Electronic Journal of Probability

Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo

Josef Dick and Daniel Rudolf

Full-text: Open access

Abstract

Markov chain Monte Carlo (MCMC) simulations are modeled as driven by true random numbers. We consider variance bounding Markov chains driven by a deterministic sequence of numbers. The star-discrepancy provides a measure of efficiency of such Markov chain quasi-Monte Carlo methods. We define a pull-back discrepancy of the driver sequence and state a close relation to the star-discrepancy of the Markov chain-quasi Monte Carlo samples. We prove that there exists a deterministic driver sequence such that the discrepancies decrease almost with the Monte Carlo rate $n^{-1/2}$. As for MCMC simulations,  a burn-in period can also be taken into account for Markov chain quasi-Monte Carlo to reduce the influence of the initial state. In particular, our discrepancy bound leads to an estimate of the error for the computation of expectations. To illustrate our theory we provide an example for the Metropolis algorithm based on a ball walk. Furthermore, under additional assumptions we prove the existence of a driver sequence such that the discrepancy of the corresponding deterministic Markov chain sample decreases with order $n^{-1+\delta}$ for every $\delta>0$.

Article information

Source
Electron. J. Probab. Volume 19 (2014), paper no. 105, 24 pp.

Dates
Accepted: 5 November 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065747

Digital Object Identifier
doi:10.1214/EJP.v19-3132

Mathematical Reviews number (MathSciNet)
MR3275857

Zentralblatt MATH identifier
1309.65004

Subjects
Primary: 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 65C40: Computational Markov chains 62F15: Bayesian inference 65C05: Monte Carlo methods 60J05: Discrete-time Markov processes on general state spaces

Keywords
Markov chain Monte Carlo Markov chain quasi-Monte Carlo variance bounding discrepancy theory spectral gap probabilistic method

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Dick, Josef; Rudolf, Daniel. Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo. Electron. J. Probab. 19 (2014), paper no. 105, 24 pp. doi:10.1214/EJP.v19-3132. https://projecteuclid.org/euclid.ejp/1465065747.


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References

  • Aistleitner, Christoph; Dick, Josef. Low-discrepancy point sets for non-uniform measures. Acta Arith. 163 (2014), no. 4, 345–369.
  • Aronszajn, N. Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, (1950). 337–404.
  • Beck, József. Some upper bounds in the theory of irregularities of distribution. Acta Arith. 43 (1984), no. 2, 115–130.
  • Brauchart, Johann S.; Dick, Josef. A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness. Constr. Approx. 38 (2013), no. 3, 397–445.
  • S. Chen, Consistency and convergence rate of Markov chain quasi-Monte Carlo with examples, Ph.D. thesis, Stanford University, 2011.
  • 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.
  • Chen, Su; Matsumoto, Makoto; Nishimura, Takuji; Owen, Art B. New inputs and methods for Markov chain quasi-Monte Carlo. Monte Carlo and quasi-Monte Carlo methods 2010, 313–327, Springer Proc. Math. Stat., 23, Springer, Heidelberg, 2012.
  • J. Dick, D. Rudolf, and H. Zhu, phDiscrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo, Preprint, Available at http://arxiv.org/abs/1303.2423 (2013).
  • Dudley, R. M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 1967 290–330.
  • Fang, K.-T.; Wang, Y. Number-theoretic methods in statistics. Monographs on Statistics and Applied Probability, 51. Chapman & Hall, London, 1994. xii+340 pp. ISBN: 0-412-46520-5.
  • Gnewuch, Michael. Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy. J. Complexity 24 (2008), no. 2, 154–172.
  • Haussler, David. Sphere packing numbers for subsets of the Boolean $n$-cube with bounded Vapnik-Chervonenkis dimension. J. Combin. Theory Ser. A 69 (1995), no. 2, 217–232.
  • Heinrich, Stefan; Novak, Erich; Wasilkowski, Grzegorz W.; Woźniakowski, Henryk. The inverse of the star-discrepancy depends linearly on the dimension. Acta Arith. 96 (2001), no. 3, 279–302.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2.
  • Kreyszig, Erwin. Introductory functional analysis with applications. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1989. xvi+688 pp. ISBN: 0-471-50459-9.
  • L. Kuipers and H. Niederreiter, phUniform distribution of sequences, Dover Publications, New York, 2006.
  • Lemieux, Christiane; Sidorsky, Paul. Exact sampling with highly uniform point sets. Math. Comput. Modelling 43 (2006), no. 3-4, 339–349.
  • León, Carlos A.; Perron, François. Optimal Hoeffding bounds for discrete reversible Markov chains. Ann. Appl. Probab. 14 (2004), no. 2, 958–970.
  • L. Liao, Variance reduction in gibbs sampler using quasi random numbers, J. Comput. Graph. Statist. 7 (1998), 253–266.
  • Liu, Jun S. Monte Carlo strategies in scientific computing. Springer Series in Statistics. Springer, New York, 2008. xvi+343 pp. ISBN: 978-0-387-76369-9; 0-387-95230-6.
  • P. Mathé and E. Novak, phSimple Monte Carlo and the Metropolis algorithm, J. Complexity 23 (2007), no. 4-6, 673–696.
  • Miasojedow, Błażej. Hoeffding's inequalities for geometrically ergodic Markov chains on general state space. Statist. Probab. Lett. 87 (2014), 115–120.
  • Owen, Art B.; Tribble, Seth D. A quasi-Monte Carlo Metropolis algorithm. Proc. Natl. Acad. Sci. USA 102 (2005), no. 25, 8844–8849 (electronic).
  • Roberts, Gareth O.; Rosenthal, Jeffrey S. Variance bounding Markov chains. Ann. Appl. Probab. 18 (2008), no. 3, 1201–1214.
  • Rudin, Walter. Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8.
  • Rudolf, Daniel. Explicit error bounds for Markov chain Monte Carlo. Dissertationes Math. (Rozprawy Mat.) 485 (2012), 1–93.
  • I. Sobol, phPseudo-random numbers for constructing discrete Markov chains by the Monte Carlo method, USSR Compat. Math. Math. Phys. 14 (1974), 36–45.
  • Steinwart, Ingo; Christmann, Andreas. Support vector machines. Information Science and Statistics. Springer, New York, 2008. xvi+601 pp. ISBN: 978-0-387-77241-7.
  • Talagrand, M. Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 (1994), no. 1, 28–76.
  • Tribble, Seth D. Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. Thesis (Ph.D.)–Stanford University. ProQuest LLC, Ann Arbor, MI, 2007. 103 pp. ISBN: 978-0549-06379-7.
  • Tribble, Seth D.; Owen, Art B. Construction of weakly CUD sequences for MCMC sampling. Electron. J. Stat. 2 (2008), 634–660.