Foss and Tweedie (2000) Perfect simulation and backward coupling. Stochastic Models. http://www.stats.bris.ac.uk/MCMC/  Home/Search   Document Details and Download   Summary   Related Articles   Check

....discussed in [24] The major issue for perfect simulation of point processes is that we must deal with an in nite state space, so that there is typically no maximal element, 3 and (crucially) uniform ergodicity of associated Markov chains is the exception rather than the rule. As pointed out in [12], the failure of uniform ergodicity implies that coalescence of the associated ow or stochastic recursive sequence cannot happen in nite time, so the standard CFTP recipe of the nite statespace work of [37] needs modi cation. It turns out that we need to introduce a special dominating process ....

....that we need to introduce a special dominating process which acts as a kind of stochastic maximum: the resulting dominated CFTP algorithm will provide viable perfect simulation if the dominating process can itself be simulated in statistical equilibrium and in reverse time. In the language of [12], vertical CFTP (coupling of realizations started at a xed time for all possible initial states) is not available if uniform ergodicity fails, but the dominating process and associated constructs provide a way to establish when horizontal CFTP (coupling of realizations started at the minimal ....

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S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stoch. Models, 14:187-203, 1998. 29

....and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [37] see also [38] and [39] and has been studied and used by a number of authors (including Kendall [26] Mller [33] Murdoch and Green [35] Foss and Tweedie [15], Kendall and Thonnes [28] Corcoran and Tweedie [4] Kendall and Mller [27] Green and Murdoch [18] and Murdoch and Rosenthal [36] By searching backwards in time until paths from all starting states have coalesced, this algorithm uses the Markov kernel K to sample exactly from #. Another ....

....(rather weak) sense that if K is uniformly ergodic, then there exists a finite integer m and a transition rule #m for the m step kernel K m such that Algorithm 2.1, applied using #m , has P(C) 0 when t is chosen su#ciently large. Compare the analogous Theorem 4. 2 for CFTP in Foss and Tweedie [15]. A similar remark applies to Algorithm 3.1. b) Just as discussed in Fill [11] see especially the end of Section 7 there) the algorithm (including its repetition of the basic routine) we have described is interruptible; that is, its running time (as measured by number of Markov chain steps) ....

Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stochastic Models 14 187--203.

....20000 that we used for the Monte Carlo maximum likelihood example in Section 2.4.2 were generated via CFTP. Recent research has resulted in many extensions of CFTP, including perfect simulation of models with non denumerable state spaces. The topic is very active: see, among others, Fill (1998) Foss and Tweedie (1998), Kendall (1998) Murdoch and Green (1998) Propp and Wilson (1998) Kendall and Th onnes (1999) H aggstr om, van Lieshout and M ller (1999) M ller (1999a,b) Th onnes (1999) and Wilson and Lund (199 ) In this section, we describe the basic idea of CFTP and apply it to the posterior ....

Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stochastic Models, 14, 187-203.

....1; 2; 1) Here the R t are random variables and is a deterministic function, called the updating function. Under mild conditions, any discrete time homogeneous Markov chain can be represented as a SRS, where the R t are IID (independent and identically distributed random variables) see Foss and Tweedie (1998) and the references therein. When making simulations, R t is generated by a vector of pseudo random numbers V t = V t1 ; V tN t ) where N t 2 N is either a constant or yet another pseudo random number; see e.g. Example 2 below. Moreover, we include negative times and let for any state x ....

....shows how monotonicity properties of the SRS make vertical CFTP feasible in practice. Finally, Wilson s read once algorithm is discussed in Section 3.3. Throughout Sections 3.1 3.3 the R t s are assumed to be IID. 3. 1 Vertical CFTP and uniform ergodicity Propp and Wilson (1996) consider what Foss and Tweedie (1998) call the smallest vertical backward coupling time , that is the rst time before 0 for coalesence of all possible chains; denoting this by T PW we have T PW = inf n t 2 N 0 : X 0 t (x) X 0 t (y) for all x; y 2 E o : 6) For example, for the falling leaves model, T PW = T f l in (2) ....

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Foss, S.G. and Tweedie, R.L. (1998). Perfect simulation and backward coupling. Stoch. Models, 14, 187-203.

....component. We call this construction, which was first used in [10] and further examined in [6] a cross over. In Figure 1 the reader can see how the minimal and the maximal path sandwich between them the paths started from all initial states. 4. 2 Dominated Coupling From The Past Foss and Tweedie [4] showed that the existence of a CFTP algorithm as developed in [23] which produces output in almost surely finite time is equivalent to the Markov chain being uniformly ergodic. However, many Markov chains of interest, in particular many spatial birthand death processes, are not uniformly but ....

S.G. Foss and R.L. Tweedie, Perfect simulation and backward coupling, Stochastic Models 14 (1998), 187--203.

....construction, which was rst used in Kendall (1998) and further examined in H#ggstr#m Nelander (1998) a cross over. In Figure 1 the reader can see how the minimal and the maximal path sandwich between them the paths started from all initial states. 4 Delta2 Dominated coupling from the past Foss Tweedie (1998) showed that the existence of a coupling from the past algorithm as in Propp Wilson (1996) which produces output in almost surely nite time implies that the target Markov chain is uniformly ergodic. However, many Markov chains of interest, in particular many spatial birth and death processes, ....

Foss, S. & Tweedie, R. (1998). Perfect simulation and backward coupling. Stochastic Models 14, 187203.

....chain Monte Carlo (MCMC) process. In it a past time is identified from which the paths of coupled Markov chains starting at every possible state would have coalesced into a single value by the present time; this value is then a sample from the steady state distribution. Foss and Tweedie [3] pointed out that for CFTP to work, the underlying Markov chain must be uniformly ergodic. Unfortunately, most of the chains in common use in Bayesian inference are not, when the state space is unbounded. However, this does not mean that CFTP can t be used; in this paper we present three ....

....coupler [4] or the multishift coupler [14] The problem of non uniform ergodicity is also illustrated in Figure 1. The lowest path started at the mode; the other two started far from it, and took dozens of steps to reach it. The effect of this on CFTP is catastrophic, as noted by Foss and Tweedie [3]. When some starting states take arbitrarily long to coalesce with others, CFTP can t ever go back far enough to find coalescence, so it fails. The remainder of this paper describes three approaches to dealing with this problem. 3 Inducing Uniform Ergodicity If the Markov chain isn t uniformly ....

S. G. Foss and R. L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

....Convergence diagnostics do not guarantee convergence, and are known to introduce bias into the results [4] The theoretical results that exist are difficult to apply in practice. Perfect simulation algorithms [18] 7] are difficult to apply except on discrete state spaces. Recent extensions [17] [8] allow the application of perfect sampling algorithms to uniformly ergodic chains on continuous state spaces. In this paper we introduce the use of the Wasserstein metric to the study of the theoretical rates of convergence of MCMC algorithms. Like total variation distance, which is the ....

Foss, S.G. and Tweedie, R.L. (1998). Perfect Simulation and Backward Coupling. Stochastic Models 14, 187--203.

....and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [34] see also [35] and [36] and has been studied and used by a number of authors (including Kendall [24] Mller [30] Murdoch and Green [32] Foss and Tweedie [14], Kendall and Thonnes [26] Corcoran and Tweedie [4] Green and Murdoch [17] Murdoch and Rosenthal [33] By searching backwards in time until paths from all starting states have coalesced, this algorithm uses the Markov kernel K to sample exactly from #. Another method of perfect simulation, ....

....(rather weak) sense that if K is uniformly ergodic, then there exists a finite integer m and a transition rule #m for the m step kernel K m such that Algorithm 1.1, applied using #m , has P(C) 0 when t is chosen su#ciently large. Compare the analogous Theorem 4. 2 for CFTP in Foss and Tweedie [14]. b) Just as discussed in Fill [11] see especially the end of Section 7 there) the algorithm (including its repetition of the basic routine) we have described is 11 interruptible; that is, its running time (as measured by number of Markov chain steps) and output are independent random ....

Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stochastic Models 14 187--203.

....and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [32] see also [33] and [34] and has been studied and used by a number of authors (including Kendall [23] Mller [28] Murdoch and Green [30] Foss and Tweedie [15], Kendall and Thonnes [25] Corcoran and Tweedie [4] Green and Murdoch [18] Murdoch and Rosenthal [31] By searching backwards in time until paths from all starting states have coalesced, this algorithm uses the Markov kernel K to sample exactly from #. Another method of perfect simulation, ....

....(rather weak) sense that if K is uniformly ergodic, then there exists a finite integer m and a transition rule #m for the m step kernel K m such that Algorithm 1.1, applied using #m , has P(C) 0 when t is chosen su#ciently large. Compare the analogous Theorem 4. 2 for CFTP in Foss and Tweedie [15]. b) Just as discussed in Fill [11] see especially the end of Section 7 there) the algorithm (including its repetition of the basic routine) we have described is interruptible; that is, its running time (as measured by number of Markov chain steps) and output are independent random variables, ....

Foss, S. G. and Tweedie, R. L. Perfect simulation and backward coupling. Comm. Statist. Stochastic Models 14 (1998), 187--203.

....is a way of running a Markov chain which ensures that the terminal value of the implementation is an exact draw from the stationary distribution of the chain. The idea was introduced by Propp and Wilson in 1996 [31] Since then, the research area has become extremely active, see for example [6, 10, 13, 15, 16, 18, 17, 19, 20, 25, 27, 28, 29, 38, 39]. The present paper is an extension and revision of [24] Perfect slice samplers for mixtures of distributions have recently been studied in [5] The paper is organised as follows. Section 2 contains a brief description of the simple slice sampler, together with a discussion of its monotonicity ....

....(xmin ;T1 ) 0 = X, that is if the maximal and minimal chain have coalesced by time zero, then their random position at time zero, X, is distributed according to the target distribution. A vertical backward coupling time is defined to be T = supft : X (xmax ;t) 0 = X (xmin ;t) 0 g. Following [13], if T is almost surely finite we call it a successful vertical backward coupling time. Strictly speaking a vertical backward coupling time is a coalescing time for the Markov chain starting from any x 2 X ; 13] But clearly if the maximal and minimal chain started at time GammaT have the same ....

[Article contains additional citation context not shown here]

S. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

....H aggstr om and Nelander [8] studied exact sampling for anti monotone chains, and M ller and Nicholls [11] proposed perfect simulated tempering aiming at handling more routine computational problems in sample based statistical inference. Important theoretical work includes Foss and Tweedie [6] and Diaconis and Freedman [3] Readers are referred to these papers, as well as several introductory and overview papers (e.g. Fismen [5] Green and Murdoch [7] and Propp and Wilson [14] for many other important developments and references. An up to date list of references in this rapidly ....

S. G. Foss and R. L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14(1-2):187-203, 1998.

....and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [32] see also [33] and [34] and has been studied and used by a number of authors (including Kendall [23] M ller [28] Murdoch and Green [30] Foss and Tweedie [15], Kendall and Thonnes [25] Corcoran and Tweedie [4] Green and Murdoch [18] Murdoch and Rosenthal [31] By searching backwards in time until paths from all starting states have coalesced, this algorithm uses the Markov kernel K to sample exactly from . Another method of perfect simulation, ....

....weak) sense that if K is uniformly ergodic, then there exists a finite integer m and a transition rule OE m for the m step kernel K m such that Algorithm 1.1, applied using OE m , has P(C) 0 when t is chosen sufficiently large. Compare the analogous Theorem 4. 2 for CFTP in Foss and Tweedie [15]. b) Just as discussed in Fill [11] see especially the end of Section 7 there) the algorithm (including its repetition of the basic routine) we have described is interruptible; that is, its running time (as measured by number of Markov chain steps) and output are independent random variables, ....

Foss, S. G. and Tweedie, R. L. Perfect simulation and backward coupling. Comm. Statist. Stochastic Models 14 (1998), 187--203.

....and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [34] see also [35] and [36] and has been studied and used by a number of authors (including Kendall [24] M ller [30] Murdoch and Green [32] Foss and Tweedie [14], Kendall and Thonnes [26] Corcoran and Tweedie [4] Green and Murdoch [17] Murdoch and Rosenthal [33] By searching backwards in time until paths from all starting states have coalesced, this algorithm uses the Markov kernel K to sample exactly from . Another method of perfect simulation, ....

....weak) sense that if K is uniformly ergodic, then there exists a finite integer m and a transition rule OE m for the m step kernel K m such that Algorithm 1.1, applied using OE m , has P(C) 0 when t is chosen sufficiently large. Compare the analogous Theorem 4. 2 for CFTP in Foss and Tweedie [14]. b) Just as discussed in Fill [11] see especially the end of Section 7 there) the algorithm (including its repetition of the basic routine) we have described is 11 interruptible; that is, its running time (as measured by number of Markov chain steps) and output are independent random ....

Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stochastic Models 14 187--203.

.... (1997a,b) Haggstrom et al. 1996) Kendall (1996) Kendall and M ller (1998) M ller (1997) and Murdoch and Green (1997) CFTP algorithms are introduced allowing e.g. for uncountable state spaces (point processes) and repulsive (or anti monotone) systems; see also Corcoran and Tweedie (1998a,b) Foss and Tweedie (1997) and Foss et al. 1997) An impatient user who stops long runs of the algorithm before termination can cause biased output of the CFTP algorithm, see Fill (1997) and Thonnes (1997) for details. Fill (1997) introduced an alternative perfect simulation algorithm, based on rejection sampling and ....

.... (1996) Kendall (1996) Kendall and M ller (1998) M ller (1997) and Murdoch and Green (1997) CFTP algorithms are introduced allowing e.g. for uncountable state spaces (point processes) and repulsive (or anti monotone) systems; see also Corcoran and Tweedie (1998a,b) Foss and Tweedie (1997) and Foss et al. 1997). An impatient user who stops long runs of the algorithm before termination can cause biased output of the CFTP algorithm, see Fill (1997) and Thonnes (1997) for details. Fill (1997) introduced an alternative perfect simulation algorithm, based on rejection sampling and unbiased for user ....

[Article contains additional citation context not shown here]

Foss, S. G. and Tweedie, R. L. (1997). Perfect simulation and backward coupling. To appear in Stochastic Models. Preprint.

....discussed in [24] The major issue for perfect simulation of point processes is that we must deal with an infinite state space, so that there is typically no maximal element, and (crucially) uniform ergodicity of associated Markov chains is the exception rather than the rule. As pointed out in [11], the failure of uniform ergodicity implies that coalescence of the associated flow or stochastic recursive sequence cannot happen in finite time, so the standard CFTP recipe of the finite state space work of [37] needs modification. It turns out that we need to introduce a special dominating ....

....that we need to introduce a special dominating process which acts as a kind of stochastic maximum: the resulting dominated CFTP algorithm will provide viable perfect simulation if the dominating process can itself be simulated in statistical equilibrium and in reverse time. In the language of [11], vertical CFTP (coupling of realizations started at a fixed time for all possible initial states) is not available if uniform ergodicity fails, but the dominating process and associated constructs provide a way to establish when horizontal CFTP (coupling of realizations started at the minimal ....

[Article contains additional citation context not shown here]

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

.... (1997a,b) H aggstr om et al. 1996) Kendall (1998) Kendall and M ller (1998) M ller (1997) and Murdoch and Green (1998) CFTP algorithms are introduced allowing e.g. for uncountable state spaces (point processes) and repulsive (or anti monotone) systems; see also Corcoran and Tweedie (1998a,b) Foss and Tweedie (1997) and Foss et al. 1997) An impatient user who stops long runs of the algorithm before termination can cause biased output of the CFTP algorithm, see Fill (1998) and Th onnes (1997) for details. Fill (1998) introduced an alternative perfect simulation algorithm, based on rejection sampling and ....

.... (1996) Kendall (1998) Kendall and M ller (1998) M ller (1997) and Murdoch and Green (1998) CFTP algorithms are introduced allowing e.g. for uncountable state spaces (point processes) and repulsive (or anti monotone) systems; see also Corcoran and Tweedie (1998a,b) Foss and Tweedie (1997) and Foss et al. 1997). An impatient user who stops long runs of the algorithm before termination can cause biased output of the CFTP algorithm, see Fill (1998) and Th onnes (1997) for details. Fill (1998) introduced an alternative perfect simulation algorithm, based on rejection sampling and unbiased for user ....

Foss, S. G. and Tweedie, R. L. (1997). Perfect simulation and backward coupling. To appear in Stochastic Models. Preprint.

....discussed in [24] The major issue for perfect simulation of point processes is that we must deal with an infinite state space, so that there is typically no maximal element, and (crucially) uniform ergodicity of associated Markov chains is the exception rather than the rule. As pointed out in [11], the failure of uniform ergodicity implies that coalescence of the associated flow or stochastic recursive sequence cannot happen in finite time, so the standard CFTP recipe of the finite statespace work of [36] needs modification. It turns out that we need to introduce a special dominating ....

....that we need to introduce a special dominating process which acts as a kind of stochastic maximum: the resulting dominated CFTP algorithm will provide viable perfect simulation if the dominating process can itself be simulated in statistical equilibrium and in reverse time. In the language of [11], vertical CFTP (coupling of realizations started at a fixed time for all possible initial states) is not available if uniform ergodicity fails, but the dominating process and associated constructs provide a way to establish when horizontal CFTP (coupling of realizations started at the minimal ....

[Article contains additional citation context not shown here]

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

....is a way of running a Markov chain which ensures that the terminal value of the implementation is an exact draw from the stationary distribution of the chain. The idea was introduced by Propp and Wilson in 1996 [20] Since then, the research area has become extremely active, see for example [3, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18]. 2 Slice Sampler Suppose f (x) x 2 X g is an unnormalized integrable density with respect to the measure and let be the corresponding probability measure: A) R A (x) dx) R X (x) dx) for all measurable A. The slice sampler is an auxiliary variables simulation method that ....

....X (xmin ;T1 ) 0 = X, that is if the maximal and minimal chain have coalesced by time zero, then their random position at time zero, X, is distributed according to the target distribution. A vertical backward coupling time is defined to be T = supft : X (xmax ;t) 0 = X (xmin ;t) 0 g. Following [7], if T is almost surely finite we say it a successful vertical backward coupling time. Strictly speaking a vertical backward coupling time is a coalescing time for the Markov chain starting from any x 2 X ; 7] But clearly if the maximal and minimal chain started at time GammaT have the same ....

[Article contains additional citation context not shown here]

S. Foss and Tweedie R. L. Perfect simulation and backward coupling. Preprint at http://www.stats.bris.ac.uk/MCMC.

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Foss, S. and Tweedie, R.L. (1998) Perfect simulation and backward coupling. Stochastic Models 14, No. 1-2, 187-204.

....measure of a Markov chain, either exactly (that is, by drawing a random sample known to be from ) or approximately, but with computable order of accuracy. These were sparked by the seminal paper of Propp and Wilson [18] and several variations and extensions of this idea have appeared since [7, 9, 10, 12, 11, 13, 14, 16, 17]. These ideas have proven effective in areas such as statistical physics [18, 7] or spatial point processes [13, 14] where they provide simple and powerful alternatives to methods based on iterating transition laws, for example. In this paper we develop an implementation of the Propp Wilson ....

.... which forms the basis of the Propp Wilson algorithm in [18] This was developed in the form below by Borovkov and Foss [2, 8, 5, 3] The SRS construction enables us to use deterministic sample path arguments which are particularly suited to a simulation environment, and more details are in [9]. We construct a probability space( Omega ; F ; P) where without loss of generality Omega = 0; 1] ZZ is a doubly infinite product space and P is Lebesgue measure; an independent and identically distributed sequence f n g 1 n= Gamma1 of uniform U[0; 1] random variables, given by n ( ....

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S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998. (Preprint at http://www.stats.bris.ac.uk/MCMC/).

....or spatial point processes, and have also been used in operations research on queueing models [4] where they provide simple and powerful alternatives to methods based on iterating transition laws, for example. The essential idea of the CFTP algorithm, which is given more formally in for example [8, 3, 2], is to find a random epoch GammaT in the past such that, if we construct sample paths from every point in the state space starting at GammaT , then all paths will have coupled successfully by time zero. If at GammaT we were to draw from then intuitively, since the paths couple regardless of ....

.... distribution [6] This suggests a method of backward coupling analogous to that in [7] Let us write the stochastic recursions for X as X n 1 = f(X n ; n ) n = Gamma1; 0; 1; where the n are a doubly infinite sequence of U[0; 1] variables: this is described in more detail in [3]. Now draw a doubly infinite set of i.i.d. U[0; 1] variables n , independent of the n . Consider the family of chains X (x) m , with X (x) m starting from x 2 X at time m. If we can find a time T such that (i) all of the values X (x) GammaT r starting at time GammaT are simultaneously ....

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

....finite. Propp and Wilson (Propp and Wilson, 1996) show that this occurs for irreducible aperiodic finite space chains, and for stochastically monotone chains as in the next section. Various extensions of this idea have appeared since the Propp Wilson paper (Corcoran and Tweedie, 1998; Fill, 1997; Foss and Tweedie, 1998; Foss et al. 1998; Haggstrom and Nelander, 1997; Haggstrom et al. 1996; Kendall, 1996; Mller, 1997; Murdoch and Green, 1998) and perfect sampling has already proven effective in areas such as statistical physics, spatial point processes, Gibbs sampling and operations research, where it ....

....(Propp and Wilson, 1996) show that this occurs for irreducible aperiodic finite space chains, and for stochastically monotone chains as in the next section. Various extensions of this idea have appeared since the Propp Wilson paper (Corcoran and Tweedie, 1998; Fill, 1997; Foss and Tweedie, 1998; Foss et al. 1998; Haggstrom and Nelander, 1997; Haggstrom et al. 1996; Kendall, 1996; Mller, 1997; Murdoch and Green, 1998) and perfect sampling has already proven effective in areas such as statistical physics, spatial point processes, Gibbs sampling and operations research, where it provides simple and ....

[Article contains additional citation context not shown here]

Foss, S. and Tweedie, R. (1998). Perfect simulation and backward coupling. Stochastic Models. To appear (Preprint at http://www.stats.bris.ac.uk/MCMC/).

.... for systems with large under up to levels [2, 4] For an overview of various perishable inventory models, see [8] Our focus is on estimating the stationary distribution of the Markov chain via simulation, using the recently developed coupling from the past or perfect simulation method [10, 6, 7]. As opposed to previous methods, this does not involve simulating a long run and then using the fact the it is reasonable to expect the the chain to be nearly stationary . Rather, it allows us to sample perfectly from , even when the form of the distribution is not known. The Markov chain ....

....that value is a draw from the stationary distribution. The rationale for this is simply that the path from 1 must have passed through some one of the points in the space at time M ; and so its value at time 0 is the common value returned by the versions. This can be made into a formal argument [10, 6], but for implementation the key requirement is to nd a random time M satisfying the criterion that all paths starting at time M coalesce by time 0. If this happens we say that coupling from the past has occurred. The following results describe certain situations that ensure such a coupling from ....

Foss, S.G. and Tweedie, R.L. 1998. Perfect Simulation and Backward Coupling. Stochastic Models 14, 187-203.

....measure of a Markov chain, either exactly (that is, by drawing a random sample known to be from ) or approximately, but with computable order of accuracy. These were sparked by the seminal paper of Propp and Wilson [18] and several variations and extensions of this idea have appeared since [6, 9, 10, 12, 11, 13, 14, 16, 17]. These ideas have proven effective in areas such as statistical physics [18, 6] or spatial point processes [13, 14] where they provide simple and powerful alternatives to methods based on iterating transition laws, for example. The essential idea of most of these approaches, which we give more ....

.... which forms the basis of the Propp Wilson algorithm in [18] This was developed in the form below by Borovkov and Foss [2, 8, 5, 3] The SRS construction enables us to use deterministic sample path arguments which are particularly suited to a simulation environment, and more details are in [9]. The SRS approach is based on the fact that one can construct a probability space ( Omega ; F ; P) an independent and identically distributed sequence f n g 1 n= Gamma1 of uniform U[0; 1] random variables, and a measurable function f : X Theta [0; 1] X such that X satisfies the ....

[Article contains additional citation context not shown here]

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 14:187--203, 1998.

....drawing a random sample known to be from ) or approximately, but with computable order of accuracy. These were sparked by the seminal paper of Propp and Wilson [18] and several variations and extensions of this idea have appeared in the literature including recent additions by Foss and Tweedie [6], Kendall [9] Murdoch and Green [17] and M ller, H aggstr om and co authors [16, 8, 7] Our approach is similar to these backwards coupling methods that enable exact draws from such invariant distributions: these have proven effective in areas such as statistical physics, spatial point ....

....as . Of course, if we followed the chain forward from to time 1, we would get a draw from : with backward coupling we go back to Gamma1 and find we get the same value as at the finite time Gamma . The formal construction showing this to be true is in Theorem 4. 1 of Foss and Tweedie [6], and more general times for such backward coupling are also described below. These coupling methods are very effective in implementation when the chain has some monotonicity properties, since then one needs only construct paths from the maximal and mininal elements, using the fact that all other ....

[Article contains additional citation context not shown here]

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models. To appear (Preprint at http://www.stats.bris.ac.uk/MCMC/). References 18

....measure of a Markov chain, either exactly (that is, by drawing a random sample known to be from ) or approximately, but with computable order of accuracy. These were sparked by the seminal paper of Propp and Wilson [17] and several variations and extensions of this idea have appeared since [5, 8, 9, 11, 10, 12, 15, 16]. These ideas have proven effective in areas such as statistical physics, spatial point processes and operations research, where they provide simple and powerful alternatives to methods based on iterating P n , for example. The essential idea of most of these approaches, which we give more ....

.... in [17] 2 Stochastic Recursive Sequences and Backward Coupling 2 This was developed in the form below by Borovkov and Foss [2, 7, 4] The SRS construction enables us to use deterministic sample path arguments which are particularly suited to a simulation environment, and more details are in [8]. The SRS approach is based on the fact that one can construct a probability space ( Omega ; F ; P) an independent and identically distributed sequence f n g 1 n= Gamma1 of Uniform[0; 1] random variables, and a measurable function f : X Theta [0; 1] X such that X satisfies the recursion ....

[Article contains additional citation context not shown here]

S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models. To appear (Preprint at http://www.stats.bris.ac.uk/MCMC/).

....finite. Propp and Wilson (Propp and Wilson, 1996) show that this occurs for irreducible aperiodic finite space chains, and for stochastically monotone chains as in the next section. Various extensions of this idea have appeared since the Propp Wilson paper (Corcoran and Tweedie, 1998; Fill, 1997; Foss and Tweedie, 1998; Foss et al. 1998; H aggstr om and Nelander, 1997; H aggstr om et al. 1996; Kendall, 1996; M ller, 1997; Murdoch and Green, 1998) and perfect sampling has already proven effective in areas such as statistical physics, spatial point processes, Gibbs sampling and operations research, where it ....

....(Propp and Wilson, 1996) show that this occurs for irreducible aperiodic finite space chains, and for stochastically monotone chains as in the next section. Various extensions of this idea have appeared since the Propp Wilson paper (Corcoran and Tweedie, 1998; Fill, 1997; Foss and Tweedie, 1998; Foss et al. 1998; H aggstr om and Nelander, 1997; H aggstr om et al. 1996; Kendall, 1996; M ller, 1997; Murdoch and Green, 1998) and perfect sampling has already proven effective in areas such as statistical physics, spatial point processes, Gibbs sampling and operations research, where it provides simple and ....

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Foss, S. and Tweedie, R. (1998). Perfect simulation and backward coupling. Stochastic Models. To appear (Preprint at http://www.stats.bris.ac.uk/MCMC/).

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Foss and Tweedie (2000) Perfect simulation and backward coupling. Stochastic Models. http://www.stats.bris.ac.uk/MCMC/

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Foss and Tweedie (2000) Perfect simulation and backward coupling. Stochastic Models. http://www.stats.bris.ac.uk/MCMC/

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Foss and Tweedie (2000) Perfect simulation and backward coupling. Stochastic Models. http://www.stats.bris.ac.uk/MCMC/

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Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stochastic Models 14 187-203.

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Foss, S. and Tweedie, R. L. (1998) Perfect simulation and backward coupling. Preprint.

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Foss, S.G. and Tweedie, R.L. (1997). Perfect simulation and backward coupling. Stochastic Models. (To appear)

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S.G. Foss and R.L. Tweedie. Perfect simulation and backward coupling. Stochastic Models, 1998. (To appear).

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S. Foss and Tweedie R. L. Perfect simulation and backward coupling. Preprint at http://www.stats.bris.ac.uk/MCMC.

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