Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing
Yongtao Guan and Stephen M. Krone
Source: Ann. Appl. Probab. Volume 17, Number 1 (2007), 284-304.
Abstract
We compare convergence rates of Metropolis–Hastings chains to multi-modal target distributions when the proposal distributions can be of “local” and “small world” type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is “slowly mixing” (in the complexity of the problem) into a chain that is “rapidly mixing.” To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.
Primary Subjects: 65C05
Secondary Subjects: 65C40
Keywords: Markov chain; Monte Carlo; small world; spectral gap; Cheeger’s inequality; state decomposition; isoperimetric inequality; Metropolis-coupled MCMC
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1171377185
Digital Object Identifier: doi:10.1214/105051606000000772
Mathematical Reviews number (MathSciNet):
MR2292588
Zentralblatt MATH identifier:
1139.65001
References
Applegate, D. and Kannan, R. (1990). Sampling and integration of near log-concave functions. In Proc. 23rd ACM STOC 156--163. ACM Press, New York.
Bhatnagar, N. and Randall, D. (2004). Torpid mixing of simulated tempering on the Potts model. In Proceedings of the 15th Annual ACM--SIAM Symposium on Discrete Algorithms (New Orleans, LA) 478--487. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet):
MR2291087
Bobkov, S. G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903--1921.
Mathematical Reviews (MathSciNet):
MR1742893
Digital Object Identifier: doi:10.1214/aop/1022677553
Project Euclid: euclid.aop/1022874820
Borell, C. (1974). Convex measures on locally convex spaces. Ark. Math. 12 239--252.
Mathematical Reviews (MathSciNet):
MR0388475
Digital Object Identifier: doi:10.1007/BF02384761
Chan, K. S. and Geyer, C. J. (1994). Discussion of the paper by Tierney. Ann. Statist. 22 1747--1758.
Geyer, C. J. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface (E. M. Keramides, ed.) 156--163. Interface Foundation, Fairfax Station.
Zentralblatt MATH:
0751.12004
Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Amer. Statist. Assoc. 90 909--920.
Guan, Y., Flei$\ss$ner, R., Joyce, P. and Krone, S. M. (2006). Markov chain Monte Carlo in small worlds. Statist. Comput. 16 193--202.
Mathematical Reviews (MathSciNet):
MR2227395
Digital Object Identifier: doi:10.1007/s11222-006-6966-6
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov Chains and their applications. Biometrika 57 97--109.
Jarner, S. F. and Yuen, W. K. (2004). Conductance bounds on the $L^2$ convergence rate of Metropolis algorithms on unbounded state spaces. Adv. in Appl. Probab. 36 243--266.
Mathematical Reviews (MathSciNet):
MR2035782
Digital Object Identifier: doi:10.1239/aap/1077134472
Project Euclid: euclid.aap/1077134472
Kannan, R. and Li, G. (1996). Sampling according to the multivariate normal density. In 37th Annual IEEE Symposium on Foundations of Computer Science (FOCS'96) 204--212. IEEE Comput. Soc. Press, Los Alamitos, CA.
Mathematical Reviews (MathSciNet):
MR1450618
Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 541--559.
Mathematical Reviews (MathSciNet):
MR1318794
Kirkpatrick, S., Jr., Gelatt, C. D. and Vecchi, M. P. (1983). Optimization by simulated annealing. Science 220 671--680.
Mathematical Reviews (MathSciNet):
MR0702485
Digital Object Identifier: doi:10.1126/science.220.4598.671
Lawler, G. F. and Sokal, A. D. (1988). Bounds on the $L^2$ spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 557--580.
Mathematical Reviews (MathSciNet):
MR0930082
Digital Object Identifier: doi:10.2307/2000925
JSTOR: links.jstor.org
Leindler, L. (1972). On a certain converse of Hölder's inequality. II. Acta Sci. Math. (Szeged) 33 217--223.
Mathematical Reviews (MathSciNet):
MR2199372
Lovász, L. and Simonovits, M. (1993). Random walks in a convex body and an improved volume algorithm. Random Structures Algorithms 4 359--412.
Mathematical Reviews (MathSciNet):
MR1238906
Lovász, L. and Vempala, S. (2003). The geometry of logconcave functions and an $O^*(n^3)$ sampling algorithm. Microsoft Research Technical Report MSR-TR-2003--4. Available at http://www-math.mit.edu/~vempala/papers/logcon-ball.pdf.
Lovász, L. and Vempala, S. (2003). Logconcave functions: Geometry and efficient sampling algorithm. In 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS'03) 1--10. IEEE Comput. Soc. Press, Los Alamitos, CA.
Madras, N. and Randall, D. (2002). Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 581--606.
Mathematical Reviews (MathSciNet):
MR1910641
Digital Object Identifier: doi:10.1214/aoap/1026915617
Project Euclid: euclid.aoap/1026915617
Marinari, E. and Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhys. Lett. 19 451--458.
Martin, R. A. and Randall, D. (2000). Sampling adsorbing staircase walks using a new Markov chain decomposition method. In 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS'01) 492--502. IEEE Comput. Soc. Press, Los Alamitos, CA.
Mathematical Reviews (MathSciNet):
MR1931846
Metropolis, N., Rosenbluth, A. E., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21 1087--1091.
Peña, J. M. (2005). Exclusion and inclusion intervals for the real eigenvalues of positive matrices. SIAM J. Matrix Anal. Appl. 26 908--917.
Mathematical Reviews (MathSciNet):
MR2178204
Digital Object Identifier: doi:10.1137/04061074X
Prékopa, A. (1973). Logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 335--343.
Mathematical Reviews (MathSciNet):
MR0404557
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 13--25.
Mathematical Reviews (MathSciNet):
MR1448322
Roberts, G. O. and Tweedie, R. L. (2001). Geometric $L^2$ and $L^1$ convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38 37--41.
Mathematical Reviews (MathSciNet):
MR1915532
Digital Object Identifier: doi:10.1239/jap/1085496589
Project Euclid: euclid.jap/1085496589
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of `small-world' networks. Nature 393 440--442.
The Annals of Applied Probability
