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May, 1994 On the Rate of Convergence of the Metropolis Algorithm and Gibbs Sampler by Geometric Bounds
Salvatore Ingrassia
Ann. Appl. Probab. 4(2): 347-389 (May, 1994). DOI: 10.1214/aoap/1177005064

Abstract

In this paper we obtain bounds on the spectral gap of the transition probability matrix of Markov chains associated with the Metropolis algorithm and with the Gibbs sampler. In both cases we prove that, for small values of $T,$ the spectral gap is equal to $1 - \lambda_2,$ where $\lambda_2$ is the second largest eigenvalue of $P$. In the case of the Metropolis algorithm we give also two examples in which the spectral gap is equal to $1 - \lambda_{\min}$, where $\lambda_{\min}$ is the smallest eigenvalue of $P$. Furthermore we prove that random updating dynamics on sites based on the Metropolis algorithm and on the Gibbs sampler have the same rate of convergence at low temperatures. The obtained bounds are discussed and compared with those obtained with a different approach.

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Salvatore Ingrassia. "On the Rate of Convergence of the Metropolis Algorithm and Gibbs Sampler by Geometric Bounds." Ann. Appl. Probab. 4 (2) 347 - 389, May, 1994. https://doi.org/10.1214/aoap/1177005064

Information

Published: May, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0802.60061
MathSciNet: MR1272731
Digital Object Identifier: 10.1214/aoap/1177005064

Subjects:
Primary: 60J10
Secondary: 15A42 , 60J15

Keywords: Gibbs sampler , Markov chains , Metropolis algorithm , rate of convergence

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 2 • May, 1994
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