #modes #map-estimators #ayanbayev #klebanov #lie

### Birzhan Ayanbayev joins the UQ Group

It is a pleasure to announce that Birzhan Ayanbayev will join the UQ research group as a postdoctoral researcher with effect from 28 February 2020. He will be working on the DFG-funded project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems”.

Published on Friday 28 February 2020 at 12:00 UTC #group #job #dfg #map-estimators #ayanbayev

### Postdoc position: Analysis of MAP estimators

There is still an opening for a full-time two-year postdoctoral research position in the UQ group at the Freie Universität Berlin. This position will be associated to the project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems” funded by the DFG.

This project aims to advance the state of the art in rigorous mathematical understanding of MAP estimators in infinite-dimensional statistical inverse problems. In particular, the research in this project will connect the “small balls” approach of Dashti et al. (2013) to the calculus of variations and hence properly link the variational and fully Bayesian points of view on inverse problems. This is an exciting opportunity for someone well-versed in the calculus of variations and tools such as Γ-convergence to make an impact on fundamental questions of non-parametric statistics and inverse problems or, vice versa, for someone with a statistical inverse problems background to advance the rigorous state of the art for such methods.

Prospective candidates are encouraged to contact me with informal enquiries. Formal applications are to be sent by post or email, by **23 December 2019**, under the heading **MAP-Analysis**, and should include a cover letter, a scientific CV including list of publications and research statement, and the contact details of two professional references.

Published on Monday 25 November 2019 at 08:00 UTC #group #job #fu-berlin #inverse-problems #dfg #map-estimators

### Weak and strong modes in Inverse Problems

The paper “Equivalence of weak and strong modes of measures on topological vector spaces” by Han Cheng Lie and myself has now appeared in *Inverse Problems*.
This paper addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in an infinite-dimensional space \(X\).
Such modes can be defined either *strongly* (a la Dashti et al. (2013), via a global maximisation) or *weakly* (a la Helin and Burger (2015), via a dense subspace \(E \subset X\)).
The question is, when are strong and weak modes equivalent?
The answer turns out to be rather subtle:
under reasonable uniformity conditions, the two kinds of modes are indeed equivalent, but finite-dimensional counterexamples exist when the uniformity conditions fail.

H. C. Lie and T. J. Sullivan. “Equivalence of weak and strong modes of measures on topological vector spaces.” *Inverse Problems* 34(11):115013, 2018.

(See also H. C. Lie and T. J. Sullivan. “Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 *Inverse Problems* 34 115013).” *Inverse Problems* 34(12):129601, 2018.

**Abstract.** A strong mode of a probability measure on a normed space \(X\) can be defined as a point \(u \in X\) such that the mass of the ball centred at \(u\) uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace \(E\) of \(X\), and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with non-atomic measures that are finite on bounded sets, the density of \(E\) and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.

Published on Saturday 22 September 2018 at 12:00 UTC #publication #inverse-problems #modes #map-estimators #lie