Bernoulli

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  • Volume 11, Number 5 (2005), 815-828.

On adaptive Markov chain Monte Carlo algorithms

Yves F. Atchadé and Jeffrey S. Rosenthal

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Abstract

We look at adaptive Markov chain Monte Carlo algorithms that generate stochastic processes based on sequences of transition kernels, where each transition kernel is allowed to depend on the history of the process. We show under certain conditions that the stochastic process generated is ergodic, with appropriate stationary distribution. We use this result to analyse an adaptive version of the random walk Metropolis algorithm where the scale parameter σ is sequentially adapted using a Robbins-Monro type algorithm in order to find the optimal scale parameter σopt. We close with a simulation example.

Article information

Source
Bernoulli, Volume 11, Number 5 (2005), 815-828.

Dates
First available in Project Euclid: 23 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.bj/1130077595

Digital Object Identifier
doi:10.3150/bj/1130077595

Mathematical Reviews number (MathSciNet)
MR2172842

Zentralblatt MATH identifier
1085.62097

Keywords
adaptive Markov chain Monte Carlo Metropolis algorithm mixingales parameter tuning Robbins-Monro algorithm

Citation

Atchadé, Yves F.; Rosenthal, Jeffrey S. On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 (2005), no. 5, 815--828. doi:10.3150/bj/1130077595. https://projecteuclid.org/euclid.bj/1130077595


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References

  • [1] Andrieu, C. and Moulines, E. (2003) Ergodicity of some adaptive Markov Chain Monte Carlo algorithms. Technical report.
  • [2] Andrieu, C., Moulines, E. and Priouret, P. (2002) Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim., To appear.
    Mathematical Reviews (MathSciNet): MR2177157
    Zentralblatt MATH: 1083.62073
    Digital Object Identifier: doi:10.1137/S0363012902417267
  • [3] Atchadé, Y.F. (2005) An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. MCMC Preprint.
  • [4] Chen, H. and Zhu, Y.-M. (1986) Stochastic approximation procedures with randomly varying truncations. Sci. Sinica Ser. A, 1, 914-926.
    Mathematical Reviews (MathSciNet): MR869196
  • [5] Davidson, J. and de Jong, R. (1997) Strong laws of large numbers for dependent heteregeneous processes: a synthesis of recent and new results. Econometric Rev., 16, 251-279.
    Mathematical Reviews (MathSciNet): MR1463083
    Zentralblatt MATH: 0903.60025
    Digital Object Identifier: doi:10.1080/07474939708800387
  • [6] Duflo, M. (1997) Random Iterative Models. Berlin: Springer-Verlag. Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (eds) (1996) Markov Chain Monte Carlo in Practice. London: Chapman & Hall.
    Mathematical Reviews (MathSciNet): MR1397966
  • [7] Gilks, W.R., Roberts, G.O. and Sahu, S.K. (1998) Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc., 93, 1045-1054.
    Mathematical Reviews (MathSciNet): MR1649199
    Zentralblatt MATH: 1064.65503
    Digital Object Identifier: doi:10.2307/2669848
    JSTOR: links.jstor.org
  • [8] Haario, H., Saksman, E. and Tamminen, J. (2001) An adaptive Metropolis algorithm. Bernoulli, 7, 223-242.
    Mathematical Reviews (MathSciNet): MR1828504
    Digital Object Identifier: doi:10.2307/3318737
    Project Euclid: euclid.bj/1080222083
  • [9] Hall, P. and Heyde, C.C. (1980) Martingale Limit Theory and Its Application. New York: Academic Press.
    Mathematical Reviews (MathSciNet): MR624435
    Zentralblatt MATH: 0462.60045
  • [10] Jarner, S.F. and Hansen, E. (2000) Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl., 85, 341-361.
    Mathematical Reviews (MathSciNet): MR1731030
    Zentralblatt MATH: 0997.60070
    Digital Object Identifier: doi:10.1016/S0304-4149(99)00082-4
  • [11] Kushner, K. and Yin, Y. (2003) Stochastic Approximation and Recursive Algorithms and Applications. New York: Springer-Verlag.
    Mathematical Reviews (MathSciNet): MR1993642
    Zentralblatt MATH: 1026.62084
  • [12] Liu, J.S. (2001) Monte Carlo Strategies in Scientific Computing. New York: Springer-Verlag.
    Mathematical Reviews (MathSciNet): MR1842342
  • [13] Meyn, S.P. and Tweedie, R.L. (1994) Computable bounds for convergence rates of Markov chains. Ann. Appl. Probab., 4, 981-1011.
    Mathematical Reviews (MathSciNet): MR1304770
    Zentralblatt MATH: 0812.60059
    Digital Object Identifier: doi:10.1214/aoap/1177004900
    Project Euclid: euclid.aoap/1177004900
  • [14] Robbins, H. and Monro, S. (1951) A stochastic approximation method. Ann. Math. Statist., 22, 400-407.
    Mathematical Reviews (MathSciNet): MR42668
    Zentralblatt MATH: 0054.05901
    Digital Object Identifier: doi:10.1214/aoms/1177729586
    Project Euclid: euclid.aoms/1177729586
  • [15] Roberts, G.O. and Rosenthal, J.S. (2001) Optimal scaling of various Metropolis-Hastings algorithms. Statist. Sci., 16, 351-367.
    Mathematical Reviews (MathSciNet): MR1888450
    Digital Object Identifier: doi:10.1214/ss/1015346320
    Project Euclid: euclid.ss/1015346320
  • [16] Roberts, G.O. and Gelman, A. and Gilks, W. (1997) Weak convergence and optimal scaling of random walk Metropolis algorithm. Ann. Applied Probab., 7, 110-120.
    Mathematical Reviews (MathSciNet): MR1428751
    Zentralblatt MATH: 0876.60015
    Digital Object Identifier: doi:10.1214/aoap/1034625254
    Project Euclid: euclid.aoap/1034625254
  • [17] Rosenthal, J.S. (2004) Adaptive MCMC Java applet. http://probability.ca/jeff/java/adapt.html.
    URL: Link to item

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