Skip to main content
SpringerLink
Log in

Micro-local analysis for the Metropolis algorithm

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove sharp rates of convergence to stationarity for a simple case of the Metropolis algorithm: the placement of a single disc of radius h randomly into the interval [ − 1 − h, 1 + h], with h > 0 small. We find good approximations for the top eigenvalues and eigenvectors. The analysis gives rigorous proof for the careful numerical work (in Exp. Math. 13, 207–213). The micro-local techniques employed offer promise for the analysis of more realistic problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allen M., Tildesly D.: Computer Simulation of Liquids. Oxford University Press, Oxford (1987)

    MATH  Google Scholar 

  2. Billera L., Diaconis P.: A geometric interpretation of the metropolis algorithm. Stat. Sci. 20, 1–5 (2001)

    MathSciNet  Google Scholar 

  3. Boutet de Monvel L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  4. Binder K., Heermann J.: Monte Carlo Simulation in Statistical Physics, 4th edn. Springer, Berlin (2002)

    MATH  Google Scholar 

  5. Burniston E., Siewert C.: Exact analytical solution of the transcendental equation \({\alpha\sin(\zeta)=\zeta}\) . SIAM J. Appl. Math. 24, 460–466 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  6. Diaconis, P., Lebeau, G., Michel, L.: Spectral analysis for the Metropolis algorithm in Euclidian space. preprint, Department of Mathematics, Université of Nice Sophia-Antipolis (2007)

  7. Diaconis P., Neuberger J.W.: Numerical results for the Metropolis algorithm. Exp. Math. 13, 207–213 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-Classical Limit. Lecture Note Series. Cambridge University Press, London (1999)

    Google Scholar 

  9. Diaconis P., Saloff-Coste L.: What do we know about the Metropolis algorithm. J. Comp. Syst. Sci. 57, 20–36 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hardy G.H.: On the zeros of the integral function x − sin(x). Mess. Math. 31, 161–165 (1902)

    Google Scholar 

  11. Hastings W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)

    Article  MATH  Google Scholar 

  12. Hammersley J., Handscomb D.: Monte Carlo Methods. Wiley, London (1964)

    MATH  Google Scholar 

  13. Hörmander. L.: The analysis of linear partial differential Operators. III. Grundl. Math. Wiss. Band 274. Pseudodifferential Operators. Springer, Berlin (1985)

  14. Jarner S., Hansen E.: Geometric ergodicity of Metropolis algorithms. Stoch. Proc. App. 85, 341–361 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jones B., Hobert J.: Honest exploration of intractable probability distributions via Markov chain monte carlo. Stat. Sci. 16, 317–334 (2001)

    Article  MathSciNet  Google Scholar 

  16. Jarner S., Yuen W.: Conductance bounds on the l 2 convergence rate of Metropolis algorithms on unbounded state spaces. Adv. Appl. Prob. 36, 243–366 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kienetz, J.: Convergence of Markov chains via analytic and isoperimetric inequalities. Ph.D diss. Univ. Bielefeld (2000)

  18. Kannan, R., Mahoney, M., Montenegro, R.: Rapid mixing of several Markov chains for a hard core model. Lectures Notes in Computer Science 2906, pp. 663–675. Springer, Heidelberg (2003)

  19. Liu J.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)

    MATH  Google Scholar 

  20. Lebeau, G., Michel, L.: Semi-classical analysis of a random walk on a manifold. preprint, Department of Mathematics, Université of Nice Sophia-Antipolis (2007)

  21. Martinez A.: An introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002)

    MATH  Google Scholar 

  22. Miclo, L., Roberto, C.: Trous spectraux pour certains algorithmes de metropolis sur \({\mathbb R}\). Lectures notes in mathematics 1729, pp. 336–352. Springer, Heidelberg (2000)

  23. Metropolis N., Rosenbluth A., Rosenbluth M., Teller A., Teller E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    Article  Google Scholar 

  24. Meyn S., Tweedie R.: Markov Chains and Stochastic Stability. Springer, New-York (1993)

    MATH  Google Scholar 

  25. Mengersen K., Tweedie R.: Rates of convergence of the hastings and Metropolis algorithms. Ann. Statist. 24, 101–121 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Peskun P.: Optimum monte carlo using Markov chains. Biometrika 60, 607–612 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  27. Roberts G., Rosenthal J.: Harris recurrence of Metropolis within Gibbs and trans-dimensional Markov chains. Ann. Appl. Prob. 16, 2123–2139 (2001)

    Article  MathSciNet  Google Scholar 

  28. Randall, D., Winkler, P.: Mixing points on an interval. 7th workshop on algorithms, engineering and experiments and the 2nd workshop on analytic algorithms and combinatorics. (2005)

  29. Saloff-Coste L.: Lectures on finite Markov chains. Lectures notes in math. 1665. Springer, New York (1997)

    Google Scholar 

  30. Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95. Soc. Math. France (1982)

  31. Sato, M., Kawai, T., Kashiwara, M.: Hyperfunctions and pseudodifferential equations. Lecture Notes in Maths 287, pp. 265–529

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Persi Diaconis & Gilles Lebeau

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Persi Diaconis

Authors
  1. Persi Diaconis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Lebeau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilles Lebeau.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diaconis, P., Lebeau, G. Micro-local analysis for the Metropolis algorithm. Math. Z. 262, 411–447 (2009). https://doi.org/10.1007/s00209-008-0383-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-008-0383-9

Keywords

Search

Navigation