MAP estimators and their consistency in Bayesian nonparametric inverse problems

Handle URI:
http://hdl.handle.net/10754/594280
Title:
MAP estimators and their consistency in Bayesian nonparametric inverse problems
Authors:
Dashti, M.; Law, K. J H; Stuart, A. M.; Voss, J.
Abstract:
We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics. © 2013 IOP Publishing Ltd.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Dashti M, Law KJH, Stuart AM, Voss J (2013) MAP estimators and their consistency in Bayesian nonparametric inverse problems. Inverse Problems 29: 095017. Available: http://dx.doi.org/10.1088/0266-5611/29/9/095017.
Publisher:
IOP Publishing
Journal:
Inverse Problems
Issue Date:
1-Sep-2013
DOI:
10.1088/0266-5611/29/9/095017
ARXIV:
arXiv:1303.4795v3
Type:
Article
ISSN:
0266-5611; 1361-6420
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorDashti, M.en
dc.contributor.authorLaw, K. J Hen
dc.contributor.authorStuart, A. M.en
dc.contributor.authorVoss, J.en
dc.date.accessioned2016-01-19T14:45:05Zen
dc.date.available2016-01-19T14:45:05Zen
dc.date.issued2013-09-01en
dc.identifier.citationDashti M, Law KJH, Stuart AM, Voss J (2013) MAP estimators and their consistency in Bayesian nonparametric inverse problems. Inverse Problems 29: 095017. Available: http://dx.doi.org/10.1088/0266-5611/29/9/095017.en
dc.identifier.issn0266-5611en
dc.identifier.issn1361-6420en
dc.identifier.doi10.1088/0266-5611/29/9/095017en
dc.identifier.urihttp://hdl.handle.net/10754/594280en
dc.description.abstractWe consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics. © 2013 IOP Publishing Ltd.en
dc.publisherIOP Publishingen
dc.titleMAP estimators and their consistency in Bayesian nonparametric inverse problemsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalInverse Problemsen
dc.contributor.institutionDepartment of Mathematics, University of Sussex, Brighton BN1 9QH, United Kingdomen
dc.contributor.institutionMathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdomen
dc.contributor.institutionSchool of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdomen
dc.identifier.arxividarXiv:1303.4795v3en
kaust.authorLaw, Kodyen
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