Abstract

We consider perfect simulation algorithms for locally stable point processes based on dominated coupling from the past, and apply these methods in two different contexts. A new version of the algorithm is developed which is feasible for processes which are neither purely attractive nor purely repulsive. Such processes include multiscale area-interaction processes, which are capable of modelling point patterns whose clustering structure varies across scales. The other topic considered is nonparametric regression using wavelets, where we use a suitable area-interaction process on the discrete space of indices of wavelet coefficients to model the notion that if one wavelet coefficient is non-zero then it is more likely that neighbouring coefficients will be also. A method based on perfect simulation within this model shows promising results compared to the standard methods which threshold coefficients independently.

Keywords coupling from the past (CFTP), dominated CFTP, exact simulation, local stability, Markov chain Monte Carlo, perfect simulation, Papangelou conditional intensity, spatial birth-and-death process

AMS subject classification (MSC2010) 62M30, 60G55, 60K35

Introduction

Markov chain Monte Carlo (MCMC) is now one of the standard approaches of computational Bayesian inference. A standard issue when using MCMC is the need to ensure that the Markov chain we are using for simulation has reached equilibrium. For certain classes of problem, this problem was solved by the introduction of coupling from the past (CFTP) (Propp and Wilson, 1996, 1998).