Article

Analysis of Convergence Rates of Some Gibbs Samplers on Continuous State Spaces

Stochastic Processes and their Applications (Impact Factor: 1.05). 08/2011; DOI: 10.1016/j.spa.2013.05.003
Source: arXiv

ABSTRACT We use a non-Markovian coupling and small modi?cations of techniques from the
theory of ?nite Markov chains to analyze some Markov chains on continuous state
spaces. The ?rst is a Gibbs sampler on narrow contingency tables, the second a
gen- eralization of a sampler introduced by Randall and Winkler.

0 Bookmarks
 · 
78 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: It is shown that the “hit-and-run” algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O *(n 2 R 2/r 2), where R and r are the radii of the inscribed and circumscribed balls of K. Thus after appropriate preprocessing, hit-and-run produces an approximately uniformly distributed sample point in time O *(n 3), which matches the best known bound for other sampling algorithms. We show that the bound is best possible in terms of R,r and n.
    Mathematical Programming 11/1999; 86(3):443-461. · 1.98 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The main technique used in algorithm design for approximating heart of the method is the study of the convergence (mixing) rates of particular Markov chains of interest. In this paper we illustrate a new approach to the coupling technique, which we call path coupling, for bounding mixing rates. Previous applications of coupling have required detailed insights into the combinatorics of the problem at hand, and this complexity can make the technique extremely difficult to apply successfully. Path coupling helps to minimize the combinatorial difficulty and in all cases provides simpler convergence proofs than does the standard coupling method. However the true power of the method is that the simplification obtained may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method. We apply the path coupling method to several hard combinatorial problems, obtaining new or improved results. We examine combinatorial problems such as graph colouring and TWICE-SAT, and problems from statistical physics, such as the antiferromagnetic Potts model and the hard-core lattice gas model. In each case we provide either a proof of rapid mixing where none was known previously, or substantial simplification of existing proofs with consequent gains in the performance of the resulting algorithms
    Foundations of Computer Science, 1997. Proceedings., 38th Annual Symposium on; 11/1997
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This is an expository paper, focussing on the following scenario. We have two Markov chains, $\mathcal {M}$ and $\mathcal {M}'$. By some means, we have obtained a bound on the mixing time of $\mathcal {M}'$. We wish to compare $\mathcal {M}$ with $\mathcal {M}'$ in order to derive a corresponding bound on the mixing time of $\mathcal {M}$. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.
    Probability Surveys 11/2004;

Preview

Download
0 Downloads
Available from