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Bayesian posterior contraction rates for linear severely ill-posed inverse problems

Agapiou, Sergios and Stuart, Andrew M. and Zhang, Yuan-Xiang (2014) Bayesian posterior contraction rates for linear severely ill-posed inverse problems. Journal of Inverse and Ill-posed Problems, 22 (3). pp. 297-321. ISSN 1569-3945. https://resolver.caltech.edu/CaltechAUTHORS:20160719-151308932

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Abstract

We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1515/jip-2012-0071DOIArticle
http://www.degruyter.com/view/j/jiip.2014.22.issue-3/jip-2012-0071/jip-2012-0071.xmlPublisherArticle
http://arxiv.org/abs/1210.1563v3arXivDiscussion Paper
Additional Information:© 2013 Walter de Gruyter GmbH. Agapiou and Stuart are supported by ERC and Yuan-Xiang Zhang is supported by China Scholarship Council and the NNSF of China (No. 11171136).
Funders:
Funding AgencyGrant Number
European Research Council (ERC)UNSPECIFIED
China Scholarship CouncilUNSPECIFIED
National Natural Science Foundation of China11171136
Subject Keywords:Gaussian prior, posterior consistency, rate of contraction, severely ill-posed problems.
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ109
Issue or Number:3
Classification Code:2010 Mathematics Subject Classification. 62G20, 62C10, 35R30, 45Q05
Record Number:CaltechAUTHORS:20160719-151308932
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160719-151308932
Official Citation:Journal of Inverse and Ill-posed Problems. Volume 22, Issue 3, Pages 297–321, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jip-2012-0071, December 2013
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69119
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:20 Jul 2016 18:07
Last Modified:03 Oct 2019 10:19

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