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Posterior consistency for Bayesian inverse problems through stability and regression results
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Vollmer, Sebastian (2013) Posterior consistency for Bayesian inverse problems through stability and regression results. Inverse Problems, Volume 29 (Number 12). Article number 125011. doi:10.1088/0266-5611/29/12/125011 ISSN 0266-5611.
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Official URL: http://dx.doi.org/10.1088/0266-5611/29/12/125011
Abstract
In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise to the posterior distribution on the unknown input. In this setting we prove posterior consistency for nonlinear inverse problems: a sequence of data is considered, with diminishing fluctuations around a single truth and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around the truth used to generate the data. Posterior consistency justifies the use of the Bayesian approach very much in the same way as error bounds and convergence results for regularization techniques do. As a guiding example, we consider the inverse problem of reconstructing the diffusion coefficient from noisy observations of the solution to an elliptic PDE in divergence form. This problem is approached by splitting the forward operator into the underlying continuum model and a simpler observation operator based on the output of the model. In general, these splittings allow us to conclude posterior consistency provided a deterministic stability result for the underlying inverse problem and a posterior consistency result for the Bayesian regression problem with the push-forward prior. Moreover, we prove posterior consistency for the Bayesian regression problem based on the regularity, the tail behaviour and the small ball probabilities of the prior.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Bayesian statistical decision theory, Inverse problems (Differential equations), Regression analysis | ||||
Journal or Publication Title: | Inverse Problems | ||||
Publisher: | Institute of Physics Publishing Ltd. | ||||
ISSN: | 0266-5611 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 29 | ||||
Number: | Number 12 | ||||
Article Number: | Article number 125011 | ||||
DOI: | 10.1088/0266-5611/29/12/125011 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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