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dc.contributor.advisor Schmidler, Scott en_US
dc.contributor.author Woodard, Dawn Banister en_US
dc.date.accessioned 2008-01-02T16:33:41Z
dc.date.available 2008-01-02T16:33:41Z
dc.date.issued 2007-09-14 en_US
dc.identifier.uri http://hdl.handle.net/10161/453
dc.description Dissertation en_US
dc.description.abstract Stochastic sampling methods are ubiquitous in statistical mechanics, Bayesian statistics, and theoretical computer science. However, when the distribution that is being sampled is multimodal, many of these techniques converge slowly, so that a great deal of computing time is necessary to obtain reliable answers. Parallel and simulated tempering are sampling methods that are designed to converge quickly even for multimodal distributions. In this thesis, we assess the extent to which this goal is acheived.We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, meaning that it converges quickly. These conditions are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models in R^M as M increases, and for the mean-field Ising model.We also obtain upper bounds on the convergence rates of parallel and simulated tempering, yielding a single set of sufficient conditions for torpid mixing of both techniques. These conditions imply torpid mixing of parallel and simulated tempering on a normal mixture model with unequal covariances in $\R^M$ as $M$ increases and on the mean-field Potts model with $q \geq 3$, regardless of the number and choice of temperatures, as well as on the mean-field Ising model if an insufficient (fixed) set of temperatures is used. The latter result is in contrast to the rapid mixing of parallel and simulated tempering on the mean-field Ising model with a linearly increasing set of temperatures. en_US
dc.format.extent 709909 bytes
dc.format.mimetype application/pdf
dc.language.iso en_US
dc.subject Mathematics en_US
dc.subject Statistics en_US
dc.subject Computer Science en_US
dc.subject tempering en_US
dc.subject rapid mixing en_US
dc.subject Markov chain en_US
dc.subject Metropolis en_US
dc.subject multimodal en_US
dc.subject convergence en_US
dc.title Conditions for Rapid and Torpid Mixing of Parallel and Simulated Tempering on Multimodal Distributions en_US
dc.type Dissertation en_US
dc.department Statistics and Decision Sciences en_US

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