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Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach. Part II: Theoretical Analysis [article]

Valentin De Bortoli, Alain Durmus, Ana F. Vidal, Marcelo Pereyra
2020 arXiv   pre-print
maximum likelihood estimation.  ...  This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal  ...  More precisely, in [49] , we adopt an empirical Bayesian approach and set θ by maximum marginal likelihood estimation, i.e. (2) To solve (2) , we aim at using gradient based optimization methods.  ... 
arXiv:2008.05793v1 fatcat:tawpjxbfozh7xjzpuxlhy4pj54

Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach. Part I: Methodology and Experiments [article]

Ana F. Vidal, Valentin De Bortoli, Marcelo Pereyra, Alain Durmus
2020 arXiv   pre-print
In this work, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image.  ...  Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed.  ...  The work of MP is supported by UKRI/EPSRC under grant EP/T007346/1.  ... 
arXiv:1911.11709v3 fatcat:hcq7a4efbjgonnhqum6i5b4nfa

Inverse Problems as Statistics [chapter]

P. B. Stark
2000 Surveys on Solution Methods for Inverse Problems  
What mathematicians, scientists, engineers, and statisticians mean by i n verse problem" di ers. For a statistician, an inverse problem is an inference or estimation problem.  ...  Canonical abstract formulations of statistical estimation problems subsume this complication by allowing probability distributions to be indexed in more-or-less arbitrary ways by parameters, which can  ...  problem maximum likelihood faces even in quite regular inverse problems is the existence of in nitely many maximizers.  ... 
doi:10.1007/978-3-7091-6296-5_13 fatcat:lirzzbzqtfdsbnplw6da2rdeey

Inverse problems as statistics

Steven N Evans, Philip B Stark
2002 Inverse Problems  
What mathematicians, scientists, engineers, and statisticians mean by i n verse problem" di ers. For a statistician, an inverse problem is an inference or estimation problem.  ...  Canonical abstract formulations of statistical estimation problems subsume this complication by allowing probability distributions to be indexed in more-or-less arbitrary ways by parameters, which can  ...  problem maximum likelihood faces even in quite regular inverse problems is the existence of in nitely many maximizers.  ... 
doi:10.1088/0266-5611/18/4/201 fatcat:2uejaww53rdb7lvy7feo6rjlkm

Solving inverse problems using data-driven models

Simon Arridge, Peter Maass, Ozan Öktem, Carola-Bibiane Schönlieb
2019 Acta Numerica  
This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.  ...  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  ...  Acknowledgements This article builds on lengthy discussions and long-standing collaborations with a large number of people.  ... 
doi:10.1017/s0962492919000059 fatcat:2f7te542wrftphdhurcdnw6dqu

Selection properties of type II maximum likelihood (empirical Bayes) in linear models with individual variance components for predictors

Tahira Jamil, Cajo J.F. ter Braak
2012 Pattern Recognition Letters  
We conclude that Type II ML is not the general answer in high dimensional prediction problems.  ...  RVM assigns individual precisions to weights of predictors which are then estimated by maximizing the marginal likelihood (type II ML or empirical Bayes).  ...  Jamil's research was supported by a grant from Higher Education Commission of Pakistan through NUFFIC (The Netherlands).  ... 
doi:10.1016/j.patrec.2012.01.004 fatcat:ikr6jwm7pjbjje6vbuw6vl75gm

Bernstein - von Mises theorems for statistical inverse problems II: Compound Poisson processes [article]

Richard Nickl, Jakob Söhl
2019 arXiv   pre-print
an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the Cramér-Rao lower bound for this inverse problem.  ...  We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form Y_t = ∑_k=1^N(t) Z_k, t > 0, where N(t) is a standard Poisson process of intensity λ,  ...  Given the sophistication of the non-linear estimators proposed so far in the 'decompounding problem' just described, one may wonder if a 'principled' Bayesian approach that just places a standard high-dimensional  ... 
arXiv:1709.07752v2 fatcat:piy6sfcaxngunjpt46ip54fsbm

Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

James G. Scott, James O. Berger
2010 Annals of Statistics  
Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations.  ...  This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression.  ...  The lemma refers to the variableselection problem, with the prior variable inclusion probability p being estimated by marginal (or Type-II) maximum likelihood in the empirical-Bayes approach. PROOF.  ... 
doi:10.1214/10-aos792 fatcat:pgz4v6b6prfwvgdvn2wn7mguei

Inverse statistical problems: from the inverse Ising problem to data science

H. Chau Nguyen, Riccardo Zecchina, Johannes Berg
2017 Advances in Physics  
Inverse problems in statistical physics are motivated by the challenges of 'big data' in different fields, in particular high-throughput experiments in biology.  ...  In inverse problems, the usual procedure of statistical physics needs to be reversed: Instead of calculating observables on the basis of model parameters, we seek to infer parameters of a model based on  ...  Maximum likelihood The inverse Ising problem is a problem of statistical inference [31, 131] .  ... 
doi:10.1080/00018732.2017.1341604 fatcat:yof3mukd6ng2rizgbvuwzckdz4

Improving landscape inference by integrating heterogeneous data in the inverse Ising problem

Pierre Barrat-Charlaix, Matteo Figliuzzi, Martin Weigt
2016 Scientific Reports  
In this paper, we extend the usual setting of the inverse Ising model by developing an integrative approach combining the equilibrium sample with (possibly noisy) measurements of the energy performed for  ...  In the standard setting, the parameters of an Ising model (couplings and fields) are inferred using a sample of equilibrium configurations drawn from the Boltzmann distribution.  ...  This work undertaken partially in the framework of CALSIMLAB, supported by the grant ANR-11-LABX-0037-01 as part of the "Investissements d'Avenir" program (ANR-11-IDEX-0004-02).  ... 
doi:10.1038/srep37812 pmid:27886273 pmcid:PMC5122905 fatcat:4j2lwni2njawzlkac4sh4zkmmi

Inverse Problems and Data Assimilation with Connections to Machine Learning [article]

Daniel Sanz-Alonso and Andrew M. Stuart and Armeen Taeb
2022 arXiv   pre-print
The first part of the notes is dedicated to studying the Bayesian framework for inverse problems.  ...  This refers to a particular class of inverse problems in which the unknown parameter is the initial condition (and/or state) of a dynamical system, and the data comprises partial and noisy observations  ...  Acknowledgments These notes were created in L A T E X by the students in ACM 159, based on lectures  ... 
arXiv:1810.06191v4 fatcat:mcj5zquwbfgebkgknildjzknzy

Regularized Ensemble Kalman Methods for Inverse Problems [article]

Xin-Lei Zhang, Carlos Michelén-Ströfer, Heng Xiao
2020 arXiv   pre-print
We demonstrate the method's ability to regularize the inverse problem with three cases of increasing complexity, starting with inferring scalar model parameters.  ...  Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation  ...  From a Bayesian perspective, both of these approaches find the maximum a posteriori (MAP) estimates.  ... 
arXiv:1910.01292v2 fatcat:72u3zgxhkngu7iolk6r52k4l4y

Assessing the relevance of fMRI-based prior in the EEG inverse problem: a bayesian model comparison approach

J. Daunizeau, C. Grova, J. Mattout, G. Marrelec, D. Clonda, B. Goulard, M. Pelegrini-Issac, J.-M. Lina, H. Benali
2005 IEEE Transactions on Signal Processing  
Therefore, the introduction of spatial priors derived from other functional modalities in the EEG/MEG inverse problem should be considered with caution.  ...  In this paper, we propose a Bayesian characterization of the relevance of fMRI-derived prior information regarding the EEG/MEG data.  ...  Bayesian approaches).  ... 
doi:10.1109/tsp.2005.853220 fatcat:kuoky6b6wfgndlw3wxho2vd4zm

Posterior contraction for empirical Bayesian approach to inverse problems under non-diagonal assumption [article]

Junxiong Jia and Jigen Peng and Jinghuai Gao
2020 arXiv   pre-print
We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise.  ...  By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter.  ...  paper, we study an empirical Bayesian approach to a family of linear inverse problems.  ... 
arXiv:1810.02221v2 fatcat:z6ysipoxizfkjdkgzwmgfa2ghe

A Bayesian Approach to Nonlinear Parameter Identification for Rigid Body Dynamics

J. Ting, M. Mistry, J. Peters, S. Schaal, J. Nakanishi
2006 Robotics: Science and Systems II  
Data driven parameter estimation offers an alternative model identification method, but it is often burdened by various other problems, such as significant noise in all measured or inferred variables of  ...  In this paper, we address all these problems by developing a Bayesian parameter identification method that can automatically detect noise in both input and output data for the regression algorithm that  ...  ACKNOWLEDGMENTS This research was supported in part by National Science Foundation grants ECS-0325383, IIS-0312802, IIS-0082995, ECS-0326095, ANI-0224419, a NASA grant AC#98 − 516, an AFOSR grant on Intelligent  ... 
doi:10.15607/rss.2006.ii.032 dblp:conf/rss/TingMPSN06 fatcat:svey7hw3qjgcdgvtjccbp3rkru
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