Rates of convergence for Gibbs sampler and other Markov chains
This thesis considers the convergences of Markov chains. It is particularly concerned with the question of how long a given Markov chain must be run until it is close to its stationary distribution. Sharp answers to this question are obtained for a wide variety of Markov chains, including several different random walks on groups, and certain versions of the Data Augmentation and Gibbs Sampler algorithms as used in Bayesian Statistics. In the first part of the thesis, random walks on finite and compact Lie groups are considered. They are analyzed using harmonic analysis. In particular, the authors analyzes a process of random rotations on the orthogonal group. The Weyl Character Formula makes it possible to obtain useful formulas for the irreducible characters of the group, and these formulas are then used to get bounds on the variation distance to normalized Harr measure after k random rotations. The existence is proven of a [open quotes]cut-off phenomenon[close quotes] for this process. This is the first such result on a non-finite group. In addition, the author considers certain families of random walks on circle groups, and proves a fairly general theorem concerning their convergence to the uniform distribution. In the second part of the thesis, the author analyzes the Data Augmentation and Gibbs Sampler algorithms. By using ideas related to Harris Recurrence for Markov chains, it is possible to obtain convergence rates for these processes in various cases. In particular, the author obtains sharp results about the convergence rates when using Data Augmentation on finite sample spaces, and also when using Gibbs Sampling with Variance Component Models. The results give an indication of how long these iterative procedures need to be run until they converge. This is a question of great importance and interest to Statisticians. The author concludes with some directions for further research.
- Research Organization:
- Harvard Univ., Cambridge, MA (United States)
- OSTI ID:
- 7204356
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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