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HOW TO GENERATE UNIFORM SAMPLES ON DISCRETE SETS USING THE SPLITTING METHOD

Published online by Cambridge University Press:  23 April 2010

Peter W. Glynn ,
Andrey Dolgin ,
Reuven Y. Rubinstein  and
Radislav Vaisman
Peter W. Glynn
Affiliation:
Faculty of Industrial Engineering and Management Technion, Israel Institute of Technology, Haifa, Israel E-mail: ierrr01@ie.technion.ac.il; iew3.technion.ac.il:8080/ierrr01.phtml
Andrey Dolgin
Affiliation:
Faculty of Industrial Engineering and Management Technion, Israel Institute of Technology, Haifa, Israel E-mail: ierrr01@ie.technion.ac.il; iew3.technion.ac.il:8080/ierrr01.phtml
Reuven Y. Rubinstein
Affiliation:
Faculty of Industrial Engineering and Management Technion, Israel Institute of Technology, Haifa, Israel E-mail: ierrr01@ie.technion.ac.il; iew3.technion.ac.il:8080/ierrr01.phtml
Radislav Vaisman
Affiliation:
Faculty of Industrial Engineering and Management Technion, Israel Institute of Technology, Haifa, Israel E-mail: ierrr01@ie.technion.ac.il; iew3.technion.ac.il:8080/ierrr01.phtml
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Abstract

The goal of this work is twofold. We show the following:

  1. 1. In spite of the common consensus on the classic Markov chain Monte Carlo (MCMC) as a universal tool for generating samples on complex sets, it fails to generate points uniformly distributed on discrete ones, such as that defined by the constraints of integer programming. In fact, we will demonstrate empirically that not only does it fail to generate uniform points on the desired set, but typically it misses some of the points of the set.

  2. 2. The splitting, also called the cloning method – originally designed for combinatorial optimization and for counting on discrete sets and presenting a combination of MCMC, like the Gibbs sampler, with a specially designed splitting mechanism—can also be efficiently used for generating uniform samples on these sets. Without introducing the appropriate splitting mechanism, MCMC fails. Although we do not have a formal proof, we guess (conjecture) that the main reason that the classic MCMC is not working is that its resulting chain is not irreducible. We provide valid statistical tests supporting the uniformity of generated samples by the splitting method and present supportive numerical results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 3 , July 2010 , pp. 405 - 422
Copyright
Copyright © Cambridge University Press 2010

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