The Annals of Applied Probability

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

Eero Saksman and Matti Vihola

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Abstract

This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223–242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462–1505].

Article information

Source
Ann. Appl. Probab., Volume 20, Number 6 (2010), 2178-2203.

Dates
First available in Project Euclid: 19 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1287494558

Digital Object Identifier
doi:10.1214/10-AAP682

Mathematical Reviews number (MathSciNet)
MR2759732

Zentralblatt MATH identifier
1209.65004

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 65C40: Computational Markov chains 60J27: Continuous-time Markov processes on discrete state spaces 93E15: Stochastic stability 93E35: Stochastic learning and adaptive control

Keywords
Adaptive Markov chain Monte Carlo convergence ergodicity Metropolis algorithm stochastic approximation

Citation

Saksman, Eero; Vihola, Matti. On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20 (2010), no. 6, 2178--2203. doi:10.1214/10-AAP682. https://projecteuclid.org/euclid.aoap/1287494558


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References

  • [1] Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 1462–1505.
  • [2] Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283–312 (electronic).
  • [3] Andrieu, C. AND Robert, C. P. (2001). Controlled MCMC for optimal sampling. Technical Report Ceremade 0125, Univ. Paris Dauphine.
  • [4] Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Statist. Comput. 18 343–373.
  • [5] Atchadé, Y. F. and Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 815–828.
  • [6] Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700–738.
  • [7] Birnbaum, Z. W. and Marshall, A. W. (1961). Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32 687–703.
  • [8] Haario, H., Laine, M., Lehtinen, M., Saksman, E. and Tamminen, J. (2004). Markov chain Monte Carlo methods for high dimensional inversion in remote sensing. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 591–607.
  • [9] Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive metropolis algorithm. Bernoulli 7 223–242.
  • [10] Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341–361.
  • [11] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics 21 1087–1092.
  • [12] Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981–1011.
  • [13] Roberts, G. O. and Rosenthal, J. S. (2007). Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44 458–475.
  • [14] Roberts, G. O. and Rosenthal, J. S. (2009). Examples of adaptive MCMC. J. Comput. Graph. Statist. 18 349–367.
  • [15] Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110.
  • [16] Saksman, E. and Vihola, M. (2010). On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Preprint. Available at arXiv:0806.2933v4.