Statistics and Computing

, Volume 11, Issue 2, pp 103–115 | Cite as

Riemann sums for MCMC estimation and convergence monitoring

  • Anne Philippe
  • Christian P. Robert


This paper develops an extension of the Riemann sum techniques of Philippe (J. Statist. Comput. Simul. 59: 295–314) in the setup of MCMC algorithms. It shows that these techniques apply equally well to the output of these algorithms, with similar speeds of convergence which improve upon the regular estimator. The restriction on the dimension associated with Riemann sums can furthermore be overcome by Rao–Blackwellization methods. This approach can also be used as a control variate technique in convergence assessment of MCMC algorithms, either by comparing the values of alternative versions of Riemann sums, which estimate the same quantity, or by using genuine control variate, that is, functions with known expectations, which are available in full generality for constants and scores.

simulation numerical integration control variate Rao–Blackwellization score 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Besag J. 1974. Spatial interaction and the statistical analysis of lattice system. J. Royal Statist. Soc B 36: 192-326.Google Scholar
  2. Brooks S.P. 1998. MCMC convergence diagnosis via multivariate bounds on log-concave densities. Ann. Statist. 26: 398-433.Google Scholar
  3. Brooks S.P. and Gelman A. 1998. Some issues in monitoring convergence of iterative simulations. In: Proceedings of the Section on Statistical Computing, ASA.Google Scholar
  4. Casella G. 1996. Statistical inference and Monte Carlo algorithms (with discussion). TEST 5: 249-344.Google Scholar
  5. Casella G. and Robert C.P. 1996. Rao-Blackwellization of sampling schemes. Biometrika 83: 81-94.Google Scholar
  6. Chen M.M. and Shao Q.M. 1997. On Monte Carlo methods for estimating ratios of normalizing constants. Ann. Statist. 25: 1563-1594.Google Scholar
  7. Cowles M.K. and Carlin B.P. 1996. Markov chain Monte Carlo convergence diagnostics: A comparison study. J. Amer. Statist. Assoc. 91: 883-904.Google Scholar
  8. Gaver D.P. and O'Muircheartaigh I.G. 1987. Robust empirical Bayes analysis of event rate. Technometrics 29(1): 1-15.Google Scholar
  9. Gelfand A.E. and Smith A.F.M. 1990. Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85: 398-409.Google Scholar
  10. Geyer C.J. 1993. Estimating normalizing constants and reweighting mixtures in Markov chain Monte-Carlo. Technical Report 568, School of Statistics, Univ. of Minnesota.Google Scholar
  11. Mengersen K.L., Robert C.P., and Guihenneuc-Jouyaux C. 1999. MCMCconvergence diagnostics: A “rewiewww”. In: Berger J.O., Bernardo J.M., Dawid A.P., Lindley D.V., and Smith A.F.M. (Eds.), Bayesian Statistics, Vol. 6. Oxford University Press, Oxford, pp. 415-440.Google Scholar
  12. Philippe A. 1997b. Simulation output by Riemann sums. J. Statist. Comput. Simul. 59: 295-314.Google Scholar
  13. Philippe A. 1997a. Importance sampling and Riemann Sums. Pub. I.R.MA. Lille 43(VI).Google Scholar
  14. Robert C.P. and Casella G. 1999. Monte-Carlo Statistical Methods. Springer-Verlag, New-York.Google Scholar
  15. Robert C.P. 1995. Convergence control techniques for MCMC algorithms. Statis. Science 10: 231-253.Google Scholar
  16. Robert C.P. 1998. Discretization and MCMC Convergence Assessment. Springer-Verlag, New York. Lecture Notes in Statistics, Vol. 135.Google Scholar
  17. Rubinstein B. 1981. Simulation and the Monte-Carlo Method. Wiley, New York.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Anne Philippe
    • 1
  • Christian P. Robert
    • 2
  1. 1.Laboratoire de Statistique et Probabilités, EP CNRS 1765 UFR Mathématiques Bât M2Université de LILLE IVilleneuve d'AscqFrance
  2. 2.Laboratoire de StatistiqueCREST, InseeMalakoff cedexFrance

Personalised recommendations