Scaling analysis of MCMC algorithms

• Main Author: Thiéry, Alexandre H.
• Published: University of Warwick 2013
• Subjects: 519.5; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.582418

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.