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Some contributions to the theory and methodology of Markov chain Monte Carlo

The general theme of this thesis is developing a better understanding of some Markov chain Monte Carlo methods. We review the literature in Chapters 1-4, including a short discussion of geometry in Markov chain Monte Carlo. In Chapter 5 we consider Langevin diffusions. First, a new class of these are derived in which the volatility is made position-dependent, using tools from stochastic analysis. Second, a complementary derivation is given, here using tools from Riemannian geometry. We hope that this work will help develop understanding of the geometric perspective among statisticians. Such derivations have been attempted previously, but solutions were not correct in general. We highlight these issues in detail. In the final part discussion is given on the use of these objects in Markov chain Monte Carlo. In Chapter 6 we consider a Metropolis-Hastings method with proposal kernel N(x,hV(x)), where x is the current state. After reviewing instances in the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of V(x) can change these compared to the Random Walk Metropolis case N(x,hS), for better or worse. In higher dimensions we show that judicious choice of V(x) can produce a geometrically converging chain when probability concentrates on an ever narrower ridge as |x| grows, something which is not true for the Random Walk Metropolis. In Chapter 7 we discuss stability of Hamiltonian Monte Carlo. For a fixed integration time we establish conditions for irreducibility and geometric ergodicity. Some results are confined to one dimension, and some further to a reference class of distributions. We find that target distributions with tails that are in between Exponential and Gaussian are needed for geometric ergodicity. Next we consider changing integration times, and show that here a geometrically ergodic chain can be constructed when tails are heavier than Exponential.
Date January 2016
CreatorsLivingstone, S. J.
PublisherUniversity College London (University of London)
Source SetsEthos UK
Detected LanguageEnglish
TypeElectronic Thesis or Dissertation

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