dc.contributor.author Ray, Kolyan dc.date.accessioned 2018-07-20T15:59:26Z dc.date.available 2018-07-20T15:59:26Z dc.date.issued 2015-07-18 dc.identifier.other PhD.38208 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/278387 dc.description.abstract The 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.sponsorship This work was supported by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1. dc.language.iso en dc.rights All rights reserved dc.subject Bayesian nonparametric statistics dc.subject Inverse problems dc.subject Adaptation dc.subject Uncertainty quantification dc.title Asymptotic theory for Bayesian nonparametric procedures in inverse problems dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.publisher.department Pure Mathematics and Mathematical Statistics dc.date.updated 2018-07-20T15:00:25Z dc.identifier.doi 10.17863/CAM.25733 dc.publisher.college St Catharine's College dc.type.qualificationtitle PhD in Mathematics cam.supervisor Nickl, Richard cam.thesis.funding true rioxxterms.freetoread.startdate 2018-07-20
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