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Perfect simulation of some point processes for the impatient user

Part of: Markov processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Elke Thönnes
Elke Thönnes*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology, School of Mathematical and Computing Sciences, S-41296 Göteborg, Sweden.
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Abstract

Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Häggström et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

Keywords

Markov chain Monte Carloperfect simulationpoint processesuser impatience biasGibbs samplerpenetrable spheres mixture processarea-interaction processcontinuum random-cluster model

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68U20: Simulation 60G57: Random measures 60D05: Geometric probability and stochastic geometry
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 69 - 87
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by EPSRC earmarked studentship and University of Warwick graduate award.

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