Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T19:10:59.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Perfect simulation of some point processes for the impatient user

Part of: Stochastic processes Geometric probability and stochastic geometry Markov 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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Baddeley, A. J. and Van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
Chayes, J. T., Chayes, L. and Kotecký, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569.CrossRefGoogle Scholar
Fill, J. A. (1998). An interruptible algorithm for perfect sampling via Mar-kov chains. Ann. Appl. Prob. 8, 131–162.Google Scholar
Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo In Practice. Chapman & Hall, London.Google Scholar
Häggström, O., Van Lieshout, M. N. M. and Møller, J. (1996). Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Aalborg Mathematics Department Research Report R-96-2040. To appear in Bernoulli.Google Scholar
Hammersley, J. M., Lewis, J. W. E. and Rowlinson, J. S. (1975). Relationships between the multinomial and Poisson models of stochastic processes, and between the canonical and grand canonical ensembles in statistical mechanics, with illustrations and Monte Carlo methods for the penetrable spheres model of liquid-vapour equilibrium. Sankhya A, 37, 457–491.Google Scholar
Kendall, W. S. (1999). Perfect simulation for the area-interaction point process. In Probability Towards 2000, ed. Accardi, L. and Heyde, C. C. World Scientific Press, Singapore. Springer, New York, pp. 218–234.Google Scholar
Kendall, W. S. and Møller, J. (1999). Perfect Metropolis-Hastings simulation of locally stable point processes. Preprint 347, Department of Statistics, University of Warwick.Google Scholar
Klein, W. (1982). Potts-model formulation of continuum percolation. Physical Review B 26, 2677–2678.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence of Markov chains. Ann. Appl. Prob. 4, 981–1011.Google Scholar
Møller, J. (1997). Markov chain Monte Carlo and spatial point processes. In Stochastic Geometry: Likelihood and Computation, ed. Barndorff-Nielsen, O., Kendall, W. S., and Van Lieshout, M. N. M. Chapman & Hall, Boca Raton, FL, pp. 141–172.Google Scholar
Propp, J. G., Wilson, D. B. (1998). Coupling from the past: a user's guide. In Miscrosurveys in Discrete Probability, ed. Aldous, D. and Propp, J. (DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41). Amer. Math. Soc., Providence, RI, pp. 181–192. Research Report, MIT, MA.Google Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.Google Scholar
Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. Royal Statist. Soc. B, 56, 377–384.Google Scholar
Rowlinson, J. S (1980). Penetrable spheres model of liquid vapour equilibrium. Adv. Chemical Phys. 41, 1–57.Google Scholar
Rowlinson, J. S. (1990). Probability densities for some one–di-mensional problems in statistical mechanics. In Disorder in Physical Systems. ed. Grimmet, G. R. and Welsh, D. J. A. Clarendon Press, Oxford, pp. 261–276.Google Scholar
Saloff-Coste, L. (1997). Simple examples of the use of Nash inequalities for finite Markov chains. In Stochastic Geometry: Likelihood and Computation, ed. Barndorff-Nielsen, O., Kendall, W. S. and Van Lieshout, M. N. M. Chapman & Hall, Boco Raton, FL, pp. 365–400.Google Scholar
Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transitions. J. Chemical Phys. 52, 1670–1684.Google Scholar