An Adaptive Version for the Metropolis Adjusted Langevin Algorithm with a Truncated Drift

  • Yves F. AtchadéEmail author


This paper extends some adaptive schemes that have been developed for the Random Walk Metropolis algorithm to more general versions of the Metropolis-Hastings (MH) algorithm, particularly to the Metropolis Adjusted Langevin algorithm of Roberts and Tweedie (1996). Our simulations show that the adaptation drastically improves the performance of such MH algorithms. We study the convergence of the algorithm. Our proves are based on a new approach to the analysis of stochastic approximation algorithms based on mixingales theory.


Adaptive Markov Chain Monte Carlo Langevin algorithms Metropolis-Hastings algorithms Stochastic approximation algorithms 

AMS 2000 Subject Classification

65C05 65C40 60J27 60J35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. C. Andrieu, and Y. F. Atchade, “On the efficiency of adaptive MCMC algorithms,” Technical Report 1, 2005.Google Scholar
  2. C. Andrieu, and E. Moulines, “On the ergodicity properties of some adaptive MCMC algorithms,” to appear Annals of Applied Probability, 2005.Google Scholar
  3. Y. F. Atchade, and J. S. Rosenthal, “On adaptive Markov chain Monte Carlo algorithm,” Bernoulli vol. 11 pp. 815–828, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  4. P. H. Baxendale, “Renewal theory and computable convergence rates for geometrically ergodic Markov chains,” Annals of Applied Probability vol. 15 pp. 700–738, 2005.zbMATHMathSciNetCrossRefGoogle Scholar
  5. A. Benveniste, M. Métivier, and P. Priouret, “Adaptive algorithms and stochastic approximations,” In Applications of Mathematics, Springer: Paris-New York, 1990.Google Scholar
  6. L. Breyer, M. Piccioni, and S. Scarlatti, “Optimal scaling of MALA for nonlinear regression,” Technical Report, 2002.Google Scholar
  7. J. Davidson, and R. de Jong, “Strong laws of large numbers for dependent heteregeneous processes: a synthesis of recent and new results,” Econometric Reviews vol. 16 pp. 251–279, 1997.zbMATHMathSciNetGoogle Scholar
  8. W. R. Gilks, G. O. Roberts, and S. K. Sahu, “Adaptive Markov chain Monte Carlo through regeneration,” Journal of the American Statistical Association vol. 93 pp. 1045–1054, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  9. H. Haario, E. Saksman, and J. Tamminen, “An adaptive metropolis algorithm,” Bernoulli vol. 7 pp. 223–242, 2001.zbMATHMathSciNetGoogle Scholar
  10. P. Hall, and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press: New York, 1980.zbMATHGoogle Scholar
  11. S. F. Jarner, and E. Hansen, “Geometric ergodicity of Metropolis algorithms,” Stochastic Processes and their Applications vol. 85 pp. 341–361, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  12. K. Kushner, and Y. Yin, Stochastic Approximation and Recursive Algorithms and Applications, Springer-Verlag: New-York, 2003.zbMATHGoogle Scholar
  13. M. Metivier, and P. Priouret, “Application of Kushner and Clark lemma to general classes of stochastic algorithms,” IEEE-IT vol. 30, 1984.Google Scholar
  14. M. Pelletier, “On the almost sure asymptotic behaviour of stochastic algorithms,” Stochastic Processes and their Applications vol. 78 pp. 217–244, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  15. C. P. Robert, and G. Casella, Monte Carlo Statistical Methods, Springer-Verlag, New York, 2004.zbMATHGoogle Scholar
  16. G. O. Roberts, and J. S. Rosenthal, “Optimal scaling of various Metropolis-Hastings algorithms,” Statistical Science vol. 16, 2001.Google Scholar
  17. G. Roberts, and R. Tweedie, “Exponential convergence of Langevin distributions and their discrete approximations,” Bernoulli vol. 2 pp. 341–363, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  18. J. S. Rosenthal, and G. O. Roberts, “Coupling and Ergodicity of adaptive MCMC,” Technical Report, MCMC preprints, 2005.Google Scholar
  19. O. Stramer, and R. L. Tweedie, “Langevin-type models (ii): Self-targeting candidates for MCMC algorithms,” Methodology and Computing in Applied Probability vol. 1 pp. 307–328, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  20. L. Tierney, “Markov chains for exploring posterior distributions,” The Annals of Statistics vol. 22 pp. 1701–1762, 1994. With discussion and a rejoinder by the author.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

Personalised recommendations