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A main problem with MCMC techniques is determining how long to run the chain/process until it is deemed to be ‘near’ equilibrium, ie. MCMC algorithms only provide anapproximatedraw from the target distribution. In some cases, however, it is possible to devise a clever algorithm which manages to ‘determine by itself’ when the chain has converged. In this instance one obtains anexact

orperfect sample, ie. an unbiased sample from the target distribution in finite time; such variants are coinedperfect simulation algorithms.

Coupling From the Past (CFTP) is one of the more widely studied and used perfect simulation protocols. It is a probabilistic algorithm or ‘simulation recipe’ for sampling Markov chains perfectly. The technique, introduced by Propp & Wilson (1996), was originally proposed for finite chains. It

has since received much attention in the literature and has been generalized to a variety of chains and unbounded state spaces including spatial point process models (Murdoch & Green 1998; Foss & Tweedie 1998; Häggströmet al. 1999; Green & Murdoch 1999; Kendall & Møller 2000; Thönnes 2000; Murdoch & Rosenthal 2000; Berthelsen & Møller 2002b). Several primers on CFTP and perfect simulation for point processes are also available (Thönnes 1997; Møller 2001; Berthelsen & Møller 2002a); Propp & Wilson (1998) gives a comprehensive overview on applications of CFTP.

An MCMC algorithm for sampling (approximately) from the target distribution π involves the construction of an ergodic Markov chain whose time-stationary (equilibrium) distribution isπ. The idea then is to initialize the chain at some arbitrary state and simulate it for a ‘long’ time. The MCMC sampler is constructed so as to ensure the transitions of the chain preserve the distribution

π. An obvious difficulty with this is how ‘long’ should the chain be run in order to ensure that it has converged or that its distribution is ‘close enough’ toπ? If the chain could be started at time −∞

then the state of the chain at time0would be an exact draw fromπ, since the chain is ergodic. An actual simulation from time−∞is infeasible; however since the interest lies predominantly in the state of the chain at time0, what if the path of the chain in avirtual simulation (Kendall & Thönnes 1999) from time−∞could be re-constructed for a finite time interval[−T,0]? This would provide a practical way of determining the state of the chain at time0. This is the essence of CFTP: it intrinsically determines how far back in the past the chain needs to be started from in order to reconstruct its infinite-time simulation path on[−T,0], for some T < ∞. Furthermore the chain at time0is guaranteed to have the same value as would have been obtained in a virtual simulation from time−∞. So letΦ∞denote the target chain started at time−∞whose stationary distribution is the target distributionπand let{0, . . . , k}represent the finite state space of the chain.

ForT > 0and−T ≤t ≤ 0, Φ∞(t)must take on one of the values0, . . . , k. So, for0≤i ≤k, let ΦT_i denote a chain started at time −T in state i. Suppose ΦT_i k_i=0 and Φ∞ are coupled and evolved using the same randomness on the interval[−T,0]. IfΦT

i (0) = ΦTj (0) for alli 6= j then

this is referred to ascoalescence, ie. all the chains started in each of thek+ 1possible states have met and take on a single common value at time0. In this case it must follow thatΦ∞(0)also has the same common value sinceΦ∞(−T)is equal toΦT

i (−T), for somei, and the coupling ensures this

equality persists for all−T ≤t≤ 0. If coalescence is not achieved by time0, the whole procedure is started off from time−S < −T. Care must be taken to ensure that the same randomness is used

over the interval [−T,0], as failure to do so will introduce bias (Propp & Wilson 1996). Such a procedure is also calledvertical CFTP, since the target chainsΦT

i are started at every statei.

Foss & Tweedie (1998) point out that CFTP can be viewed in the context of “backward couplings in stochastic recursive sequences theory”; such methods have been around for a while and indeed Propp & Wilson (1998) comment that “the conceptual ingredients were up in the air even before the versatility of the method was made clear”. Foss & Tweedie thus attempt to embed CFTP in a general probabilistic framework by describing the theoretical framework that leads to forward and backward coupling constructions. Further theory and historical origins can be found in their paper as well as in Propp & Wilson (1998) who identify some precursors to the ideas behind CFTP.

Vertical CFTP can be employed for any finite ergodic Markov chain. However as the state space gets larger it becomes impractical and computationally burdensome to keep track of all the ΦT_i

chains. Suppose there exists a partial order₄on the state space with maximal and minimal elements 1and0respectively. In addition suppose also that the transitions of the target chainΦpreserve the partial ordering, ie. the transition function is monotone (Definition 1.30). In this case CFTP can be implemented more efficiently by keeping track of only two chains: ΦT

1 andΦT0. This is because

coalescence of all theΦT_i chains is equivalent to the coalescence ofΦT₁ andΦT₀, since the transitions of the chain maintain the ordering with respect to₄. This is referred to asMonotone CFTP.

A natural extension is to the case of continuous or even unbounded state spaces. Such a consid- eration is both natural and important, especially when dealing with point processes since the state space is unbounded. It turns out that there is a neat extension of CFTP called Dominated CFTP (domCFTP) developed by Kendall (1998) for the area-interaction point process, and generalized by Kendall & Møller (2000) for any point process with a locally stable density (Definition 1.11). Wilson (2000) also use the termCoupling Into And From The Past (CIAFTP)for domCFTP.

Häggström & Nelander (1998) consider perfect sampling of anti-monotone systems, such as point processes with repulsive densities (cf. Eq. 1.11). Häggström et al. (1999) describe an exact Gibbs sampler for the bivariate Widom & Rowlinson (1970) penetrable spheres model, employing quasi-minimal and quasi-maximal states in order to devise a CFTP-based construction. Another addition,read-once CFTP(Wilson 2000), involves running the Markov chain forwards in time and never restarting it in the past. It has the advantage of not needing to store random numbers for reuse, whereas conventional CFTP does. David Wilson’s web site contains an annotated bibliography on

perfect simulation:http://dimacs.rutgers.edu/˜dbwilson/exact.html/.