Convergence of Slice Sampler Markov Chains

We analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastically monotone, and we deduce analytic bounds on the total variation distance from stationarity of the method by using Foster–Lyapunov drift condition methodology.

[1]  D. Daley Stochastically monotone Markov Chains , 1968 .

[2]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[3]  J. Marsden,et al.  Elementary classical analysis , 1974 .

[4]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[5]  E. Nummelin General irreducible Markov chains and non-negative operators: Embedded renewal processes , 1984 .

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

[7]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[8]  A comparison theorem for conditioned Markov processes , 1991 .

[9]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[10]  Peter Green,et al.  Spatial statistics and Bayesian computation (with discussion) , 1993 .

[11]  S. Meyn,et al.  Computable Bounds for Geometric Convergence Rates of Markov Chains , 1994 .

[12]  J. Rosenthal Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo , 1995 .

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

[14]  R. Tweedie,et al.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms , 1996 .

[15]  Richard L. Tweedie,et al.  Geometric Convergence Rates for Stochastically Ordered Markov Chains , 1996, Math. Oper. Res..

[16]  Radford M. Neal Markov Chain Monte Carlo Methods Based on `Slicing' the Density Function , 1997 .

[17]  Antonietta Mira,et al.  On the use of auxiliary variables in Markov chain Monte Carlo sampling , 1997 .

[18]  L. Tierney A note on Metropolis-Hastings kernels for general state spaces , 1998 .

[19]  Mary Kathryn Cowles,et al.  A simulation approach to convergence rates for Markov chain Monte Carlo algorithms , 1998, Stat. Comput..

[20]  D. Higdon Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications , 1998 .

[21]  J. Rosenthal,et al.  On convergence rates of Gibbs samplers for uniform distributions , 1998 .

[22]  P. Damlen,et al.  Gibbs sampling for Bayesian non‐conjugate and hierarchical models by using auxiliary variables , 1999 .

[23]  G. S. Fishman An Analysis of Swendsen–Wang and Related Sampling Methods , 1999 .

[24]  R. Tweedie,et al.  Bounds on regeneration times and convergence rates for Markov chains fn1 fn1 Work supported in part , 1999 .

[25]  R. L. Tweedie,et al.  Rates of convergence of stochastically monotone and continuous time Markov models , 2000 .

[26]  Alison L. Gibbs,et al.  Convergence of Markov chain Monte Carlo algorithms with applications to image restoration , 2000 .

[27]  Gareth O. Roberts,et al.  Corrigendum to : Bounds on regeneration times and convergence rates for Markov chains , 2001 .