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Exponential convergence of adaptive importance sampling for Markov chains

Part of:Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Keith Baggerly,
Dennis Cox and
Rick Picard
Keith Baggerly*
Affiliation:
Rice University
Dennis Cox*
Affiliation:
National Center for Atmospheric Research
Rick Picard*
Affiliation:
Los Alamos National Laboratory
*
∗ Postal address: Department of Statistics, Rice University, 610 South Main St., Houston, TX 77005, USA
∗∗ Postal address: Geophysics Statistics project, National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307-3000, USA. Email address: dcox@cgd.ucar.edu
∗∗∗ Postal address: Statistical Sciences Group, MS F600, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Abstract

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.

Keywords

Adaptive proceduresbiased random walkslearning algorithmsMonte Carlo simulationparticle transportzero-variance solution

MSC classification

Primary:65C05: Monte Carlo methods

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000, pp. 342 - 358
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

This research was supported as part of an ongoing effort to improveMonte Carlo Methods by the Los AlamosNationalLaboratory Directed Research and Development Program.

References

Apostol, T. A. (1957). Mathematical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Baggerly, K., Cox, D., and Picard, R. (1998). Adaptive importance sampling for random walks on continuous state spaces. Los Alamos National Laboratory Tech. Rep. LA-UR-98-1214. Available at http://www-xdiv.lanl.gov/XTM/projects/mc21/docs/ as publication 25.Google Scholar
Booth, T. E. (1985). Exponential convergence for Monte Carlo particle transport? Trans. Amer. Nucl. Soc. 50, 267–268.Google Scholar
Booth, T. E. (1989). Zero-variance solutions for linear Monte Carlo. Nucl. Sci. Eng. 102, 332–340.CrossRefGoogle Scholar
Booth, T. E. (1997). Exponential convergence on a continuous Monte Carlo transport problem. Nucl. Sci. Eng. 127, 338–345.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Glasserman, P. (1993). Stochastic monotonicity and conditional Monte Carlo for likelihood ratios. Adv. Appl. Prob. 25, 103–115.Google Scholar
Kollman, C. (1993). Rare event simulation in radiation transport. Unpublished Ph.D. thesis, Stanford University.Google Scholar
Kollman, C., Baggerly, K., Cox, D., and Picard, R. (1999). Adaptive importance sampling on discrete Markov chains. Ann. Appl. Prob. 9, 391–412.Google Scholar
Lux, I., and Koblinger, L. (1991). Monte Carlo Particle Transport Methods: Neutron and Photon Calculations. CRC Press, Boca Raton.Google Scholar
Parzen, E. (1954). On uniform convergence of families of sequences of random variables. University of California Publications in Statistics 2, 23–54.Google Scholar
Revuz, D. (1975). Markov Chains. North-Holland, Amsterdam.Google Scholar
Rubinstein, R. Y. (1981). Simulation and the Monte Carlo Method. Wiley, New York.Google Scholar