Convergence Rate Analysis of Markov Chains

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dc.contributor.advisorMadras, Neal
dc.creatorJovanovski, Oliver
dc.date.accessioned2015-01-26T14:21:25Z
dc.date.available2015-01-26T14:21:25Z
dc.date.copyright2014-05-22
dc.date.issued2015-01-26
dc.date.updated2015-01-26T14:21:24Z
dc.degree.disciplineMathematics & Statistics
dc.degree.levelDoctoral
dc.degree.namePhD - Doctor of Philosophy
dc.description.abstractWe consider a number of Markov chains and derive bounds for the rate at which convergence to equilibrium occurs. For our main problem, we establish results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. The Bayesian hierarchical method involves statistical models that incorporate prior beliefs about the likelihood of observed data to arrive at posterior interpretations, and appears in applications for information technology, statistical genetics, market research and others. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate. In addition, we also consider a stochastic image restoration model proposed by A. Gibbs, and give an upper bound on the time it takes for a Markov chain defined by this model to be arbitrarily close in total variation to equilibrium. We use Gibbs' result for convergence in the Wasserstein metric to arrive at our result. Our bound for the time to equilibrium is of similar order to that of Gibbs.
dc.identifier.urihttp://hdl.handle.net/10315/28188
dc.language.isoen
dc.rightsAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
dc.subjectMathematics
dc.subject.keywordsBayesianen_US
dc.subject.keywordsProbabilityen_US
dc.subject.keywordsConvergenceen_US
dc.subject.keywordsEquilibriumen_US
dc.subject.keywordsMarkov chainsen_US
dc.subject.keywordsMCMCen_US
dc.subject.keywordsGibbs sampleren_US
dc.subject.keywordsHierarchicalen_US
dc.titleConvergence Rate Analysis of Markov Chains
dc.typeElectronic Thesis or Dissertation

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