Annals of Applied Probability

Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

Yongtao Guan and Stephen M. Krone

Full-text: Open access

Enhanced PDF (198 KB)

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.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 1 (2007), 284-304.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1171377185

Digital Object Identifier
doi:10.1214/105051606000000772

Mathematical Reviews number (MathSciNet)
MR2292588

Zentralblatt MATH identifier
1139.65001

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 65C40: Computational Markov chains

Keywords
Markov chain Monte Carlo small world spectral gap Cheeger’s inequality state decomposition isoperimetric inequality Metropolis-coupled MCMC

Citation

Guan, Yongtao; Krone, Stephen M. Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing. Ann. Appl. Probab. 17 (2007), no. 1, 284--304. doi:10.1214/105051606000000772. https://projecteuclid.org/euclid.aoap/1171377185


Export citation

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
    Zentralblatt MATH: 1318.82023
  • 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
    Zentralblatt MATH: 0964.60013
  • 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
    Zentralblatt MATH: 0297.60004
    Project Euclid: euclid.afm/1485896206
  • 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
    Digital Object Identifier: doi:10.1007/BF02950753
  • 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.
    Zentralblatt MATH: 0850.62834
    Digital Object Identifier: doi:10.1080/01621459.1995.10476590
  • 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.
    Zentralblatt MATH: 0219.65008
    Digital Object Identifier: doi:10.1093/biomet/57.1.97
  • 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
    Zentralblatt MATH: 1042.60040
  • 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
    Zentralblatt MATH: 0824.52012
  • 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
    Zentralblatt MATH: 1225.90162
  • 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
    Zentralblatt MATH: 0716.60073
  • Leindler, L. (1972). On a certain converse of Hölder's inequality. II. Acta Sci. Math. (Szeged) 33 217–223.
    Mathematical Reviews (MathSciNet): MR2199372
    Zentralblatt MATH: 0245.26011
  • 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
    Zentralblatt MATH: 0788.60087
  • 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
    Zentralblatt MATH: 1017.60080
  • 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
    Zentralblatt MATH: 1082.15031
  • 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
    Zentralblatt MATH: 0890.60061
  • 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
    Zentralblatt MATH: 1011.60050
  • Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of `small-world' networks. Nature 393 440–442.
    Zentralblatt MATH: 1368.05139
    Digital Object Identifier: doi:10.1038/30918

The Institute of Mathematical Statistics

Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more