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dc.contributor.authorRay, Kolyan
dc.date.accessioned2018-07-20T15:59:26Z
dc.date.available2018-07-20T15:59:26Z
dc.date.issued2015-07-18
dc.identifier.otherPhD.38208
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/278387
dc.description.abstractThe main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.
dc.description.sponsorshipThis work was supported by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1.
dc.language.isoen
dc.rightsAll rights reserved
dc.subjectBayesian nonparametric statistics
dc.subjectInverse problems
dc.subjectAdaptation
dc.subjectUncertainty quantification
dc.titleAsymptotic theory for Bayesian nonparametric procedures in inverse problems
dc.typeThesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridge
dc.publisher.departmentPure Mathematics and Mathematical Statistics
dc.date.updated2018-07-20T15:00:25Z
dc.identifier.doi10.17863/CAM.25733
dc.publisher.collegeSt Catharine's College
dc.type.qualificationtitlePhD in Mathematics
cam.supervisorNickl, Richard
cam.thesis.fundingtrue
rioxxterms.freetoread.startdate2018-07-20


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