Bernoulli

  • Bernoulli
  • Volume 18, Number 3 (2012), 975-1001.

Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II

Yves F. Atchadé and Gersende Fort

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Abstract

We prove a central limit theorem for a general class of adaptive Markov Chain Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We discuss in detail the special case of stochastic approximation. We use the result to analyze the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin algorithm with a heavy tailed target density.

Article information

Source
Bernoulli, Volume 18, Number 3 (2012), 975-1001.

Dates
First available in Project Euclid: 28 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1340887010

Digital Object Identifier
doi:10.3150/11-BEJ360

Mathematical Reviews number (MathSciNet)
MR2948909

Zentralblatt MATH identifier
1244.60072

Keywords
adaptive Markov chain Monte Carlo Markov chain Metropolis adjusted Langevin algorithms stochastic approximations subgeometric ergodicity

Citation

Atchadé, Yves F.; Fort, Gersende. Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II. Bernoulli 18 (2012), no. 3, 975--1001. doi:10.3150/11-BEJ360. https://projecteuclid.org/euclid.bj/1340887010


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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.
  • [3] Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343–373.
  • [4] Atchadé, Y. and Fort, G. (2010). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. Bernoulli 16 116–154.
  • [5] Atchade, Y.F. and Fort, G. (2009). Limit theorems for some adaptive mcmc algorithms with sub-geometric kernels: Part ii. Technical report. Univ. Michigan. Available at arXiv:0911.0221.
  • [6] Atchade, Y.F., Fort, G., Moulines, E. and Priouret, P. (2009). Adaptive Markov chain Monte Carlo: Theory and methods. Technical report, Univ. Michigan.
  • [7] Atchadé, Y.F. and Rosenthal, J.S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 815–828.
  • [8] Bai, Y. (2008). The simultaneous drift conditions for Adaptive Markov Chain Monte Carlo algorithms. Technical report, Univ. Toronto. (Personal communication).
  • [9] Baxendale, P.H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700–738.
  • [10] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. New York: Academic Press [Harcourt Brace Jovanovich Publishers].
  • [11] Jarner, S.F. and Roberts, G.O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224–247.
  • [12] Jarner, S.F. and Roberts, G.O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Statist. 34 781–815.
  • [13] Kamatani, K. (2009). Metropolis–Hastings algorithms with acceptance ratios of nearly 1. Ann. Inst. Statist. Math. 61 949–967.
  • [14] Kipnis, C. and Varadhan, S.R.S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [15] Kushner, H.J. and Yin, G.G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Applications of Mathematics (New York) 35. New York: Springer.
  • [16] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
  • [17] Roberts, G.O. and Rosenthal, J.S. (2001). Optimal scaling for various Metropolis–Hastings algorithms. Statist. Sci. 16 351–367.
  • [18] Roberts, G.O. and Rosenthal, J.S. (2007). Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44 458–475.
  • [19] Roberts, G.O. and Tweedie, R.L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2 341–363.
  • [20] Saksman, E. and Vihola, M. (2010). On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20 2178–2203.