Chapter 1 Perfect Sampling for Pointprocess Cluster

When disease incidence locations are observed in a region, there is often interest in studying whether there is clustering about landmarks representing possible centralized sources of the disease. In this article we study a Bayesian approach to the detection and estimation of such landmarks. Spatial point processes are used to specify both the observation process and the prior distribution of the landmarks. We develop a perfect sampling algorithm for the posterior distribution of landmarks under various conditions on the prior and likelihood. Bayesian cluster models of the type we consider were introduced by Bad-deley and van Lieshout (1993), primarily for applications in computer vision. The dissertation of van Lieshout (1995) focused on the special case of the Neyman{Scott cluster process, in which the observations arise from a superposition of inhomogeneous Poisson processes associated with each landmark (Neyman and Scott, 1972), and she applied it to the well known redwood seedling data used by Strauss (1975). Hurn (1998) applied the Baddeley{van Lieshout approach to the study of changes in the size and shape of living cells. Lawson and Clarke (1999) survey the statistical literature on disease clustering models. Markov chain Monte Carlo (MCMC) techniques are indispensable for the application of point process models in statistics, see, e.g., the survey of MMller (1999). The typical MCMC sampler obtains draws that are at best only approximately from the target distribution, and are often plagued by convergence problems, even when a long \burn-in" period is used. Moreover , if independent draws are required, then every draw must be produced by a separate chain. However, using an algorithm developed by Kendall and MMller (2000), it is possible to sample perfectly from the posterior distribution in the Bayesian cluster model. Perfect samplers originate in the seminal work of Propp and Wilson (1996), whose coupling from the past (CFTP) algorithm delivers an exact draw from the target distribution. The most