Perfect simulation for marked point processes

CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ABSTRACT This paper extends some recently proposed exact simulation algorithms for point processes to marked patterns and reports on a simulation study into the relative efficiency for a range of Markov marked point processes. marked point process. 1. Introduction One of the most exciting developments in computational statistics in recent years has been the introduction of exact (or perfect) simulation methods following the ground breaking paper by Propp and Wilson [30]. During the last two decades, statistical inference for complex models tended to be based on Markov Chain Monte Carlo (MCMC) techniques, as exemplified by the review papers in [2]. The difficulty with such methods is the need for careful burn-in and convergence diagnostics to assess whether the underlying Markov chain has reached its stationary distribution. Exact simulation methods in contrast determine by themselves during run time if and when equilibrium is reached, hence their appeal. In their 1996 paper, Propp and Wilson presented exact samplers for a range of discrete distributions. Their method, coupling from the past, requires an order relation on the state space which has to be preserved by the transition kernel to be practical. Modifications that do not require such an order were studied in [17, 18] and further extended in [26]. All papers cited above use the Gibbs sampler dynamics. For specific models, faster convergence may be obtained by exploiting salient properties of the model. Examples include Fill and Huber's randomness recycler [10, 11]. Murdoch and Green [27] proposed coupling from the past algorithms for continuous state spaces. They do not rely on order relations nor on the Gibbs sampler. Instead, bounds and rejection sampling ideas are exploited. Rejection sampling is also the driving principle behind the FMMR method [8, 9], but, as the method is closely related to coupling from the past, monotonicity properties are helpful. For point process models, MCMC methods are typically based on spatial birth-and-death processes [25, 29, 31] or the Metropolis-Hastings paradigm [13, 15, 28]. One may think of the first approach as the natural analogue of the Gibbs sampler, as both methods change a