Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T05:12:12.687Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometric ergodicity of hybrid samplers

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

G. Fort ,
E. Moulines ,
G. O. Roberts  and
J. S. Rosenthal
G. Fort*
Affiliation:
LMC-IMAG, Grenoble
E. Moulines*
Affiliation:
ENST, Paris
G. O. Roberts*
Affiliation:
Lancaster University
J. S. Rosenthal*
Affiliation:
University of Toronto
*
∗ Postal address: LMC-IMAG, 51 rue des Mathématiques, BP 53, 38041 Grenoble Cedex 9, France. Email address: [email protected]
∗∗ Postal address: ENST, 46 rue Barrault, 75634 Paris Cedex 13, France.
∗∗∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK.
∗∗∗∗ Postal address: Department of Statistics, University of Toronto, 100 St George Street, Toronto, Ontario, Canada M5S 3G3.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

Keywords

Markov chain Monte Carlohybrid samplergeometric ergodicity

MSC classification

Primary: 65C05: Monte Carlo methods 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 123 - 146
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Chan, K. S., and Geyer, C. J. (1994). Discussion: Markov chains for exploring posterior distributions. Ann. Statist. 22, 1747–1758.Google Scholar
[2] Chan, K. S., and Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 90, 242–252.Google Scholar
[3] Douc, E., Moulines, R., and Rosenthal, J. S. (2002). Quantitative convergence rates for inhomogeneous Markov chains. Submitted.Google Scholar
[4] Fort, G. (2002). Computable bounds for V-geometric ergodicity of Markov transition kernels. Tech. Rep. LMC, Université J. Fourier, Grenoble.Google Scholar
[5] Fort, G., and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings–Metropolis algorithm. Statist. Prob. Lett. 49, 401–410.Google Scholar
[6] Fort, G., and Moulines, E. (2003). Computable bounds for subgeometrical ergodicity. To appear in Stoch. Process. Appl.Google Scholar
[7] Fort, G., and Moulines, E. (2003). Convergence of the Monte-Carlo EM for curved exponential families. To appear in Ann. Statist. 31, No. 4.CrossRefGoogle Scholar
[8] Hobert, J. P., and Geyer, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects models. J. Multivariate Anal. 67, 414–430.CrossRefGoogle Scholar
[9] Jarner, S. F., and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl. 85, 341–361.Google Scholar
[10] Laugwitz, D. (1965). Differential and Riemannian Geometry. Academic Press, New York.Google Scholar
[11] Mengersen, K. L., and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24, 101–121.CrossRefGoogle Scholar
[12] Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
[13] Meyn, S. P., and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4, 981–1011.Google Scholar
[14] Robert, C. P., and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.Google Scholar
[15] Roberts, G. O, and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Prob. 2, 13–25.CrossRefGoogle Scholar
[16] Roberts, G. O., and Rosenthal, J. S. (1998). Two convergence properties of hybrid samplers. Ann. Appl. Prob. 8, 397–407.Google Scholar
[17] Roberts, G. O., and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110.Google Scholar
[18] Roberts, G. O., and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stoch. Process. Appl. 80, 211–229.Google Scholar
[19] Roberts, G. O., and Tweedie, R. L. (2002). Understanding MCMC. to appear.Google Scholar
[20] Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558–566.Google Scholar
[21] Sahu, S. K., and Roberts, G. O. (1999). On convergence of the EM algorithm and the Gibbs sampler. Statist. Comput. 9, 55–64.Google Scholar
[22] Smith, A. F. M., and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. B 55, 3–24.Google Scholar
[23] Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion and a rejoinder by the author). Ann. Statist. 22, 1701–1762.Google Scholar
[24] Zeger, S. L. (1988). A regression model for time series of counts. Biometrika 75, 621–629.CrossRefGoogle Scholar