Abstract
Decomposition theorems are useful tools for bounding the convergence rates of Markov chains. The theorems relate the mixing rate of a Markov chain to smaller, derivative Markov chains, defined by a partition of the state space, and can be useful when standard, direct methods fail. Not only does this simplify the chain being analyzed, but it allows a hybrid approach whereby different techniques for bounding convergence rates can be used on different pieces. We demonstrate this approach by giving bounds on the mixing time of a chain on circuits of length 2n in ℤd.
Supported in part by NSF Grant No. CCR-9703206.
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C. Borgs, J. T. Chayes, A. Frieze, J. H. Kim, P. Tetali, E. Vigoda, and V. H. Vu. Torpid mixing of some MCMC algorithms in statistical physics. Proc. 40th IEEE Symposium on Foundations of Computer Science, 218–229, 1999.
R. Bubley and M. Dyer. Faster random generation of linear extensions. Discrete Mathematics, 201:81–88, 1999.
R. Bubley and M. Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. Proc. 38th Annual IEEE Symposium on Foundations of Computer Science 223–231, 1997.
P. Diaconis and L. Saloff-Coste. Comparison theorems for reversible Markov chains. Annals of Applied Probability, 3:696–730, 1993.
M. Dyer and C. Greenhill. On Markov chains for independent sets. Journal of Algorithms, 35: 17–49, 2000.
M. Dyer and C. Greenhill. A more rapidly mixing Markov chain for graph colorings. Random Structures and Algorithms, 13:285–317, 1998.
C. J. Geyer Markov Chain Monte Carlo Maximum Likelihood. Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface (E. M. Keramidas, ed.), 156–163. Interface Foundation, Fairfax Station, 1991.
C. J. Geyer and E. A. Thompson. Annealing Markov Chain Monte Carlo with Applications to Ancestral Inference. J. Amer. Statist. Assoc. 90 909–920, 1995.
M. Luby and E. Vigoda. Fast Convergence of the Glauber dynamics for sampling independent sets. Random Structures and Algorithms 15: 229–241, 1999.
N. Madras and M. Piccioni. Importance sampling for families of distributions. Ann. Appl. Probab. 9: 1202–1225, 1999.
N. Madras and D. Randall. Factoring graphs to bound mixing rates. Proc. 37th Annual IEEE Symposium on Foundations of Computer Science, 194–203, 1996.
N. Madras and D. Randall. Markov chain decomposition for convergence rate analysis. Annals of Applied Probability, (to appear), 2001.
N. Madras and Z. Zheng. On the swapping algorithm. Preprint, 2001.
E. Marinari and G. Parisi. Simulated tempering: a new Monte Carlo scheme. Europhys. Lett. 19 451–458, 1992.
R. A. Martin and D. Randall Sampling adsorbing staircase walks using a new Markov chain decomposition method. Proceedings of the 41st Symposium on the Foundations of Computer Science (FOCS 2000), 492–502, 2000.
R. A. Martin and D. Randall Disjoint decomposition with applications to sampling circuits in some Cayley graphs. Preprint, 2001.
D. Randall and P. Tetali. Analyzing Glauber dynamics by comparison of Markov chains. Journal of Mathematical Physics, 41:1598–1615, 2000.
A. J. Sinclair. Algorithms for random generation & counting: a Markov chain approach. Birkhäuser, Boston, 1993.
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Randall, D. (2001). Decomposition Methods and Sampling Circuits in the Cartesian Lattice. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_8
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DOI: https://doi.org/10.1007/3-540-44683-4_8
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