ABSTRACT
Spatial point processes arise naturally in many contexts, including population studies, forestry, epidemiology, agriculture, and material science; for more examples, see Ripley (1977), StoyanandStoyan (1995), andMøller andWaagepetersen (2007). Typically, statistical models for these data sets are given by densities with respect to a Poisson point process. In Section 9.2 these Poisson point processes and densities are described in detail, together with several examples. As in many applications, these densities are often unnormalized, and calculating the normalizing constant exactly is computationally unfeasible. Therefore Monte Carlo methods are used instead. Many of these methods involve construction of a Markov chain whose stationary dis-
tribution matches the target density. There are two primary types of chains used for these point processes. In Section 9.3, the Metropolis-Hastings and reversible jump (Green, 1995) methods are described. Section 9.4 shows how to build continuous-time spatial birth and death chains for these problems. Next, in Section 9.5 perfect sampling techniques are introduced. These methods draw
samples exactly drawn from the target distribution. Acceptance/rejection methods can be used for small problems, while larger problems require methods such as Kendall and Møller’s (2000) dominated coupling from the past. Sections 9.2 through9.5develop techniques for sampling from the statisticalmodel.When
the interest is in sampling from the posterior, Section 9.6 goes further and shows how these methods can bemodified in order to accomplish this task and carry out Bayesian inference. Finally, Section 9.7 examines what is known about the running time of these methods,
and strategies for improving the convergence of these chains.
