×

Batch means and spectral variance estimators in Markov chain Monte Carlo. (English) Zbl 1184.62161

Summary: Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

MSC:

62M15 Inference from stochastic processes and spectral analysis
60J22 Computational methods in Markov chains
65C05 Monte Carlo methods
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, T. W. (1994). The Statistical Analysis of Time Series . Wiley, New York. · Zbl 0835.62074
[2] Bednorz, W. and Latuszyński, K. (2007). A few remarks on “Fixed-width output analysis for Markov chain Monte Carlo” by Jones et al. J. Amer. Statist. Assoc. 102 1485-1486.
[3] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002
[4] Bratley, P., Fox, B. L. and Schrage, L. E. (1987). A Guide to Simulation . Springer, New York. · Zbl 0515.68070
[5] Chan, K. S. and Geyer, C. J. (1994). Comment on “Markov chains for exploring posterior distributions.” Ann. Statist. 22 1747-1758. · Zbl 0829.62080 · doi:10.1214/aos/1176325750
[6] Chien, C.-H., Goldsman, D. and Melamed, B. (1997). Large-sample results for batch means. Manag. Sci. 43 1288-1295. · Zbl 1043.90512 · doi:10.1287/mnsc.43.9.1288
[7] Csáki, E. and Csörgő, M. (1995). On additive functionals of Markov chains. J. Theoret. Probab. 8 905-919. · Zbl 0834.60073 · doi:10.1007/BF02410117
[8] Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics . Academic Press, New York. · Zbl 0539.60029
[9] Damerdji, H. (1991). Strong consistency and other properties of the spectral variance estimator. Manag. Sci. 37 1424-1440. · Zbl 0741.62086 · doi:10.1287/mnsc.37.11.1424
[10] Damerdji, H. (1994). Strong consistency of the variance estimator in steady-state simulation output analysis. Math. Oper. Res. 19 494-512. JSTOR: · Zbl 0803.65147 · doi:10.1287/moor.19.2.494
[11] Damerdji, H. (1995). Mean-square consistency of the variance estimator in steady-state simulation output analysis. Oper. Res. 43 282-291. JSTOR: · Zbl 0830.62077 · doi:10.1287/opre.43.2.282
[12] Flegal, J. M., Haran, M. and Jones, G. L. (2008). Markov chain Monte Carlo: Can we trust the third significant figure? Statist. Sci. 23 250-260. · Zbl 1327.62017 · doi:10.1214/08-STS257
[13] Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry: Likelihood and Computation (O. E. Barndorff-Nielsen, W. S. Kendall and M. N. M. van Lieshout, eds.) 79-140. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 0809.62089
[14] Gilks, W. R., Roberts, G. O. and Sahu, S. K. (1998). Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc. 93 1045-1054. JSTOR: · Zbl 1064.65503 · doi:10.2307/2669848
[15] Glynn, P. W. and Iglehart, D. L. (1990). Simulation output analysis using standardized time series. Math. Oper. Res. 15 1-16. JSTOR: · Zbl 0704.65110 · doi:10.1287/moor.15.1.1
[16] Glynn, P. W. and Whitt, W. (1991). Estimating the asymptotic variance with batch means. Oper. Res. Lett. 10 431-435. · Zbl 0744.62113 · doi:10.1016/0167-6377(91)90019-L
[17] Glynn, P. W. and Whitt, W. (1992). The asymptotic validity of sequential stopping rules for stochastic simulations. Ann. Appl. Probab. 2 180-198. · Zbl 0792.68200 · doi:10.1214/aoap/1177005777
[18] Häggström, O. (2004). On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Related Fields 132 74-82. · Zbl 1082.60013 · doi:10.1007/s00440-004-0390-7
[19] Hobert, J. P. and Geyer, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model. J. Multivariate Anal. 67 414-430. · Zbl 0922.60069 · doi:10.1006/jmva.1998.1778
[20] 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. JSTOR: · Zbl 1035.60080 · doi:10.1093/biomet/89.4.731
[21] Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. · Zbl 0997.60070 · doi:10.1016/S0304-4149(99)00082-4
[22] Johnson, A. A. and Jones, G. L. (2008). Gibbs sampling for a Bayesian hierarchical version of the general linear mixed model. Technical report, School of Statistics, Univ. Minnesota.
[23] Jones, G. L. (2004). On the Markov chain central limit theorem. Probab. Surv. 1 299-320. · Zbl 1189.60129 · doi:10.1214/154957804100000051
[24] Jones, G. L., Haran, M., Caffo, B. S. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101 1537-1547. · Zbl 1171.62316 · doi:10.1198/016214506000000492
[25] Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312-334. · Zbl 1127.60309 · doi:10.1214/ss/1015346317
[26] 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. · Zbl 1048.62069 · doi:10.1214/009053604000000184
[27] Kendall, M. G. and Stuart, A. (1977). The Advanced Theory of Statistics . Vol. I: Distribution Theory , 4th ed.; Vol. 2: Inference and Relationship , 3rd ed. Charles Griffin, London. · Zbl 0353.62013
[28] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RVs and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111-131. · Zbl 0308.60029 · doi:10.1007/BF00533093
[29] Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33-58. · Zbl 0307.60045 · doi:10.1007/BF00532688
[30] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing . Springer, New York. · Zbl 0991.65001
[31] Liu, J. S. and Wu, Y. N. (1999). Parameter expansion for data augmentation. J. Amer. Statist. Assoc. 94 1264-1274. JSTOR: · Zbl 1069.62514 · doi:10.2307/2669940
[32] Major, P. (1976). The approximation of partial sums of independent RV’s. Z. Wahrsch. Verw. Gebiete 35 213-220. · Zbl 0338.60031 · doi:10.1007/BF00532673
[33] Marchev, D. and Hobert, J. P. (2004). Geometric ergodicity of van Dyk and Meng’s algorithm for the multivariate Student’s t model. J. Amer. Statist. Assoc. 99 228-238. · Zbl 1089.60518 · doi:10.1198/016214504000000223
[34] Meketon, M. S. and Schmeiser, B. (1984). Overlapping batch means: Something for nothing? In WSC ’84: Proceedings of the 16th Conference on Winter Simulation 226-230. IEEE Press, Piscataway, NJ.
[35] Mengersen, K. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121. · Zbl 0854.60065 · doi:10.1214/aos/1033066201
[36] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001
[37] Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011. · Zbl 0812.60059 · doi:10.1214/aoap/1177004900
[38] Mykland, P., Tierney, L. and Yu, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90 233-241. JSTOR: · Zbl 0819.62082 · doi:10.2307/2291148
[39] Neath, R. and Jones, G. L. (2009). Variable-at-a-time implementations of Metropolis-Hastings. Technical report, School of Statistics, Univ. Minnesota.
[40] Philipp, W. and Stout, W. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2 1-140. · Zbl 0361.60007
[41] Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods . Springer, New York. · Zbl 0935.62005
[42] Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. Roy. Statist. Soc. Ser. B 56 377-384. JSTOR: · Zbl 0796.62029
[43] Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 643-660. JSTOR: · Zbl 0929.62098 · doi:10.1111/1467-9868.00198
[44] Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. · Zbl 1189.60131 · doi:10.1214/154957804100000024
[45] Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558-566. JSTOR: · Zbl 0824.60077 · doi:10.2307/2291067
[46] Rosenthal, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Stat. Comput. 6 269-275.
[47] Roy, V. and Hobert, J. P. (2007). Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 607-623. · doi:10.1111/j.1467-9868.2007.00602.x
[48] Song, W. T. and Schmeiser, B. W. (1995). Optimal mean-squared-error batch sizes. Manag. Sci. 41 110-123. · Zbl 0819.62076 · doi:10.1287/mnsc.41.1.110
[49] Tan, A. and Hobert, J. P. (2009). Block Gibbs sampling for Bayesian random effects models with improper priors: Convergence and regeneration. J. Comput. Graph. Statist.
[50] Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. · Zbl 0829.62080 · doi:10.1214/aos/1176325750
[51] van Dyk, D. A. and Meng, X.-L. (2001). The art of data augmentation (with discussion). J. Comput. Graph. Statist. 10 1-50. JSTOR: · Zbl 04565162 · doi:10.1198/10618600152418584
[52] Welch, P. D. (1987). On the relationship between batch means, overlapping means and spectral estimation. In WSC ’87: Proceedings of the 19th Conference on Winter Simulation 320-323. ACM, New York.
[53] Whittle, P. (1960). Bounds for the moments of linear and quadratic forms in independent variables. Theory Probab. Appl. 5 302-305. · Zbl 0101.12003
[54] Zeidler, E. (1990). Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators , 2nd ed. Spring, New York. · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.