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342
NonUniform Random Variate Generation
, 1986
"... Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various ..."
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Cited by 618 (21 self)
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Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
The Markov Chain Monte Carlo method: an approach to approximate counting and integration
, 1996
"... In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stocha ..."
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Cited by 231 (12 self)
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In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of nonasymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 182 (15 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
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Cited by 131 (1 self)
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Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
General state space Markov chains and MCMC algorithm
 PROBABILITY SURVEYS
, 2004
"... This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform e ..."
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Cited by 106 (26 self)
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This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for MetropolisHastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
Generating Random Spanning Trees More Quickly than the Cover Time
 PROCEEDINGS OF THE TWENTYEIGHTH ANNUAL ACM SYMPOSIUM ON THE THEORY OF COMPUTING
, 1996
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Finite Markov Chains and Algorithmic Applications
 IN LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
, 2001
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Markov Chain Algorithms for Planar Lattice Structures
, 1995
"... Consider the following Markov chain, whose states are all domino tilings of a 2n x 2n chessboard: starting from some arbitrary tiling, pick a 2 x 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes 90° in place. Repeat ..."
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Cited by 90 (9 self)
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Consider the following Markov chain, whose states are all domino tilings of a 2n x 2n chessboard: starting from some arbitrary tiling, pick a 2 x 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes 90° in place. Repeat many times. This process is used in practice to generate a random tiling, and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various twodimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution.
Exact sampling from a continuous state space, Scandinavian
 Journal of Statistics
, 1998
"... ABSTRACT. Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than foll ..."
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Cited by 87 (7 self)
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ABSTRACT. Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than following the monotone sampling device of Propp & Wilson, our approach uses methods related to gammacoupling and rejection sampling to simulate the chain, and direct accounting of sample paths.