Markov chain Monte Carlo: Can we trust the third significant figure?

Statistical Science

Volumn 23, Issue 2, 2008, Pages 250-260

  Flegal, James M ^a   Haran, Murali ^b   Jones, Galin L ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Convergence diagnostic; Markov chain; Monte Carlo; Standard errors

Indexed keywords

EID: 50849118196     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-STS257     Document Type: Article

References (49)

1. BRATLEY, P., FOX, B. L. and SCHRAGE, L. E. (1987). A Guide to Simulation. Springer, New York.
2. BROOKS, S. P. and GELMAN, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434-455. MR1665662
3. CHEN, M.-H., SHAO, Q.-M. and IBRAHIM, J. G. (2000). Monte Carlo Methods in Bayesian Computation. Springer, New York. MR1742311
4. CHRISTENSEN, O. F., MOLLER, J. and W AAGEPETERSEN, R. P. (2001). Geometric ergodicity of Metropolis-Hastings algorithms for conditional simulation in generalized linear mixed models. Methodol. Comput. Appl. Probab. 3 309-327. MR1891114
5. COWLES, M. K. and CARLIN, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. J. Amer. Statist. Assoc. 91 883-904. MR1395755
6. COWLES, M. K., ROBERTS, G. O. and ROSENTHAL, J. S. (1999). Possible biases induced by MCMC convergence diagnostics. J. Statist. Comput. Simul. 64 87-104. MR1741840
7. DOUC, R., FORT, G., MOULINES, E. and SOULIER, P. (2004). Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 1353-1377. MR2071426
8. FINLEY, A. O., BANERJEE, S. and CARLIN, B. P. (2007). sp-Bayes: an R package for univariate and multivariate hierarchical point-referenced spatial models. J. Statist. Software 19.
9. FISHMAN, G. S. (1996). Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York. MR1392474
10. FORT, G. and MOULINES, E. (2000). V-subgeometric ergodicity for a Hastings-Metropolis algorithm. Statist. Probab. Lett. 49 401-410. MR1796485
11. FORT, G. and MOULINES, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57-99. MR1947960
12. GELMAN, A., CARLIN, J. B., STERN, H. S. and RUBIN, D. B. (2004). Bayesian Data Analysis, 2nd ed. Chapman and Hall/CRC, Boca Raton, FL. MR2027492
13. GELMAN, A. and RUBIN, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457-472.
14. GEYER, C. J. (1992). Practical Markov chain Monte Carlo (with discussion). Statist. Sci. 7 473-511.
15. 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 and Hall/CRC, Boca Raton, FL. MR1673118
16. 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.
17. GLYNN, P. W. and IGLEHART, D. L. (1990). Simulation output analysis using standardized time series. Math. Oper. Res. 15 1-16. MR1038232
18. GLYNN, P. W. and WHITT, W. (1991). Estimating the asymptotic variance with batch means. Oper. Res. Lett. 10 431-435. MR1141337
19. GLYNN, P. W. and WHITT, W. (1992). The asymptotic validity of sequential stopping rules for stochastic simulations. Ann. Appl. Probab. 2 180-198. MR1143399
20. HARAN, M., BHAT, K., MOLINEROS, J. and DE WOLF, E. (2007). Estimating the risk of a crop epidemic from coincident spatiotemporal processes. Technical report, Dept. Statistics, Pennsylvania State Univ.
21. HOAGLIN, D. C. and ANDREWS, D. F. (1975). The reporting of computation-based results in statistics. Amer. Statist. 29 122-126.
22. 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. MR1659196
23. IHAKA, R. and GENTLEMAN, R. (1996). R: A language for data analysis and graphics. J. Comput. Graph. Statist. 5 299-314.
24. JARNER, S. F. and HANSEN, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. MR1731030
25. JARNER, S. F. and ROBERTS, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224-247. MR1890063
26. 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.
27. JONES, G. L. (2004). On the Markov chain central limit theorem. Probab. Surv. 1 299-320. MR2068475
28. 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. MR2279478
29. JONES, G. L. and HOBERT, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312-334. MR1888447
30. 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. MR2060178
31. L'ECUYER, P., SIMARD, R., CHEN, E. J. and KELTON, W. D. (2002). An objected-oriented random-number package with many long streams and substreams. Oper. es. 50 1073-1075.
32. LIU, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York. MR1842342
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. MR2054301
34. MARINARI, E. and PARISI, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhys. Lett. 19 451-458.
35. MENGERSEN, K. and TWEEDIE, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121. MR1389882
36. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. MR1287609
37. MEYN, S. P. and TWEEDIE, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011. MR1304770
38. MIRA, A. and TIERNEY, L. (2002). Efficiency and convergence properties of slice samplers. Scand. J. Statist. 29 1-12. MR1894377
39. MYKLAND, P., TIERNEY, L. and YU, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90 233-241. MR1325131
40. ROBERT, C. P. (1995). Convergence control methods for Markov chain Monte Carlo algorithms. Statist. Sci. 10 231-253. MR1390517
41. ROBERT, C. P. and CASELLA, G. (1999). Monte Carlo Statistical Methods. Springer, New York. MR1707311
42. ROBERTS, G. O. (1996). Markov chain concepts related to sampling algorithms. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. E. Spiegelhalter, eds.) 45-57. Chapman and Hall, London. MR1397967
43. 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. MR1281941
44. ROBERTS, G. O. and ROSENTHAL, J. S. (1999). Convergence of slice sampler Markov chains. J. Roy. Statist. Soc. Ser. B 61 643-660. MR1707866
45. ROBERTS, G. O. and ROSENTHAL, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. MR2095565
46. ROSENTHAL, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558-566. MR1340509
47. ROSENTHAL, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Statist. Comput. 6 269-275.
48. ROY, V. and HOBERT, J. P. (2007). Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression. J. Roy. Statist. Soc. Ser. B 69 607-623. MR2370071
49. TIERNEY, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. MR1329166