The Annals of Applied Probability

Weak convergence of Metropolis algorithms for non-i.i.d. target distributions

Mylène Bédard

Full-text: Open access


In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234.

Article information

Ann. Appl. Probab., Volume 17, Number 4 (2007), 1222-1244.

First available in Project Euclid: 10 August 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 65C40: Computational Markov chains

Metropolis algorithm weak convergence optimal scaling diffusion Markov chain Monte Carlo


Bédard, Mylène. Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Probab. 17 (2007), no. 4, 1222--1244. doi:10.1214/105051607000000096.

Export citation


  • Bédard, M. (2006). Optimal acceptance rates for metropolis algorithms: Moving beyond 0.234. Technical report, Univ. Toronto. Available at
  • Bédard, M. (2006). Efficient sampling using metropolis algorithms: Applications of optimal scaling results. Technical report, Univ. Toronto. Available at
  • Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 25–38.
  • Besag, J., Green, P. J., Higdon, D. and Mergensen, K. (1995). Bayesian computation and stochastic systems. Statist. Sci. 10 3–66.
  • Breyer, L. A., Piccioni, M. and Scarlatti, S. (2002). Optimal scaling of MALA for nonlinear regression. Ann. Appl. Probab. 14 1479–1505.
  • Breyer, L. A. and Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Process. Appl. 90 181–206.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika. 57 97–109.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21 1087–1092.
  • Neal, P. and Roberts, G. O. (2006). Optimal scaling for partially updating MCMC algorithms. Ann. Appl. Probab. 16 475–515.
  • Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk metropolis algorithms. Ann. Appl. Probab. 7 110–120.
  • Roberts, G. O. and Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 255–68.
  • Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci. 16 351–367.
  • Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surveys 1 20–71.
  • Rosenthal, J. S. (2000). A First Look at Rigorous Probability Theory. World Scientific, Singapore.