Markov Chain Monte Carlo and Spatial Point Processes

Currently, Markov chain Monte Carlo methods attract much attention among statisticians, cf. e.g. Smith and Roberts (1993), Besag and Green (1993), Besag et al. (1994), Tierney (1994) and the accompaning discussions and references. The genesis of most ideas lies in statistical physics following the early work by Metropolis et al. (1953). In that paper the first Markov chain Monte Carlo algorithm for simulating a Gibbsian point process with a fixed and finite number of points was developed. Hastings (1970) introduced a general class of Markov chain Monte Carlo algorithms which covers nearly any algorithm considered so far. In statistics some of the earliest and most important applications of Markov chain Monte Carlo appear to be within spatial statistics, see e.g. the discussion in Besag (1974), and especially the Gibbs sampler (Geman and Geman, 1984) have been frequently used. The Gibbs sampler has earlier been introduced in statistical physics (e.g. Creutz, 1979) where it is known as the ’heat bath algorithm’; it appeared implicit in spatial statistics in Suomela’s (1976) thesis and in Ripley (1977, 1979) too. Following Preston (1977), Ripley (1977) discussed also other ’birth-death’ techniques for simulating finite point processes with a fixed or random number of points. Indeed Markov chain Monte Carlo methods apply on a broad spectrum of complex problems in statistics, particulary in Bayesian inference, but in this contribution we restrict attention to spatial (marked) point processes as used in spatial statistics and stochastic geometry, see e.g. Diggle (1983), Ripley (1988), Baddeley and Møller (1989), and Stoyan, Kendall and Mecke (1995). The objective is both to review those algorithms which are used for simulating point processes with a finite and either fixed or random number of points, and to discuss the problem of modelling spatial (marked) point processes which exhibit regular or clustered patterns. This contribution is a revised version of Møller (1994b). Section 2 provides a general setup for finite processes which covers ordinary spatial point processes and marked point processes. In Section 3