ABSTRACT
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. (1995), 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.
