Markov chain Monte Carlo and spatial point processes

Abstract

This chapter explains some general background material on Markov chain Monte Carlo. It discusses quantitative bounds for the rate of convergence. The chapter focuses on some empirical results for simulating models for either clustered or regular point patterns. Though a Markov chain algorithm may have good convergence properties in theory, it is pointed out that it may converge extremely slowly in practice in the case of strong interaction. The chapter provides a general setup for finite processes which covers ordinary spatial point processes and marked point processes. It aims to study general Metropolis–Hastings single point updating type algorithms following Hastingsand relate these algorithms to particular examples like the Metropolis algorithm, the Gibbs sampler, and various spatial birth-and-death process techniques. The chapter is concerned with some practical aspects of using such algorithms in cases where different types of Gibbsian or Markovian point processes are used as models for either regular or clustered patterns.