Statistical Science

Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?

James M. Flegal, Murali Haran, and Galin L. Jones

Full-text: Open access


Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Article information

Statist. Sci. Volume 23, Number 2 (2008), 250-260.

First available: 21 August 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Flegal, James M.; Haran, Murali; Jones, Galin L. Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?. Statistical Science 23 (2008), no. 2, 250--260. doi:10.1214/08-STS257.

Export citation


  • Bratley, P., Fox, B. L. and Schrage, L. E. (1987). A Guide to Simulation. Springer, New York.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Chen, M.-H., Shao, Q.-M. and Ibrahim, J. G. (2000). Monte Carlo Methods in Bayesian Computation. Springer, New York.
  • Christensen, O. F., Moller, J. and Waagepetersen, R. P. (2001). Geometric ergodicity of Metropolis–Hastings algorithms for conditional simulation in generalized linear mixed models. Methodol. Comput. Appl. Probab. 3 309–327.
  • Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. J. Amer. Statist. Assoc. 91 883–904.
  • 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.
  • Douc, R., Fort, G., Moulines, E. and Soulier, P. (2004). Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 1353–1377.
  • Finley, A. O., Banerjee, S. and Carlin, B. P. (2007). spBayes: an R package for univariate and multivariate hierarchical point-referenced spatial models. J. Statist. Software 19.
  • Fishman, G. S. (1996). Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York.
  • Fort, G. and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings–Metropolis algorithm. Statist. Probab. Lett. 49 401–410.
  • Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57–99.
  • 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.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • Geyer, C. J. (1992). Practical Markov chain Monte Carlo (with discussion). Statist. Sci. 7 473–511.
  • 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.
  • 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.
  • Glynn, P. W. and Iglehart, D. L. (1990). Simulation output analysis using standardized time series. Math. Oper. Res. 15 1–16.
  • Glynn, P. W. and Whitt, W. (1991). Estimating the asymptotic variance with batch means. Oper. Res. Lett. 10 431–435.
  • Glynn, P. W. and Whitt, W. (1992). The asymptotic validity of sequential stopping rules for stochastic simulations. Ann. Appl. Probab. 2 180–198.
  • 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.
  • Hoaglin, D. C. and Andrews, D. F. (1975). The reporting of computation-based results in statistics. Amer. Statist. 29 122–126.
  • 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.
  • Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. J. Comput. Graph. Statist. 5 299–314.
  • Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341–361.
  • Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224–247.
  • 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.
  • Jones, G. L. (2004). On the Markov chain central limit theorem. Probab. Surv. 1 299–320.
  • 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.
  • Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312–334.
  • 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.
  • 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. Res. 50 1073–1075.
  • Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • 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.
  • Marinari, E. and Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhys. Lett. 19 451–458.
  • Mengersen, K. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981–1011.
  • Mira, A. and Tierney, L. (2002). Efficiency and convergence properties of slice samplers. Scand. J. Statist. 29 1–12.
  • Mykland, P., Tierney, L. and Yu, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90 233–241.
  • Robert, C. P. (1995). Convergence control methods for Markov chain Monte Carlo algorithms. Statist. Sci. 10 231–253.
  • Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.
  • 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.
  • 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.
  • Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. J. Roy. Statist. Soc. Ser. B 61 643–660.
  • Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20–71.
  • Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558–566.
  • Rosenthal, J. S. (1996). Analysis of the Gibbs sampler for a model related to James–Stein estimators. Statist. Comput. 6 269–275.
  • 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.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762.