Agapiou, Sergios, Stuart, A. M. and Zhang, Yuan-Xiang (2013) Bayesian posterior contraction rates for linear severely ill-posed inverse problems. Journal of Inverse and Ill-posed Problems, 22 (3). pp. 297-321. doi:10.1515/jip-2012-0071

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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:	Journal Article
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Journal of Inverse and Ill-posed Problems
Publisher:	De Gruyter
ISSN:	0928-0219
Official Date:	5 December 2013
Dates:	Date Event 5 December 2013 Accepted 5 December 2013 Published 1 October 2012 Submitted
Volume:	22
Number:	3
Page Range:	pp. 297-321
DOI:	10.1515/jip-2012-0071
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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