Perfect sampling using bounding chains
Mark Huber
Source: Ann. Appl. Probab. Volume 14, Number 2
(2004), 734-753.
Abstract
Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph.
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Keywords: Monte Carlo; Markov chains; perfect simulation; coupling from the past; mixing times; proper colorings; Potts model; sink free orientations
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1082737109
Digital Object Identifier: doi:10.1214/105051604000000080
Mathematical Reviews number (MathSciNet): MR2052900
Zentralblatt MATH identifier: 02100752
References
Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilités XVII. Lecture Notes in Math. 986 243--297. Springer, New York.
Mathematical Reviews (MathSciNet): MR770418
Zentralblatt MATH: 0514.60067
Bubley, R. and Dyer, M. (1997). Graph orientations with no sink and an approximation for a hard case of $\sharp$SAT. In Proc. 8th ACM--SIAM Sympos. Discrete Algorithms 248--257. ACM, New York.
Mathematical Reviews (MathSciNet): MR1447671
Bubley, R. and Dyer, M. (1997). Path coupling: A technique for proving rapid mixing in Markov chains. In 38th Annual Sympos. Foundations Comp. Sci. 223--231.
Cohn, H., Pemantle, R. and Propp, J. (2002). Generating a random sink-free orientation in quadratic time. Electron. J. Combin. 9 $\sharp$R10.
Mathematical Reviews (MathSciNet): MR1946148
Diaconis, P. and Saloff-Coste, L. (1995). What do we know about the Metropolis algorithm? In Proc. 27th ACM Sympos. Theory Computing 112--129. ACM, New York.
Diaconis, P. and Shashahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159--179.
Mathematical Reviews (MathSciNet): MR626813
Doeblin, W. (1933). Exposé de la théorie des chains simples constantes de Markov à un nombre fini d'états. Rev. Math. de l'Union Interbalkanique 2 77--105.
Dyer, M., Frieze, A. and Jerrum, M. (1998). On counting independent sets in sparse graphs. Technical Report ECS-LFCS-98-391, Univ. Edinburgh.
Mathematical Reviews (MathSciNet): MR1917561
Dyer, M. and Greenhill, C. (2000). On Markov chains for independent sets. J. Algorithms 35 17--49.
Mathematical Reviews (MathSciNet): MR1747721
Digital Object Identifier: doi:10.1006/jagm.1999.1071
Zentralblatt MATH: 0961.05063
Fill, J. A., Machida, M., Murdoch, D. J. and Rosenthal, J. S. (2000). Extension of Fill's perfect rejection sampling algorithm to general chains. Random Structures Algorithms 17 290--316.
Mathematical Reviews (MathSciNet): MR1801136
Fishman, G. S. (1996). Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York.
Mathematical Reviews (MathSciNet): MR1392474
Zentralblatt MATH: 0859.65001
Fortuin, C. and Kasteleyn, P. (1972). On the random cluster model I: Introduction and relation to other models. Phys. 57 536--564.
Mathematical Reviews (MathSciNet): MR359655
Digital Object Identifier: doi:10.1016/0031-8914(72)90045-6
Häggström, O. and Nelander, K. (1998). Exact sampling from antimonotone systems. Statist. Neerlandica 52 360--380.
Mathematical Reviews (MathSciNet): MR1670194
Digital Object Identifier: doi:10.1111/1467-9574.00090
Zentralblatt MATH: 0948.60069
Häggström, O. and Nelander, K. (1999). On exact simulation from Markov random fields using coupling from the past. Scand. J. Statist. 26 395--411.
Mathematical Reviews (MathSciNet): MR1712047
Digital Object Identifier: doi:10.1111/1467-9469.00156
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97--109.
Huber, M. L. (1998). Exact sampling and approximate counting techniques. In Proc. 30th Sympos. Theory Computing 31--40.
Mathematical Reviews (MathSciNet): MR1731559
Huber, M. L. (1999). Exact sampling using Swendsen--Wang. In Proc. 10th Sympos. Discrete Algorithms 921--922.
Huber, M. L. (1999). Perfect sampling with bounding chains. Ph.D. dissertation, Cornell Univ.
Huber, M. L. (2000). A faster method for sampling independent sets. In Proc. 11th ACM--SIAM Sympos. Discrete Algorithms 625--626. ACM, New York.
Mathematical Reviews (MathSciNet): MR1755521
Zentralblatt MATH: 0954.65005
Jerrum, M., Valiant, L. and Vazirani, V. (1986). Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43 169--188.
Mathematical Reviews (MathSciNet): MR855970
Digital Object Identifier: doi:10.1016/0304-3975(86)90174-X
Zentralblatt MATH: 0597.68056
Kelly, F. (1991). Loss networks. Ann. Appl. Probab. 1 319--378.
Mathematical Reviews (MathSciNet): MR1111523
JSTOR: links.jstor.org
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953). Equation of state calculation by fast computing machines. J. Chem. Phys. 21 1087--1092.
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.
Mathematical Reviews (MathSciNet): MR1611693
Sinclair, A. (1993). Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR1201590
Zentralblatt MATH: 0780.68096
van den Berg, J. and Steif, J. (1994). Percolation and the hard-core lattice gas model. Stochastic Process. Appl. 49 179--197.
Mathematical Reviews (MathSciNet): MR1260188
Digital Object Identifier: doi:10.1016/0304-4149(94)90132-5
Zentralblatt MATH: 0787.60125
Vigoda, E. (2000). Improved bounds for sampling colorings. J. Math. Phys. 41 1555--1569.
Mathematical Reviews (MathSciNet): MR1757969
Digital Object Identifier: doi:10.1063/1.533196
Zentralblatt MATH: 0978.60083
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