On the efficiency of adaptive MCMC algorithms

Christophe Andrieu (University of Bristol)
Yves Atchade (University of Michigan)

Abstract


We study a class of adaptive Markov Chain Monte Carlo (MCMC) processes which aim at behaving as an ``optimal'' target process via a learning procedure. We show, under appropriate conditions, that the adaptive MCMC chain and the ``optimal'' (nonadaptive) MCMC process share many asymptotic properties. The special case of adaptive MCMC algorithms governed by stochastic approximation is considered in details and we apply our results to the adaptive Metropolis algorithm of [Haario, Saksman, Tamminen].

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Pages: 336-349

Publication Date: October 12, 2007

DOI: 10.1214/ECP.v12-1320

References

  1. C. Andrieu. Discussion, ordinary meeting on inverse problems, wednesday 10th december, 2003, london. Journal of the Royal Statistical Society B 66 (2004), 627-652.
  2. Andrieu, Christophe; Moulines, Éric. On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 (2006), no. 3, 1462--1505. MR2260070
  3. Andrieu, Christophe; Moulines, Éric; Priouret, Pierre. Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 (2005), no. 1, 283--312 (electronic). MR2177157 (2006f:62074)
  4. C. Andrieu and C. P. Robert. Controlled mcmc for optimal sampling. Technical Report Universit'e Paris Dauphine, Ceremade 0125. (2001).
  5. Atchadé, Yves F.; Rosenthal, Jeffrey S. On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 (2005), no. 5, 815--828. MR2172842 (2006h:62077)
  6. Baxendale, Peter H. Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 (2005), no. 1B, 700--738. MR2114987 (2005m:60164)
  7. Benveniste, Albert; Métivier, Michel; Priouret, Pierre. Adaptive algorithms and stochastic approximations.Translated from the French by Stephen S. Wilson.Applications of Mathematics (New York), 22. Springer-Verlag, Berlin, 1990. xii+365 pp. ISBN: 3-540-52894-6 MR1082341 (92h:62137)
  8. Markov chain Monte Carlo in practice.Edited by W. R. Gilks, S. Richardson and D. J. Spiegelhalter.Interdisciplinary Statistics. Chapman & Hall, London, 1996. xviii+486 pp. ISBN: 0-412-05551-1 MR1397966 (97d:62006)
  9. Gilks, Walter R.; Roberts, Gareth O.; Sahu, Sujit K. Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc. 93 (1998), no. 443, 1045--1054. MR1649199 (2000f:62063)
  10. Goldstein, Sheldon. Maximal coupling. Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 2, 193--204. MR0516740 (80k:60041)
  11. Haario, Heikki; Saksman, Eero; Tamminen, Johanna. An adaptive metropolis algorithm. Bernoulli 7 (2001), no. 2, 223--242. MR1828504 (2002c:65008)
  12. H. Kushner and G. Yin. Stochastic approximation and recursive algorithms and Applications. Springer, Springer-Verlag, New-York (2003).
  13. J.S. Rosenthal and G.O. Roberts. Coupling and ergodicity of adaptive mcmc.
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