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

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. ...

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

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. ...

##
###
Inverse Problems as Statistics
[chapter]

2000
*
Surveys on Solution Methods for Inverse Problems
*

What mathematicians, scientists, engineers, and statisticians mean by i n verse

doi:10.1007/978-3-7091-6296-5_13
fatcat:lirzzbzqtfdsbnplw6da2rdeey
*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. ...##
###
Inverse problems as statistics

2002
*
Inverse Problems
*

What mathematicians, scientists, engineers, and statisticians mean by i n verse

doi:10.1088/0266-5611/18/4/201
fatcat:2uejaww53rdb7lvy7feo6rjlkm
*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. ...##
###
Solving inverse problems using data-driven models

2019
*
Acta Numerica
*

This survey paper aims to give

doi:10.1017/s0962492919000059
fatcat:2f7te542wrftphdhurcdnw6dqu
*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. ...##
###
Selection properties of type II maximum likelihood (empirical Bayes) in linear models with individual variance components for predictors

2012
*
Pattern Recognition Letters
*

We conclude that Type

doi:10.1016/j.patrec.2012.01.004
fatcat:ikr6jwm7pjbjje6vbuw6vl75gm
*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). ...##
###
Bernstein - von Mises theorems for statistical inverse problems II: Compound Poisson processes
[article]

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*...

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

2010
*
Annals of Statistics
*

Our second goal is to contrast

doi:10.1214/10-aos792
fatcat:pgz4v6b6prfwvgdvn2wn7mguei
*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. ...##
###
Inverse statistical problems: from the inverse Ising problem to data science

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] . ...

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

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). ...

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

2022
*
arXiv
*
pre-print

The first

arXiv:1810.06191v4
fatcat:mcj5zquwbfgebkgknildjzknzy
*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 ...##
###
Regularized Ensemble Kalman Methods for Inverse Problems
[article]

2020
*
arXiv
*
pre-print

We demonstrate the method's ability to

arXiv:1910.01292v2
fatcat:72u3zgxhkngu7iolk6r52k4l4y
*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*. ...##
###
Assessing the relevance of fMRI-based prior in the EEG inverse problem: a bayesian model comparison approach

2005
*
IEEE Transactions on Signal Processing
*

Therefore, the introduction

doi:10.1109/tsp.2005.853220
fatcat:kuoky6b6wfgndlw3wxho2vd4zm
*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*). ...##
###
Posterior contraction for empirical Bayesian approach to inverse problems under non-diagonal assumption
[article]

2020
*
arXiv
*
pre-print

We investigate

arXiv:1810.02221v2
fatcat:z6ysipoxizfkjdkgzwmgfa2ghe
*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*. ...##
###
A Bayesian Approach to Nonlinear Parameter Identification for Rigid Body Dynamics

2006
*
Robotics: Science and Systems II
*

Data driven

doi:10.15607/rss.2006.ii.032
dblp:conf/rss/TingMPSN06
fatcat:svey7hw3qjgcdgvtjccbp3rkru
*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 ...
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