Convergence of Markov chain Monte Carlo algorithms with applications to image restoration

Markov chain Monte Carlo algorithms, such as the Gibbs sampler and Metropolis-Hastings algorithm, are widely used in statistics, computer science, chemistry and physics for exploring complicated probability distributions. A critical issue for users of these algorithms is the determination of the number of iterations required so that the result will be approximately a sample from the distribution of interest. In this thesis, we give precise bounds on the convergence time of the Gibbs sampler used in the Bayesian restoration of a degraded image. We consider convergence as measured by both the usual choice of metric, total variation distance, and the Wasserstein metric. In both cases we exploit the coupling characterisation of the metric to get our results. Our results can also be applied to the coupling-from-the-past algorithm of Propp and Wilson (1996) to get bounds on its running time. The application of our theoretical results requires the computation of parameters of the algorithm. These computations may be prohibitively difficult in many situations. We discuss how our results can be applied in these situations through the use of auxiliary simulation to estimate these parameters. We also give a summary of probability metrics and the relationships between them, including several new relationships.

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