Bernstein--von Mises Theorems and Uncertainty Quantification for Linear Inverse Problems

Giordano, Matteo; Kekkonen, Hanne

doi:10.1137/18m1226269

Cited by 19 publications

(29 citation statements)

References 52 publications

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“…For example, Monard, Nickl and Paternain (2019) consider the case of inverting the (generalized) ray transform, whereas Nickl (2017 a ) considers PDE parameter estimation problems. The case with general linear forward problem is treated by Giordano and Kekkonen (2018), who build upon the techniques of Monard et al. (2019).…”

Section: Statistical Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

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Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

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Section: Statistical Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

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“…Since the parameter θ in many UQ applications is an unknown function, one may use a Bayesian nonparametric approach to estimate θ (BNP, see, for example Müller et al, 2015, chap. 4), using a prior measure on a set of regressors with the machinery of BNP (see Giordano and Kekkonen, 2020, and references therein).…”

Section: Discussionmentioning

confidence: 99%

Error Control of the Numerical Posterior with Bayes Factors in Bayesian Uncertainty Quantification

et al. 2022

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In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of Capistrán et al. (2016) to (a priori) expected Bayes factors (BF) and in a more general, infinitedimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.

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“…Bernstein-von Mises theorems for general inverse problems: This paper builds on key ideas for nonparametric Bernstein-von Mises theorems in direct models [4][5][6][7][8]. For inverse problems previous work on Bernstein-von Mises theorems treated regression-type problems where the likelihood has a more explicit Gaussian structure, see [21,24] and also the more recent contributions [19,25]. In our jump process setting, the log-likelihood function does not have the form of a Gaussian process, but we show how empirical process methods [18] can be used to obtain exact Gaussian posterior asymptotics in such situations as well.…”

Section: Discussionmentioning

confidence: 99%

Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes

Nickl¹,

Söhl²

2019

Electron. J. Statist.

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We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the formwhere N (t) is a standard Poisson process of intensity λ, and Z k are drawn i.i.d. from jump measure µ. A high-dimensional wavelet series prior for the Lévy measure ν = λµ is devised and the posterior distribution arises from observing discrete samples Y ∆ , Y 2∆ , . . . , Y n∆ at fixed observation distance ∆, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true Lévy density that are optimal up to logarithmic factors over Hölder classes, as sample size n increases. We prove a functional Bernstein-von Mises theorem for the distribution functions of both µ and ν, as well as for the intensity λ, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence posterior based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.MSC 2000 subject classification: 62G20, 65N21, 60G51, 60J75

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Bernstein--von Mises Theorems and Uncertainty Quantification for Linear Inverse Problems

Cited by 19 publications

References 52 publications

Solving inverse problems using data-driven models

Solving inverse problems using data-driven models

Error Control of the Numerical Posterior with Bayes Factors in Bayesian Uncertainty Quantification

Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes

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