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Convergence of Conditional Metropolis-Hastings Samplers

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

Published online by Cambridge University Press:  22 February 2016

Galin L. Jones ,
Gareth O. Roberts  and
Jeffrey S. Rosenthal
Galin L. Jones*
Affiliation:
University of Minnesota
Gareth O. Roberts*
Affiliation:
University of Warwick
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
∗ Postal address: School of Statistics, University of Minnesota, Minneapolis, MN 55455, USA. Email address: [email protected]
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
∗∗∗ Postal address: Department of Statistics, University of Toronto, Toronto, Ontario, M5S 3G3, Canada. Email address: [email protected]
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Abstract

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We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in a conditional Metropolis-Hastings sampler (CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.

Keywords

Markov chain Monte Carlo algorithmindependence samplerGibbs samplergeometric ergodicityconvergence rate

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains 65C40: Computational Markov chains 62F15: Bayesian inference
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 422 - 445
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by the National Institutes for Health.

Partially supported by the Natural Sciences and Engineering Research Council of Canada.

References

Brooks, S., Gelman, A., Jones, G. L. and Meng, X.-L. (eds) (2011). Handbook of Markov Chain Monte Carlo. CRC Press, Boca Raton, FL.CrossRefGoogle Scholar
Chan, K. S. and Geyer, C. J. (1994). Discussion: Markov chains for exploring posterior distributions. Ann. Statist. 22, 1747–1758.CrossRefGoogle Scholar
Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Prob. 3, 696–730.CrossRefGoogle Scholar
Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69, 959–993.CrossRefGoogle Scholar
Flegal, J. M., Haran, M. and Jones, G. L. (2008). Markov chain Monte Carlo: can we trust the third significant figure? Statist. Sci. 23, 250–260.CrossRefGoogle Scholar
Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85, 398–409.CrossRefGoogle Scholar
Hobert, J. P. and Geyer, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model. J. Multivariate Anal. 67, 414–430.CrossRefGoogle Scholar
Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl. 85, 341–361.CrossRefGoogle Scholar
Johnson, A. A. and Jones, G. L. (2010). Gibbs sampling for a Bayesian hierarchical general linear model. Electron. J. Statist. 4, 313–333.CrossRefGoogle Scholar
Johnson, A. A., Jones, G. L. and Neath, R. C. (2013). Component-wise Markov chain Monte Carlo: uniform and geometric ergodicity under mixing and composition. Statist. Sci. 28, 360–375.CrossRefGoogle Scholar
Jones, G. L. (2004). On the Markov chain central limit theorem. Prob. Surveys 1, 299–320.CrossRefGoogle Scholar
Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16, 312–334.Google Scholar
Jones, G. L. and Hobert, J. P. (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. Ann. Statist. 32, 784–817.CrossRefGoogle Scholar
Jones, G. L., Haran, M., Caffo, B. S. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101, 1537–1547.CrossRefGoogle Scholar
Lawler, G. F. and Sokal, A. D. (1988). Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309, 557–580.Google Scholar
Liu, J. S. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statist. Comput. 6, 113–119.CrossRefGoogle Scholar
Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81, 27–40.CrossRefGoogle Scholar
Marchev, D. and Hobert, J. P. (2004). Geometric ergodicity of van Dyk and Meng's algorithm for the multivariate Student's t model. J. Amer. Statist. Assoc. 99, 228–238.CrossRefGoogle Scholar
Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24, 101–121.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Papaspiliopoulos, O. and Roberts, G. (2008). Stability of the Gibbs sampler for Bayesian hierarchical models. Ann. Statist. 36, 95–117.CrossRefGoogle Scholar
Peskun, P. H. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60, 607–612.CrossRefGoogle Scholar
Robert, C. P. (1995). Convergence control methods for Markov chain Monte Carlo algorithms. Statist. Sci. 10, 231–253.CrossRefGoogle Scholar
Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. R. Statist. Soc. B 56, 377–384.Google Scholar
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Prob. 2, 13–25.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1998). Two convergence properties of hybrid samplers. Ann. Appl. Prob. 8, 397–407.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. J. R. Statist. Soc. B 61, 643–660.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (2001). Markov chains and de-initializing processes. Scand. J. Statist. 28, 489–504.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Prob. Surveys 1, 20–71.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (2011). Quantitative non-geometric convergence bounds for independence samplers. Methodol. Comput. Appl. Prob. 13, 391–403.CrossRefGoogle Scholar
Roberts, G. O. and Sahu, S. K. (1997). Updating schemes, correlation structure, blocking and parametrization for the Gibbs sampler. J. R. Statist. Soc. B 59, 291–317.CrossRefGoogle Scholar
Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88, 603–621.CrossRefGoogle Scholar
Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales, Vol. 1, Foundations, 2nd edn. John Wiley, Chichester.Google Scholar
Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558–566. (Correction: 90 (1995), 1136.)CrossRefGoogle Scholar
Rosenthal, J. S. (1996). Analysis of the Gibbs sampler for a model related to James–Stein estimators. Statist. Comput. 6, 269–275.CrossRefGoogle Scholar
Roy, V. and Hobert, J. P. (2007). Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression. J. R. Statist. Soc. B 69, 607–623.CrossRefGoogle Scholar
Schervish, M. J. and Carlin, B. P. (1992). On the convergence of successive substitution sampling. J. Comput. Graph. Statist. 1, 111–127.Google Scholar
Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Prob. Comput. 1, 351–370.CrossRefGoogle Scholar
Smith, R. L. and Tierney, L. (1996). Exact transition probabilities for the independence Metropolis sampler. Tech. Rep., University of North Carolina.Google Scholar
Tan, A. and Hobert, J. P. (2009). Block Gibbs sampling for Bayesian random effects models with improper priors: convergence and regeneration. J. Comput. Graph. Statist. 18, 861–878.CrossRefGoogle Scholar
Tierney, L. (1994). Markov chains for exploring posterior distributions. With discussion and a rejoinder by the author. Ann. Statist. 22, 1701–1762.Google Scholar
Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.CrossRefGoogle Scholar