Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T09:15:55.609Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Asmussen, S. & Glynn, P.W. (2007). Stochastic simulation: Algorithms and analyses. New York: Springer.CrossRefGoogle Scholar
2.Botev, Z.I. & Kroese, D.P. (2008). An efficient algorithm for rare-event probability estimation, combinatorial optimization, and counting. Methodology and Computing in Applied Probability 10: 471–505.CrossRefGoogle Scholar
3.Garvels, M.J.J. (2000). The splitting method in rare-event simulation. Ph.D. thesis, University of Twente.Google Scholar
4.Garvels, M.J.J. & Rubinstein, R.Y. (2000). A combined splitting–cross entropy method for rare event probability estimation of single queues and ATM networks. Unpublished manuscriptGoogle Scholar
5.Lagnoux-Renaudie, A. (2009). A two-steps branching splitting model under cost constraint to. Journal of Applied Probability 46: 429–452.CrossRefGoogle Scholar
6.Melas, V.B. (1997). On the efficiency of the splitting and roulette approach for sensitivity analysis. Winter Simulation Conference, Atlanta, GA, pp. 269–274.CrossRefGoogle Scholar
7.Gryazina, E. & Polyak, B. (2010). Randomized methods based on new Monte Carlo schemes for control and optimizations. Annals of Operations Research.Google Scholar
8.L'Ecuyer, P., Demers, V. & Tuffin, B. (2007). Rare-events, cloning, and quasi-Monte Carlo. ACM Transactions on Modeling and Computer Simulation, 17(2).CrossRefGoogle Scholar
9.Ross, S.M. (2006). Simulation. New York: Wiley.Google Scholar
10.Rubinstein, R.Y. (1999). The cross-entropy method for combinatorial and continuous optimization. Methodology and Computing in Applied Probability 1: 127–190.CrossRefGoogle Scholar
11.Rubinstein, R.Y. (2008). The Gibbs cloner for combinatorial optimization, counting and sampling. Methodology and Computing in Applied Probability 11: 491–549.CrossRefGoogle Scholar
12.Rubinstein, R.Y. (2010). Randomized algorithms with splitting: Why the classic randomized algorithms do not work and how to make them work. Methodology and Computing in Applied Probability 12: 1–41.CrossRefGoogle Scholar
13.Rubinstein, R.Y. & Kroese, D.P. (2004). The cross-entropy method: A unified approach to combinatorial optimization, Monte-Carlo simulation and machine learning. New York: Springer.CrossRefGoogle Scholar
14.Rubinstein, R.Y. & Kroese, D.P. (2007). Simulation and the Monte Carlo method; 2nd ed.New York: Wiley.CrossRefGoogle Scholar
15.Smith, R.L. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Operations Research, 32: 1296–1308.CrossRefGoogle Scholar