Perfect Simulation of Conditionally Specified ModelsJesper MollerJournal of the Royal Statistical Society. Series B (Statistical Methodology), Vol. 61, No. 1.(1999), pp. 251-264.Stable URL:http://links.jstor.org/sici?sici=1369-7412%281999%2961%3A1%3C251%3APSOCSM%3E2.0.CO%3B2-ZJournal of the Royal Statistical Society. Series B (Statistical Methodology) is currently published by Royal Statistical Society.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/rss.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. Formore information regarding JSTOR, please contact [email protected]://www.jstor.orgWed Jun 6 11:56:05 2007J. R. Statist. Soc. B (1999) 61, Part 1, pp. 251-264 Perfect simulation of conditionally specified models Jesper Maller Aalborg University, Denmark [Received June 1997. Final revision May 19981 Summary. We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as autogamma, auto-Poisson and autonegative binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes. Keywords: Coupling from the past; Exact simulation; Gibbs sampling; Locally specified exponential family distributions; Markov chain Monte Carlo methods; Metropolis-Hastings algorithm; Spatial statistics 1. Introduction Since Propp and Wilson's (1996) seminal work on perfect simulation there has been exten- sive interest in developing and applying their ideas in different contexts (see the survey in Propp and Wilson (1998)). Briefly, the main idea is to use coupling from the past (CFTP) and repeatedly to use the same sampler for generating upper and lower Markov chains started increasingly further back in time until a pair of upper and lower chains coalesce at time 0, and then to return the result as a perfect (or exact) simulation from a given target distribution (in Kendall (1998) and Kendall and M0ller (1998) it is argued why the terminology 'perfect' is preferable). To do this Propp and Wilson (1996) assumed that the state space is finite and equipped with a partial ordering so that the sampler is monotone and there are a unique minimal element, in which the lower processes are started, and a unique maximal element, in which the upper processes are started. Then a chain produced by the sampler, started at any time n d 0 in an arbitrary initial state, sandwiches between that pair of lower and upper chains which was started at the same time n. Thereby it can be established under weak (ergodicity) conditions for the sampler that coalescence will happen for all sufficiently large n, and by considering a 'virtual simulation from time minus infinity' it follows that the output is a simulation from the target distribution. These ideas have now been extended in various ways. Kendall (1998) and Haggstrijm et al. (1996) outlined how to do perfect simulation for point processes, where the state space is uncountable. In particular, as Propp and Wilson (1996) in their examples required the target distribution to be attractive in a certain sense so that the Gibbs sampler becomes monotone, Kendall's work showed how to handle the opposite repulsive case. This has been further generalized in Kendall and M0ller (1998), where the role of the minimal element (the empty point configuration) is emphasized (there is no maximal element in a point process setting); Addressfor correspondence: Jesper M~ller, Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7E, DK-9220 Aalborg 0,Denmark. E-mail: [email protected] O1999 Royal Statistical Society 1369-741 2/99/61 251see also Kendall (1997) and Haggstrom and Nelander (1997). An even more general approach, but for simulating multivariate continuous distributions, has recently been studied in Murdoch and Green (1998). This and the other papers mentioned will be commented further on in this paper. The purpose of this paper is to show how these ideas can be further extended to produce perfect simulation of multivariate discrete (Section 2) and continuous (Section 3) target distributions, where the target distribution is naturally specified through its conditional distributions of one component given the others so that Gibbs sampling is the obvious way of producing samples. The examples of such distributions to be discussed will mainly be locally specified exponential family distributions (Besag, 1974; Cressie, 1993) with applications in spatial statistics such as autobinomial, auto-Poisson, autonegative binomial, autogamma models and certain pairwise difference interaction models (Sections 2.3 and 3.3). Indeed many other examples of models could be included; for example combinations of the local char- acteristics from different types of models may specify a joint distribution from which we can make perfect simulations. The autogamma model has also been used in other papers on Markov chain Monte Carlo methods, in particular in connection with a Bayesian analysis of a data set on pump reliability (Gelfand and Smith, 1990; Murdoch and Green, 1998). In relation to this some empirical results will be reported in Section 3.3.2. The techniques used in Section 2 are much inspired by Kendall and M~ller (1998) and some of the terminology and notation will be borrowed from that work. Compared with Propp and Wilson (1996, 1998) and Haggstrom and Nelander (1997) the extension is mainly that infinite discrete state spaces are covered as well provided that the model is repulsive. For