The Annals of Statistics

Consistency of Markov chain quasi-Monte Carlo on continuous state spaces

S. Chen, J. Dick, and A. B. Owen

Full-text: Open access


The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0, 1) random variables. Tribble [Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences (2007) Stanford Univ.] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The previous theoretical justification for using anything other than i.i.d. U(0, 1) points shows consistency for estimated means, but only applies for discrete stationary distributions. We extend those results to some MCMC algorithms for continuous stationary distributions. The main motivation is the search for quasi-Monte Carlo versions of MCMC. As a side benefit, the results also establish consistency for the usual method of using pseudo-random numbers in place of random ones.

Article information

Ann. Statist. Volume 39, Number 2 (2011), 673-701.

First available in Project Euclid: 9 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C40: Computational Markov chains 62F15: Bayesian inference
Secondary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 65C05: Monte Carlo methods

Completely uniformly distributed coupling iterated function mappings Markov chain Monte Carlo


Chen, S.; Dick, J.; Owen, A. B. Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. Ann. Statist. 39 (2011), no. 2, 673--701. doi:10.1214/10-AOS831.

Export citation


  • [1] Albert, J. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • [2] Alsmeyer, G. and Fuh, C.-D. (2001). Limit theorems for iterated function mappings. Stochastic Process. Appl. 96 123–142.
  • [3] Andrieu, C. and Moulines, E. (2006). On the ergodicity properties of some MCMC algorithms. Ann. Appl. Probab. 16 1462–1505.
  • [4] Ash, R. B. (1972). Real Analysis and Probability. Academic Press, New York.
  • [5] Billingsley, P. (1999). Convergence of Probability Measures. Wiley, New York.
  • [6] Chaudary, S. (2004). Acceleration of Monte Carlo methods using low discrepancy sequences. Ph.D. thesis, UCLA.
  • [7] Chentsov, N. N. (1967). Pseudorandom numbers for modelling Markov chains. Comput. Math. Math. Phys. 7 218–2332.
  • [8] Craiu, R. V. and Lemieux, C. (2007). Acceleration of the multiple-try Metropolis algorithm using antithetic and stratified sampling. Stat. Comput. 17 109–120.
  • [9] Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.
  • [10] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45–76.
  • [11] Dick, J. (2007). A note on the existence of sequences with small star discrepancy. J. Complexity 23 649–652.
  • [12] Dick, J. (2009). On quasi-Monte Carlo rules achieving higher order convergence. In Monte Carlo and Quasi-Monte Carlo Methods 2008 (P. L’Ecuyer and A. B. Owen, eds.) 73–96. Springer, Heidelberg.
  • [13] Doerr, B. and Friedrich, T. (2009). Deterministic random walks on the two-dimensional grid. Combin. Probab. Comput. 18 123–144.
  • [14] Finney, D. J. (1947). The estimation from individual records of the relationship between dose and quantal response. Biometrika 34 320–334.
  • [15] Gaver, D. and O’Murcheartaigh, I. (1987). Robust empirical Bayes analysis of event rates. Technometrics 29 1–15.
  • [16] Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
  • [17] Gelman, A. and Shirley, K. (2010). Inference from simulations and monitoring convergence. In Handbook of Markov Chain Monte Carlo: Methods and Applications. (S. Brooks, A. Gelman, G. Jones and X.-L. Meng, eds.) 131–143. Chapman and Hall/CRC Press, Boca Raton, FL.
  • [18] Ghorpade, S. R. and Limaye, B. V. (2006). A Course in Calculus and Real Analysis. Springer, New York.
  • [19] Gnewuch, M., Srivastav, A. and Winzen, C. (2008). Finding optimal volume subintervals with k points and computing the star discrepancy are NP-hard. J. Complexity 24 154–172.
  • [20] Gordon, R. D. (1941). Value of Mill’s ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. Ann. Math. Statist. 18 364–366.
  • [21] Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223–242.
  • [22] Knuth, D. E. (1998). The Art of Computer Programming, 3rd ed. Seminumerical Algorithms 2. Addison-Wesley, Reading, MA.
  • [23] Lebesgue, H. L. (1902). Intégrale, longueur, aire. Ph.D. thesis, Univ. de Paris.
  • [24] L’Ecuyer, P., Lecot, C. and Tuffin, B. (2008). A randomized quasi-Monte Carlo simulation method for Markov chains. Oper. Res. 56 958–975.
  • [25] L’Ecuyer, P. and Lemieux, C. (1999). Quasi-Monte Carlo via linear shift-register sequences. In Proceedings of the 1999 Winter Simulation Conference (P. A. Farrington, H. B. Nembhard, D. T. Sturrock and G. W. Evans, eds.) 632–639. IEEE Press, Piscataway, NJ.
  • [26] Lemieux, C. and Sidorsky, P. (2006). Exact sampling with highly uniform point sets. Math. Comput. Modelling 43 339–349.
  • [27] Liao, L. G. (1998). Variance reduction in Gibbs sampler using quasi random numbers. J. Comput. Graph. Statist. 7 253–266.
  • [28] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • [29] Matsumoto, M. and Nishimura, T. (1998). Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions on Modeling and Computer Simulation 8 3–30.
  • [30] Marsden, J. E. and Hoffman, M. J. (1993). Elementary Classical Analysis, 2nd ed. Macmillan, New York.
  • [31] Morokoff, W. and Caflisch, R. E. (1993). A quasi-Monte Carlo approach to particle simulation of the heat equ ation. SIAM J. Numer. Anal. 30 1558–1573.
  • [32] Neal, R. M. (2003). Slice sampling. Ann. Statist. 31 705–767.
  • [33] Niederreiter, H. (1986). Multidimensional integration using pseudo-random numbers. Math. Programming Stud. 27 17–38.
  • [34] Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, PA.
  • [35] Owen, A. B. (1995). Randomly permuted (t, m, s)-nets and (t, s)-sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. Jau-Shyong Shiue, eds.) 299–317. Springer, New York.
  • [36] Owen, A. B. (2005). Multidimensional variation for quasi-Monte Carlo. In Contemporary Multivariate Analysis and Design of Experiments: In Celebration of Prof. Kai-Tai Fang’s 65th Birthday (J. Fan and G. Li, eds.). World Sci. Publ., Hackensack, NJ.
  • [37] Owen, A. B. and Tribble, S. D. (2005). A quasi-Monte Carlo Metropolis algorithm. Proc. Natl. Acad. Sci. USA 102 8844–8849.
  • [38] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains. Random Structures and Algorithms 9 223–252.
  • [39] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [40] Roberts, G. O., Rosenthal, J. S. and Schwartz, P. O. (1998). Convergence properties of perturbed Markov chains. J. Appl. Probab. 35 1–11.
  • [41] Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23 470–472.
  • [42] Sobol’, I. M. (1974). Pseudo-random numbers for constructing discrete Markov chains by the Monte Carlo method. USSR Comput. Math. Math. Phys. 14 36–45.
  • [43] Tribble, S. D. (2007). Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. Ph.D. thesis, Stanford Univ.
  • [44] Tribble, S. D. and Owen, A. B. (2008). Construction of weakly CUD sequences for MCMC sampling. Electron. J. Stat. 2 634–660.
  • [45] Weyl, H. (1916). Über die gleichverteilung von zahlen mod. eins. Math. Ann. 77 313–352.