Abstract
Markov Chain Monte Carlo has long become a very useful, established tool in statistical physics and spatial statistics. Recent years have seen the development of a new and exciting generation of Markov Chain Monte Carlo methods: perfect simulation algorithms. In contrast to conventional Markov Chain Monte Carlo, perfect simulation produces samples which are guaranteed to have the exact equilibrium distribution. In the following we provide an example-based introduction into perfect simulation focussed on the method called Coupling From The Past.
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References
Brooks, S.P., G.O. Roberts (1999): ‘Assessing convergence of Markov Chain Monte Carlo algorithms’, Statist. Comput. 8, pp. 319–335
Cowles, M.K., B.P. Carlin (1996): ‘Markov chain convergence diagnostics: a comparative review’, J. Amer. Statist. Assoc. 91, pp. 883–904
Creutz, M. (1979): ‘Confinement and critical dimensionality of space-time’, Phys. Rev. Lett. 43, pp. 553–556
Daley, D.J., D. Vere-Jones (1988): Introduction to the Theory of Point Processes (Springer, New York)
Fill, J.A. (1998): ‘An interruptible algorithm for exact sampling via Markov chains’, Ann. Appl. Probab. 8, pp. 131–162
Fill, J.A., M. Machida, D. J. Murdoch, J.S. Rosenthal (1999): ‘Extension of Fill’s perfect rejection sampling algorithm to general chains’, preprint available on http://www.mts.jhu.edu/~fill/.
Fill, J.A. (1998): ‘The Move-To-Front rule: A case study for two perfect sampling algorithms’, Probab. Engrg. Inform. Sci. 12, pp. 283–302
Foss, S.G., R.L. Tweedie (1998) ‘Perfect simulation and backward coupling’, Stochastic Models 14, pp. 187–203
Geman, S., D. Geman (1984): ‘Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images’, IEEE Trans. PAMI 6,pp. 721–741
Gilks, W.R., S. Richardson, D.J. Spiegelhalter (1996): Markov Chain Monte Carlo In Practice (Chapman & Hall, London)
Green, P.J., D.J. Murdoch (1999): ‘Exact sampling for Bayesian inference: towards general purpose algorithms’. In: Bayesian Statistics 6, ed. by Bernardo, J.M., J.O. Berger, A.P. Dawid, A.F.M. Smith (Oxford University Press, Oxford), pp. 301–321
Häggström, O., K. Nelander (1998): ‘Exact sampling from anti-monotone systems’, Statist. Neerlandica 52, pp. 360–380
Häggström, O., J.E. Steif (1999): ‘Propp-Wilson algorithms and finitary codings for high noise random fields’, to appear in Combin. Probab. Comput.
Kendall, W.S. (1997): ‘Perfect simulation for spatial point processes’. In: Proc. ISI 51st session, Istanbul August 1997, volume 3, pp. 163–166
Kendall, W.S. (1998): ‘Perfect simulation for the area-interaction point process’. In: Probability Towards 2000, ed. by L. Accardi, C.C. Heyde (Springer, New York), pp. 218–234
Kendall, W.S., J. Møller (1999): ‘Perfect Metropolis-Hastings simulation of locally stable point processes’, Research report 347, University of Warwick
Kendall, W.S., E. Thönnes (1999): ‘Perfect simulation in stochastic geometry’, Pattern Recognition bf 32, pp. 1569–1586
Lindvall, T. (1992): Lectures On The Coupling Method. Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons)
Meyn, S.P., R.L. Tweedie (1993): Markov Chains and Stochastic Stability (Springer Verlag, New York)
Mira, A., J. Møller, G. O. Roberts (1998): ‘Perfect slice samplers’, preprint, available on http://www.dimacs.rutgers.edu/~dbwilson/exact
Møller, J. (1999): ‘Markov Chain Monte Carlo and spatial point processes’. In: Stochastic Geometry: Likelihood And Computation, ed. by W.S. Kendall, O.E. Barndor.-Nielsen, M.N.M. van Lieshout. Proceedings Séminaire Européen de Statistique (Chapman & Hall/CRC, Boca Raton), pp. 141–172
Møller, J., K. Schladitz (1998): ‘Extensions of Fill’s algorithm for perfect simulation’, preprint, to appear in J. Roy. Statist. Soc. B
Møller, J. (1999): ‘Perfect simulation of conditionally specified models’, J. Roy. Statist. Soc. B 61, pp. 251–264
Murdoch, D.J., P.J. Green (1998): ‘Exact sampling from a continuous state space’, Scand. J. Statist. 25,pp. 483–502
Murdoch, D.J., J.S. Rosenthal (1998): ‘An extension of Fill’s exact sampling algorithm to non-monotone chains’, preprint
Norris, J.R. (1997): Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge)
Propp, J.G., D.B. Wilson (1996) ‘Exact sampling with coupled Markov chains and applications to statistical mechanics’, Random Structures Algorithms 9, pp. 223–252
Ripley, B.D. (1987): Stochastic Simulation (John Wiley & Sons, New York)
Roberts, G.O., N.G. Polson (1994): ‘On the geometric convergence of the Gibbs sampler’, J. Roy. Statist. Soc. B 56, pp. 377–384
Rosenthal, J.S. (1995): ‘Minorization conditions and convergence rates for Markov chain Monte Carlo’, J. Amer. Statist. Assoc. 90, pp. 558–566
Saloff-Coste, L. (1999): ‘Simple examples of the use of Nash inequalities for finite Markov chains’. In: Stochastic Geometry: Likelihood and Computation, ed. by O. Barndor.-Nielsen, W.S. Kendall, M.N.M van Lieshout (Chapman & Hall/CRC, Boca Raton), pp. 365–400
Stoyan, D. (1983): Comparison Methods for Queues and Other Stochastic Models. Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons, Chichester)
Stoyan, D., W.S. Kendall, J. Mecke (1995) Stochastic Geometry and its Applications. Wiley Series in Probability and Mathematical Statistics (John Wiley & Sons, Chichester), 2nd ed.
Stoyan, D., H. Stoyan (1994): Fractals, Random Shapes and Point Fields (John Wiley & Sons, Chichester)
Thönnes, E. (1999): ‘Perfect simulation of some point processes for the impatient user’, Adv. Appl. Probab. 31, pp. 69–87
van den Berg, J., J.E. Steif (1999): ‘On the existence and non-existence of finitary codings for a class of random fields’, to appear in Ann. Probab.
Winkler, G. (1991): Image Analysis, Random Fields and Monte Carlo Methods. Applications of Mathematics (Springer, Berlin)
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Thönnes, E. (2000). A Primer on Perfect Simulation. In: Mecke, K.R., Stoyan, D. (eds) Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol 554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45043-2_13
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DOI: https://doi.org/10.1007/3-540-45043-2_13
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