Abstract
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.
This is a preview of subscription content, access via your institution.
References
Besag J. 1974. Spatial interaction and the statistical analysis of lattice system. J. Royal Statist. Soc B 36: 192-326.
Brooks S.P. 1998. MCMC convergence diagnosis via multivariate bounds on log-concave densities. Ann. Statist. 26: 398-433.
Brooks S.P. and Gelman A. 1998. Some issues in monitoring convergence of iterative simulations. In: Proceedings of the Section on Statistical Computing, ASA.
Casella G. 1996. Statistical inference and Monte Carlo algorithms (with discussion). TEST 5: 249-344.
Casella G. and Robert C.P. 1996. Rao-Blackwellization of sampling schemes. Biometrika 83: 81-94.
Chen M.M. and Shao Q.M. 1997. On Monte Carlo methods for estimating ratios of normalizing constants. Ann. Statist. 25: 1563-1594.
Cowles M.K. and Carlin B.P. 1996. Markov chain Monte Carlo convergence diagnostics: A comparison study. J. Amer. Statist. Assoc. 91: 883-904.
Gaver D.P. and O'Muircheartaigh I.G. 1987. Robust empirical Bayes analysis of event rate. Technometrics 29(1): 1-15.
Gelfand A.E. and Smith A.F.M. 1990. Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85: 398-409.
Geyer C.J. 1993. Estimating normalizing constants and reweighting mixtures in Markov chain Monte-Carlo. Technical Report 568, School of Statistics, Univ. of Minnesota.
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.
Philippe A. 1997b. Simulation output by Riemann sums. J. Statist. Comput. Simul. 59: 295-314.
Philippe A. 1997a. Importance sampling and Riemann Sums. Pub. I.R.MA. Lille 43(VI).
Robert C.P. and Casella G. 1999. Monte-Carlo Statistical Methods. Springer-Verlag, New-York.
Robert C.P. 1995. Convergence control techniques for MCMC algorithms. Statis. Science 10: 231-253.
Robert C.P. 1998. Discretization and MCMC Convergence Assessment. Springer-Verlag, New York. Lecture Notes in Statistics, Vol. 135.
Rubinstein B. 1981. Simulation and the Monte-Carlo Method. Wiley, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Philippe, A., Robert, C.P. Riemann sums for MCMC estimation and convergence monitoring. Statistics and Computing 11, 103–115 (2001). https://doi.org/10.1023/A:1008926514119
Issue Date:
DOI: https://doi.org/10.1023/A:1008926514119
- simulation
- numerical integration
- control variate
- Rao–Blackwellization
- score