Annals of Applied Probability

Importance sampling for families of distributions

Neal Madras and Mauro Piccioni

Full-text: Open access

Abstract

This paper analyzes the performance of importance sampling distributions for computing expectations with respect to a whole family of probability laws in the context of Markov chain Monte Carlo simulation methods. Motivations for such a study arise in statistics as well as in statistical physics. Two choices of importance sampling distributions are considered in detail: mixtures of the distributions of interest and distributions that are "uniform over energy levels" (motivated by physical applications). We analyze two examples, a "witch's hat" distribution and the mean field Ising model, to illustrate the advantages that such simulation procedures are expected to offer in a greater generality. The connection with the recently proposed simulated tempering method is also examined.

Article information

Source
Ann. Appl. Probab., Volume 9, Number 4 (1999), 1202-1225.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1029962870

Digital Object Identifier
doi:10.1214/aoap/1029962870

Mathematical Reviews number (MathSciNet)
MR1728560

Zentralblatt MATH identifier
0966.60061

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 65C05: Monte Carlo methods 82B80: Numerical methods (Monte Carlo, series resummation, etc.) [See also 65-XX, 81T80]

Keywords
Markov chain Monte Carlo importance sampling simulated tempering Metropolis algorithm spectral gap Ising model

Citation

Madras, Neal; Piccioni, Mauro. Importance sampling for families of distributions. Ann. Appl. Probab. 9 (1999), no. 4, 1202--1225. doi:10.1214/aoap/1029962870. https://projecteuclid.org/euclid.aoap/1029962870


Export citation

References

  • Arnold, S. F. (1993). Gibbs Sampling. In Handbook of Statistics (C. R. Rao ed.) 10 North-Holland, Amsterdam.
  • Bennett, C. H. (1976). Efficient estimation of free energy differences from Monte Carlo data. J. Comput. Phy s. 22 245-268.
  • Berg, B. A. and Neuhaus, T. (1991). Multicanonical algorithms for first order phase transitions. Phy s. Lett. B 267 249-253.
  • Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. J. Roy. Statist. Soc. Ser. B 55 25-37.
  • Binder, K. and Heermann, D. W. (1992). Monte Carlo Simulation in Statistical physics, 2nd ed. Springer, New York.
  • Bucklew, J. A. (1990). Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York.
  • Caracciolo, S., Pelissetto, A. and Sokal, A. D. (1990). Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints. J. Statist. Phy s. 60 1-53.
  • Chan, K. S. and Gey er, C. J. (1994). Discussion of "Markov chains for exploring posterior distributions" by L. Tierney. Ann. Statist. 22 1747-1758.
  • Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750.
  • Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36-61.
  • Duflo, M. (1996). Algorithmes Stochastiques. Springer, Berlin.
  • Ellis, R. (1985). Entropy, Large Deviations and Statistical Mechanics. Springer, New York.
  • Evans, M. and Swartz, T. (1995). Methods for approximating integrals in statistics with special emphasis on Bayesian integration problems. Statist. Sci. 10 254-272.
  • Ferguson, T. S. (1996). A Course in Large Sample Theory. Chapman and Hall, London.
  • Ferrenberg, A. M. and Swendsen, R. H. (1988). New Monte Carlo technique for studying phase transitions. Phy s. Rev. Lett. 61 2635-2638.
  • Gamerman, D. (1997). Monte Carlo Markov Chains: Stochastic Simulation for Bayesian Inference. Chapman and Hall, London.
  • Gey er, C. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics: Proceedings of the 23rd Sy mposium on the Interface (E. M. Keramidas, ed.) 156-163. Interface Foundation, Fairfax Station.
  • Gey er, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Amer. Statist. Assoc. 90 909-920.
  • Gilks, W. R., Richardson, S. and Spigelhalter, D. J. (eds.) (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall, London.
  • Gross, L. (1979). Decay of correlations in classical lattice models at high temperature. Comm. Math. Phy s. 68 9-27.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109.
  • Janse van Rensburg, E. J. and Madras, N. (1997). Monte Carlo study of the -point for collapsing trees. J. Statist. Phy s. 86 1-36.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phy s. 104 1-19.
  • Liu, J., Wong, W. H. and Kong, A. (1994). A covariance structure of the Gibbs sampler with applications to the comparison of estimators and augmentation schemes. Biometrika 81 27-40.
  • Madras, N. and Randall, D. (1999). Markov chain decomposition for convergence rate analysis. Preprint.
  • Marinari, E. and Parisi, G. (1992). Simulated tempering: a new Monte Carlo scheme. Europhy s. Lett. 19 451-458.
  • Meng, X. L. and Wong, W. H. (1996). Simulating ratios of normalizing constants via a simple identity: a theoretical explanation. Statist. Sinica 6 831-860.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller E. (1953). Equations of state calculation by fast computing machines. J. Chem. Phy s. 21 1087- 1092.
  • Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Neal, R. (1996). Sampling from multimodal distributions using tempered transitions. Statist. Comput. 6 353-366.
  • Peskun, P. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60 607- 612.
  • Robert, C. (1996). Methodes de Monte Carlo par cha ines de Markov. Economica, Paris.
  • Sinclair, A. (1993). Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkh¨auser, Boston.
  • Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B 55 3-23.
  • Sokal, A. D. (1989). Monte Carlo methods in statistical mechanics: Foundations and new algorithms. Lecture notes: Cours de Troisi eme Cy cle de la physique en Suisse Romande
  • (Lausanne, June 1989). Unpublished manuscript
  • Swendsen, R. H. and Ferrenberg, A. M. (1990). Histogram methods for Monte Carlo data analysis. In Computer Studies in Condensed Matter physics II (D. P. Landau, K. K. Man and H. B. Sch ¨uttler, eds.) 179-183. Springer, Berlin.
  • Tanner, M. A. (1993). Tools for Statistical Inference. Springer, New York.
  • Thompson, C. J. (1972). Mathematical Statistical Mechanics. Macmillan, New York.
  • Tierney, L. (1998). A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8 1-9.
  • Torrie, G. M. and Valleau, J. P. (1977). Nonphysical sampling distributions in Monte Carlo free energy estimation: Umbrella sampling. J. Comput. Phy s. 23 187-199.
  • Trotter, H. F. and Tukey, J. W. (1956). Conditional Monte Carlo for normal samples. In Sy mposium on Monte Carlo Methods (H. A. Meyer, ed.) 64-79. Wiley, New York.
  • Valleau, J. P. (1991). Density-scaling: a new Monte Carlo technique in statistical mechanics. J. Comput. Phy s. 96 193-216.