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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
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.
An Interruptible Algorithm for Perfect Sampling via Markov Chains
 Annals of Applied Probability
, 1998
"... For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate nor ..."
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Cited by 84 (7 self)
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For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time. The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with longrun distribution #. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult. Recently, Jim Propp and David Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from #. However, the running t...
Perfect simulation for interacting point processes, loss networks and Ising models
, 1999
"... We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birthanddeath processes. Examples include area and perimeterinteracting point processes (with stochastic grains), invariant meas ..."
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Cited by 41 (12 self)
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We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birthanddeath processes. Examples include area and perimeterinteracting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve any coupling hence it is not tied up to monotonicity requirements and it directly provides samples of the infinitevolume measure. The algorithm is based on a twostep procedure: (i) a perfectsimulation scheme for (spacetime) marked Poisson processes (free birthanddeath process, free loss networks), and (ii) a "cleaning" algorithm that trims out this process according to the interaction rules of the target process. The first step involves the perfect generation of "ancestors" of a given object, that is of predecessors 1 that may have an influence on the birthrate under the targe...
Exact Sampling From AntiMonotone Systems
 Statistica Neerlandica
, 1998
"... A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yields samples which have exactly the desired distribution. The ProppWilson algorithm requires this distribution to have a certain structure called monotonicity. ..."
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Cited by 39 (1 self)
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A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yields samples which have exactly the desired distribution. The ProppWilson algorithm requires this distribution to have a certain structure called monotonicity. In this paper an idea of Kendall is applied to show how the algorithm can be extended to the case where monotonicity is replaced by antimonotonicity. As illustrating examples, simulations of the hardcore model and the randomcluster model are presented.
How to Couple from the Past Using a ReadOnce Source of Randomness
, 1999
"... We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related ..."
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Cited by 34 (1 self)
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We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a readonce stream of randomness, we call it readonce CFTP. The memory and time requirements of readonce CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a readonce version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative...
Perfect simulation of conditionally specified models
, 1999
"... . JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publi ..."
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Cited by 32 (4 self)
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. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
On some weighted Boolean models.
 Advances in Theory and Applications of Random Sets
, 1997
"... An overview is given of some recent work (joint with Adrian Baddeley of Perth and Colette van Lieshout of Warwick) on a new class of random point and set processes, obtained using a rather natural weighting procedure employing quermass integrals. The concept of exact (or perfect) simulation of point ..."
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Cited by 30 (10 self)
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An overview is given of some recent work (joint with Adrian Baddeley of Perth and Colette van Lieshout of Warwick) on a new class of random point and set processes, obtained using a rather natural weighting procedure employing quermass integrals. The concept of exact (or perfect) simulation of point processes is then introduced, and a discussion is given of possibilities for perfect simulation of quermass weighted processes. 1 Introduction. This short paper is a progress report on some recent work of mine (partly in collaboration with Adrian Baddeley of Perth and Colette van Lieshout of Warwick); concerning new models for point processes and random sets, and the application to them of a new technique for "exact" or "perfect" simulation. 2 A brief overview of quermassinteraction processes In this section we review the idea of weighting point processes and random sets using quermass integrals. Recall Baddeley and Van Lieshout's definition [BvL95] of an areainteraction point process: ...
Coupling from the Past: a User's Guide
, 1997
"... . The Markov chain Monte Carlo method is a general technique for obtaining samples from a probability distribution. In earlier work, we showed that for many applications one can modify the Markov chain Monte Carlo method so as to remove all bias in the output resulting from the biased choice of an i ..."
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Cited by 27 (2 self)
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. The Markov chain Monte Carlo method is a general technique for obtaining samples from a probability distribution. In earlier work, we showed that for many applications one can modify the Markov chain Monte Carlo method so as to remove all bias in the output resulting from the biased choice of an initial state for the chain; we have called this method Coupling From The Past (CFTP). Here we describe this method in a fashion that should make our ideas accessible to researchers from diverse areas. Our expository strategy is to avoid proofs and focus on sample applications. 1. Introduction In Markov chain Monte Carlo studies, one attempts to sample from a distribution ß by running a Markov chain whose unique steadystate distribution is ß. Ideally, one has proved a theorem that guarantees that the time for which one plans to run the chain is substantially greater than the mixing time of the chain, so that the distribution ~ ß that one's procedure actually samples from is known to be cl...