Thiéry, Alexandre H. (2013) Scaling analysis of MCMC algorithms. PhD thesis, University of Warwick.

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Abstract

Markov Chain Monte Carlo (MCMC) methods have become a workhorse
for modern scientific computations. Practitioners utilize MCMC in many different
areas of applied science yet very few rigorous results are available for justifying
the use of these methods. The purpose of this dissertation is to analyse random
walk type MCMC algorithms in several limiting regimes that frequently occur in
applications. Scaling limits arguments are used as a unifying method for studying
the asymptotic complexity of these MCMC algorithms. Two distinct strands of
research are developed: (a) We analyse and prove diffusion limit results for MCMC
algorithms in high or infinite dimensional state spaces. Contrarily to previous results
in the literature, the target distributions that we consider do not have a product
structure; this leads to Stochastic Partial Differential Equation (SPDE) limits. This
proves among other things that optimal proposals results already known for product
form target distributions extend to much more general settings. We then show
how to use these MCMC algorithms in an infinite dimensional Hilbert space in
order to imitate a gradient descent without computing any derivative. (b) We
analyse the behaviour of the Random Walk Metropolis (RWM) algorithm when
used to explore target distributions concentrating on the neighbourhood of a low
dimensional manifold of Rn. We prove that the algorithm behaves, after being
suitably rescaled, as a diffusion process evolving on a manifold.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Monte Carlo method, Markov processes, Algorithms
Official Date:	February 2013
Dates:	Date Event February 2013 Submitted
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Roberts, Gareth; Stuart, A. M.
Extent:	vii, 135 leaves.
Language:	eng

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