Notes on perfect simulation

Markov Chain Monte Carlo (MCMC) originated in statistical physics, but has spilled over into various application areas, leading to a corresponding variety of techniques and methods. That variety stimulates new ideas and developments from many different places, and there is much to be gained from cross-fertilization. This book presents five expository essays by leaders in the field, drawing from perspectives in physics, statistics and genetics, and showing how different aspects of MCMC come to the fore in different contexts. The essays derive from tutorial lectures at an interdisciplinary program at the Institute for Mathematical Sciences, Singapore, which exploited the exciting ways in which MCMC spreads across different disciplines

[1]  Michael Cranston,et al.  Efficient coupling on the circle , 2000 .

[2]  W. Kendall,et al.  Ising models and multiresolution quad-trees , 2003, Advances in Applied Probability.

[3]  Giovanni Montana,et al.  Small sets and Markov transition densities , 2002 .

[4]  Bin Yu,et al.  Regeneration in Markov chain samplers , 1995 .

[5]  J. Rosenthal.  QUANTITATIVE CONVERGENCE RATES OF MARKOV CHAINS: A SIMPLE ACCOUNT , 2002 .

[6]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[7]  A Sample Path Proof of the Duality for Stochastically Monotone Markov Processes , 1985 .

[8]  David Bruce Wilson.  Layered Multishift Coupling for use in Perfect Sampling Algorithms (with a primer on CFTP) , 1999 .

[9]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[10]  J. A. Fill.  An interruptible algorithm for perfect sampling via Markov chains , 1998 .

[11]  R. Tweedie,et al.  Perfect simulation and backward coupling , 1998 .

[12]  David Bruce Wilson,et al.  How to get an exact sample from a generic Markov chain and sample a random spanning tree from a directed graph, both within the cover time , 1996, SODA '96.

[13]  V. Johnson.  Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths , 1996 .

[14]  Jesper Møller,et al.  Extensions of Fill's algorithm for perfect simulation , 1999 .

[15]  Gareth O. Roberts,et al.  Catalytic Perfect Simulation , 2001 .

[16]  Ward Whitt,et al.  Comparison methods for queues and other stochastic models , 1986 .

[17]  A New Method for Coupling Random Fields , 2002 .

[18]  Jakob Gulddahl Rasmussen,et al.  A note on a perfect simulation algorithm for marked Hawkes processes , 2004 .

[19]  P. Diaconis,et al.  The cutoff phenomenon in finite Markov chains. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[21]  Olle Häggström,et al.  Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some , 1996 .

[22]  Olle Häggström.  Finite Markov Chains and Algorithmic Applications , 2002 .

[23]  G. Letac.  A contraction principle for certain Markov chains and its applications , 1986 .

[24]  David Bruce Wilson,et al.  How to couple from the past using a read-once source of randomness , 1999, Random Struct. Algorithms.

[25]  D. Griffeath.  A maximal coupling for Markov chains , 1975 .

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

[27]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[28]  Wilfrid S. Kendall,et al.  Perfect simulation for correlated Poisson random variables conditioned to be positive , 2002, Stat. Comput..

[29]  Y. Kifer.  Ergodic theory of random transformations , 1986 .

[30]  Wilfrid S. Kendall,et al.  DEPARTMENT OF STATISTICS UNIVERSITY OF WARWICK Simulation of cluster point processes without edge effects , 2002 .

[31]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[32]  James Allen Fill,et al.  Extension of Fill's perfect rejection sampling algorithm to general chains , 2000, Random Struct. Algorithms.

[33]  James Allen Fill,et al.  An interruptible algorithm for perfect sampling via Markov chains , 1997, STOC '97.

[34]  Jorge Mateu,et al.  Spatial point process modelling and its applications , 2004 .

[35]  E. Thönnes.  Perfect simulation of some point processes for the impatient user , 1999, Advances in Applied Probability.

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

[37]  W. Kendall.  Coupling methods and the storage equation , 1983, Journal of Applied Probability.

[38]  Nicholas G. Polson,et al.  On the Geometric Convergence of the Gibbs Sampler , 1994 .

[39]  Y. Gousseau,et al.  The dead leaves model : general results and limits at small scales , 2003, math/0312035.

[40]  G. Roberts,et al.  Perfect slice samplers , 2001 .

[41]  Stephen P. Brooks,et al.  Assessing Convergence of Markov Chain Monte Carlo Algorithms , 2007 .

[42]  David Bruce Wilson,et al.  Exact sampling with coupled Markov chains and applications to statistical mechanics , 1996, Random Struct. Algorithms.

[43]  R. Tweedie,et al.  Perfect sampling of ergodic Harris chains , 2001 .

[44]  J. N. Corcoran,et al.  SHIFT AND SCALE COUPLING METHODS FOR PERFECT SIMULATION , 2003, Probability in the Engineering and Informational Sciences.

[45]  Gareth O. Roberts,et al.  SMALL AND PSEUDO-SMALL SETS FOR MARKOV CHAINS , 2001 .

[46]  Mark Sweeny.  Monte Carlo study of weighted percolation clusters relevant to the Potts models , 1983 .

[47]  Luc Devroye,et al.  Simulating Perpetuities , 2001 .

[48]  W. Kendall.  Perfect Simulation for Spatial Point Processes , 1997 .

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

[50]  Olle Häggström,et al.  Propp–Wilson Algorithms and Finitary Codings for High Noise Markov Random Fields , 2000, Combinatorics, Probability and Computing.

[51]  W. Doeblin.  Sur les propriétés asymptotiques de mouvements régis par certains types de chaînes simples , 1938 .

[52]  Jesper Møller,et al.  A Bayesian MCMC method for point process models with intractable normalising constants , 2004 .

[53]  Duncan J. Murdoch,et al.  Efficient use of exact samples , 2000, Stat. Comput..

[54]  Stephen P. Brooks,et al.  Perfect Forward Simulation via Simulated Tempering , 2006 .

[55]  A. A. Borovkov,et al.  STOCHASTICALLY RECURSIVE SEQUENCES AND THEIR GENERALIZATIONS , 1992 .

[56]  David Bruce Wilson,et al.  Generating random spanning trees more quickly than the cover time , 1996, STOC '96.

[57]  Nancy L. Garcia,et al.  Loss Network Representation of Peierls Contours , 1998, math/9806131.

[58]  Olle Häggström,et al.  Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes , 1999 .

[59]  David Wilson,et al.  Coupling from the past: A user's guide , 1997, Microsurveys in Discrete Probability.

[60]  Mark Huber,et al.  A bounding chain for Swendsen‐Wang , 2003, Random Struct. Algorithms.

[61]  Wilfrid S. Kendall,et al.  Perfect simulation in stochastic geometry , 1999, Pattern Recognit..

[62]  D. Siegmund.  The Equivalence of Absorbing and Reflecting Barrier Problems for Stochastically Monotone Markov Processes , 1976 .

[63]  W. Kendall.  Geometric ergodicity and perfect simulation , 2004, math/0410012.

[64]  A. Kolmogoroff.  Zur Theorie der Markoffschen Ketten , 1936 .

[65]  James Allen Fill,et al.  Speeding up the FMMR perfect sampling algorithm: A case study revisited , 2003, Random Struct. Algorithms.

[66]  B. Silverman,et al.  PERFECT SIMULATION OF SPATIAL POINT PROCESSES USING DOMINATED COUPLING FROM THE PAST WITH APPLICATION TO A MULTISCALE AREA-INTERACTION POINT PROCESS , 2009, 0903.2651.

[67]  Peter Green,et al.  Exact sampling for Bayesian inference: towards general purpose algorithms , 1998 .

[68]  G. Roberts,et al.  Corrigendum: Perfect slice samplers , 2002 .

[69]  Aleksandr Alekseevich Borovkov,et al.  Asymptotic methods in queuing theory , 1984 .

[70]  David B. Wilson.  How to couple from the past using a read-once source of randomness , 2000 .

[71]  C. Robert,et al.  A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling , 2004, math/0407120.

[72]  D. Murdoch.  Exact Sampling for Bayesian Inference: Unbounded State Spaces , 2000 .

[73]  W. Kendall,et al.  Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes , 2000, Advances in Applied Probability.

[74]  Krishna B. Athreya,et al.  Perfect sampling for Doeblin chains , 2000 .

[75]  R. Tweedie,et al.  Exponential Convergence of Langevin Diiusions and Their Discrete Approximations , 1997 .

[76]  W. Kendall.  On some weighted Boolean models , 1997 .

[77]  D. Wilson.  Mixing times of lozenge tiling and card shuffling Markov chains , 2001, math/0102193.

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

[79]  Geoff K. Nicholls,et al.  Perfect simulation for sample-based inference , 1999 .

[80]  James Allen Fill,et al.  The randomness recycler: a new technique for perfect sampling , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[81]  R. M. Loynes,et al.  The stability of a queue with non-independent inter-arrival and service times , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[82]  John S. Rowlinson,et al.  New Model for the Study of Liquid–Vapor Phase Transitions , 1970 .

[83]  W. Kendall,et al.  Efficient Markovian couplings: examples and counterexamples , 2000 .

[84]  V. S. Anil Kumar,et al.  Coupling vs. conductance for the Jerrum-Sinclair chain , 2001 .

[85]  P. Diaconis,et al.  Strong Stationary Times Via a New Form of Duality , 1990 .

[86]  David Mumford,et al.  Occlusion Models for Natural Images: A Statistical Study of a Scale-Invariant Dead Leaves Model , 2004, International Journal of Computer Vision.

[87]  D. Jeulin.  On some weighted Boolean models , 1997 .

[88]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .

[89]  G. Casella,et al.  Explaining the Perfect Sampler , 2001 .

[90]  V. S. Anil Kumar,et al.  Coupling vs. conductance for the Jerrum-Sinclair chain , 2001, Random Struct. Algorithms.

[91]  P. Ferrari,et al.  Perfect simulation for interacting point processes, loss networks and Ising models , 1999, math/9911162.

[92]  Wilfrid S. Kendall,et al.  Perfect Simulation for the Area-Interaction Point Process , 1998 .

[93]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[94]  N. L. Lawrie,et al.  Comparison Methods for Queues and Other Stochastic Models , 1984 .

[95]  J. Rosenthal,et al.  Convergence of Slice Sampler Markov Chains , 1999 .

[96]  Xiao-Li Meng,et al.  Antithetic Coupling for Perfect Sampling , 2000 .

[97]  David Aldous,et al.  The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees , 1990, SIAM J. Discret. Math..

[98]  Xeni K. Dimakos,et al.  A Guide to Exact Simulation , 2001 .