Variance bounding Markov chains
Gareth O. Roberts and Jeffrey S. Rosenthal
Source: Ann. Appl. Probab. Volume 18, Number 3
(2008), 1201-1214.
Abstract
We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all L2 functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis–Hastings algorithms.
First Page:
Show
Hide
Keywords: Markov chain Monte Carlo; Metropolis–Hastings algorithm; central limit theorem; variance; Peskun order; geometric ergodicity; spectrum
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1211819798
Digital Object Identifier: doi:10.1214/07-AAP486
Mathematical Reviews number (MathSciNet): MR2418242
Zentralblatt MATH identifier: 1142.60047
References
[1] Baxter, J. R. and Rosenthal, J. S. (1995). Rates of convergence for everywhere-positive Markov chains. Statist. Probab. Lett. 22 333–338.
Mathematical Reviews (MathSciNet): MR1333192
[2] Bradley, R. C. (2005). Basic properties of strong mixing conditions: A survey and some open questions. Probab. Surv. 2 107–144.
Mathematical Reviews (MathSciNet): MR2178042
Digital Object Identifier: doi:10.1214/154957805100000104
Project Euclid: euclid.ps/1115386870
[3] Chan, K. S. and Geyer, C. J. (1994). Discussion to “Markov chains for exploring posterior distributions” by L. Tierney. Ann. Statist. 22 1747–1758.
Mathematical Reviews (MathSciNet): MR1329166
Digital Object Identifier: doi:10.1214/aos/1176325750
Project Euclid: euclid.aos/1176325750
Zentralblatt MATH: 0829.62080
[4] Conway, J. B. (1985). A Course in Functional Analysis. Springer, New York.
Mathematical Reviews (MathSciNet): MR768926
Zentralblatt MATH: 0558.46001
[5] Diaconis, P., Holmes, S. and Neal, R. M. (2000). Analysis of a non-reversible Markov chain sampler. Ann. Appl. Probab. 10 726–752.
Mathematical Reviews (MathSciNet): MR1789978
Digital Object Identifier: doi:10.1214/aoap/1019487508
Project Euclid: euclid.aoap/1019487508
Zentralblatt MATH: 1083.60516
[6] Fill, J. A. (1991). Eigenvalue bounds on convergence to stationarity for non-reversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1 62–87.
Mathematical Reviews (MathSciNet): MR1097464
Digital Object Identifier: doi:10.1214/aoap/1177005981
Project Euclid: euclid.aoap/1177005981
Zentralblatt MATH: 0726.60069
[7] Geyer, C. J. (1992). Practical Markov chain Monte Carlo. Statist. Sci. 7 473–483.
[8] Gilks, W. R. and Roberts, G. O. (1995). Strategies for improving MCMC. In MCMC in Practice (W. R. Gilks, D. J. Spiegelhalter and S. Richardson, eds.) 89–114. Chapman and Hall, London.
[9] Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
[10] Hobert, J. P., Jones, G. L., Presnell, B. and Rosenthal, J. S. (2002). On the applicability of regenerative simulation in Markov chain Monte carlo. Biometrika 89 731–743.
Mathematical Reviews (MathSciNet): MR1946508
Zentralblatt MATH: 1035.60080
Digital Object Identifier: doi:10.1093/biomet/89.4.731
JSTOR: links.jstor.org
[11] Hobert, J. P. and Rosenthal, J. S. (2007). Norm comparisons for data augmentation. Preprint.
Mathematical Reviews (MathSciNet): MR2396335
[12] Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
Mathematical Reviews (MathSciNet): MR322926
Zentralblatt MATH: 0219.60027
[13] Jones, G. L. (2004). On the Markov chain central limit theorem. Probab. Surv. 1 299–320.
Mathematical Reviews (MathSciNet): MR2068475
Digital Object Identifier: doi:10.1214/154957804100000051
Project Euclid: euclid.ps/1104335301
[14] Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312–334.
Mathematical Reviews (MathSciNet): MR1888447
Digital Object Identifier: doi:10.1214/ss/1015346317
Project Euclid: euclid.ss/1015346317
Zentralblatt MATH: 1127.60309
[15] Jones, G. L. and Hobert, J. P. (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. Ann. Statist. 32 784–817.
Mathematical Reviews (MathSciNet): MR2060178
Digital Object Identifier: doi:10.1214/009053604000000184
Project Euclid: euclid.aos/1083178947
Zentralblatt MATH: 1048.62069
[16] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
Mathematical Reviews (MathSciNet): MR834478
Digital Object Identifier: doi:10.1007/BF01210789
Project Euclid: euclid.cmp/1104114929
Zentralblatt MATH: 0588.60058
[17] Lawler, G. F. and Sokal, A. D. (1988). Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 557–580.
Mathematical Reviews (MathSciNet): MR930082
Digital Object Identifier: doi:10.2307/2000925
JSTOR: links.jstor.org
Zentralblatt MATH: 0716.60073
[18] Liu, J. S., Wong, W. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81 27–40.
Mathematical Reviews (MathSciNet): MR1279653
Zentralblatt MATH: 0811.62080
Digital Object Identifier: doi:10.1093/biomet/81.1.27
JSTOR: links.jstor.org
[19] Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
Mathematical Reviews (MathSciNet): MR1389882
Digital Object Identifier: doi:10.1214/aos/1033066201
Project Euclid: euclid.aos/1033066201
Zentralblatt MATH: 0854.60065
[20] Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21 1087–1091.
[21] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
Mathematical Reviews (MathSciNet): MR1287609
Zentralblatt MATH: 0925.60001
[22] Mira, A. (2001). Ordering and improving the performance of Monte Carlo Markov chains. Statist. Sci. 16 340–350.
Mathematical Reviews (MathSciNet): MR1888449
Digital Object Identifier: doi:10.1214/ss/1015346319
Project Euclid: euclid.ss/1015346319
Zentralblatt MATH: 1127.60312
[23] Mira, A. and Geyer, C. J. (2000). On non-reversible Markov chains. In Fields Institute Communications 26. Monte Carlo Methods (N. Madras, ed.) 93–108. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1772309
Zentralblatt MATH: 0969.60071
[24] Mira, A., Møller, J. and Roberts, G. O. (2001). Perfect slice samplers. J. Roy. Statist. Soc. Ser. B 63 593–606.
[25] Neal, R. M. (2003). Slice sampling (with discussion). Ann. Statist. 31 705–767.
Mathematical Reviews (MathSciNet): MR1994729
Digital Object Identifier: doi:10.1214/aos/1056562461
Project Euclid: euclid.aos/1056562461
Zentralblatt MATH: 1051.65007
[26] Peskun, P. H. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60 607–612.
Mathematical Reviews (MathSciNet): MR362823
Zentralblatt MATH: 0271.62041
Digital Object Identifier: doi:10.1093/biomet/60.3.607
JSTOR: links.jstor.org
[27] Roberts, G. O. (1999). A note on acceptance rate criteria for CLTs for Metropolis–Hastings algorithms. J. Appl. Probab. 36 1210–1217.
Mathematical Reviews (MathSciNet): MR1742161
Digital Object Identifier: doi:10.1239/jap/1032374766
Project Euclid: euclid.jap/1032374766
Zentralblatt MATH: 0966.65006
[28] Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 13–25.
Mathematical Reviews (MathSciNet): MR1448322
[29] Roberts, G. O. and Rosenthal, J. S. (1998). Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canad. J. Statist. 26 5–31.
Mathematical Reviews (MathSciNet): MR1624414
Digital Object Identifier: doi:10.2307/3315667
JSTOR: links.jstor.org
[30] Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. J. Roy. Statist. Soc. Ser. B 61 643–660.
Mathematical Reviews (MathSciNet): MR1707866
Digital Object Identifier: doi:10.1111/1467-9868.00198
JSTOR: links.jstor.org
Zentralblatt MATH: 0929.62098
[31] Roberts, G. O. and Rosenthal, J. S. (2006). Examples of adaptive MCMC. Preprint.
[32] Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110.
Mathematical Reviews (MathSciNet): MR1399158
Zentralblatt MATH: 0888.60064
Digital Object Identifier: doi:10.1093/biomet/83.1.95
JSTOR: links.jstor.org
[33] Roberts, G. O. and Tweedie, R. L. (1996). Exponential convergence of Langevin diffusions and their discrete approximations. Bernoulli 2 341–364.
Mathematical Reviews (MathSciNet): MR1440273
Digital Object Identifier: doi:10.2307/3318418
Project Euclid: euclid.bj/1178291835
Zentralblatt MATH: 0870.60027
[34] Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558–566.
Mathematical Reviews (MathSciNet): MR1340509
Digital Object Identifier: doi:10.2307/2291067
JSTOR: links.jstor.org
Zentralblatt MATH: 0824.60077
[35] Rosenthal, J. S. (2002). Quantitative convergence rates of Markov chains: A simple account. Electron. Comm. Probab. 7 123–128.
Mathematical Reviews (MathSciNet): MR1917546
[36] Rosenthal, J. S. (2003). Asymptotic variance and convergence rates of nearly-periodic MCMC algorithms. J. Amer. Statist. Assoc. 98 169–177.
Mathematical Reviews (MathSciNet): MR1965683
Digital Object Identifier: doi:10.1198/016214503388619193
Zentralblatt MATH: 1048.60057
[37] Häggström, O. and Rosenthal, J. S. (2007). On variance conditions for Markov chain CLTs. Electron. Comm. Probab. To appear.
Mathematical Reviews (MathSciNet): MR2365647
[38] Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet): MR1157815
Zentralblatt MATH: 0867.46001
[39] Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B 55 3–24.
Mathematical Reviews (MathSciNet): MR1210421
JSTOR: links.jstor.org
[40] Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762.
Mathematical Reviews (MathSciNet): MR1329166
Digital Object Identifier: doi:10.1214/aos/1176325750
Project Euclid: euclid.aos/1176325750
Zentralblatt MATH: 0829.62080
[41] Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. Ann. Appl. Probab. 8 1–9.
Mathematical Reviews (MathSciNet): MR1620401
Digital Object Identifier: doi:10.1214/aoap/1027961031
Project Euclid: euclid.aoap/1027961031
Zentralblatt MATH: 0935.60053
The Annals of Applied Probability
