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Aspects of Bayesian inverse problems
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Agapiou, Sergios (2013) Aspects of Bayesian inverse problems. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2716991~S1
Abstract
The goal of this thesis is to contribute to the formulation and understanding
of the Bayesian approach to inverse problems in function space. To this end
we examine two important aspects of this approach: the frequentist asymptotic
properties of the posterior, and the extraction of information from the posterior
via sampling. We work in a separable Hilbert space setting and consider Gaussian
priors on the unknown in conjugate Gaussian models. In the first part of this
work we consider linear inverse problems with Gaussian additive noise and study
the contraction in the small noise limit of the Gaussian posterior distribution to
a Dirac measure centered on the true parameter underlying the data. In a wide
range of situations, which include both mildly and severely ill-posed problems, we
show how carefully calibrating the scaling of the prior as a function of the size of
the noise, based on a priori known information on the regularity of the truth, yields
optimal rates of contraction. In the second part we study the implementation in
RN of hierarchical Bayesian linear inverse problems with Gaussian noise and priors,
and with hyper-parameters introduced through the scalings of the prior and noise
covariance operators. We use function space intuition to understand the large N
behaviour of algorithms designed to sample the posterior and show that the two
scaling hyper-parameters evolve under these algorithms in contrasting ways: as N
grows the prior scaling slows down while the noise scaling speeds up. We propose
a reparametrization of the prior scaling which is robust with respect to the increase
in dimension. Our theory on the slowing down of the evolution of the prior scaling
extends to hierarchical approaches in more general conjugate Gaussian settings,
while our intuition covers other parameters of the prior covariance operator as well.
Throughout the thesis we use a blend of results from measure theory and probability
theory with tools from the theory of linear partial differential equations and
numerical analysis.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Inverse problems (Differential equations), Bayesian statistical decision theory | ||||
Official Date: | September 2013 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Stuart, Andrew; O'Connell, Neil | ||||
Extent: | vii, 158 leaves. | ||||
Language: | eng |
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