Regeneration in Markov chain samplers

Abstract Markov chain sampling has recently received considerable attention, in particular in the context of Bayesian computation and maximum likelihood estimation. This article discusses the use of Markov chain splitting, originally developed for the theoretical analysis of general state-space Markov chains, to introduce regeneration into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance. The approach is applied to several samplers, including certain Metropolis samplers that can be used on their own or in hybrid samplers, and is illustrated in several examples.

[1]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[4]  E. Nummelin,et al.  A splitting technique for Harris recurrent Markov chains , 1978 .

[5]  K. Athreya,et al.  A New Approach to the Limit Theory of Recurrent Markov Chains , 1978 .

[6]  Linus Schrage,et al.  A guide to simulation , 1983 .

[7]  Paul Bratley,et al.  A guide to simulation , 1983 .

[8]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Notes and comments , 1984 .

[9]  E. Nummelin.  General irreducible Markov chains and non-negative operators: List of symbols and notation , 1984 .

[10]  Paul Bratley,et al.  A guide to simulation (2nd ed.) , 1986 .

[11]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[12]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[13]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[14]  A. O'Hagan,et al.  The Calculation of Posterior Distributions by Data Augmentation: Comment , 1987 .

[15]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[16]  D. Gaver,et al.  Robust empirical bayes analyses of event rates , 1987 .

[17]  J. Geweke,et al.  Bayesian Inference in Econometric Models Using Monte Carlo Integration , 1989 .

[18]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[19]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[20]  Peter W. Glynn,et al.  Stationarity detection in the initial transient problem , 1992, TOMC.

[21]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[22]  L. Tierney.  Exploring Posterior Distributions Using Markov Chains , 1992 .

[23]  D. Spiegelhalter,et al.  Modelling Complexity: Applications of Gibbs Sampling in Medicine , 1993 .

[24]  J. Besag,et al.  Spatial Statistics and Bayesian Computation , 1993 .

[25]  Bin Yu.  Density Estimation in the $L^\infty$ Norm for Dependent Data with Applications to the Gibbs Sampler , 1993 .

[26]  C. Geyer,et al.  Discussion: Markov Chains for Exploring Posterior Distributions , 1994 .

[27]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[28]  C. Geyer.  On the Convergence of Monte Carlo Maximum Likelihood Calculations , 1994 .