The Annals of Applied Probability

Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo

Josef Dick, Daniel Rudolf, and Houying Zhu

Full-text: Open access

Abstract

Markov chains can be used to generate samples whose distribution approximates a given target distribution. The quality of the samples of such Markov chains can be measured by the discrepancy between the empirical distribution of the samples and the target distribution. We prove upper bounds on this discrepancy under the assumption that the Markov chain is uniformly ergodic and the driver sequence is deterministic rather than independent $U(0,1)$ random variables. In particular, we show the existence of driver sequences for which the discrepancy of the Markov chain from the target distribution with respect to certain test sets converges with (almost) the usual Monte Carlo rate of $n^{-1/2}$.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 3178-3205.

Dates
Received: March 2013
Revised: February 2015
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884315

Digital Object Identifier
doi:10.1214/16-AAP1173

Mathematical Reviews number (MathSciNet)
MR3563205

Zentralblatt MATH identifier
1351.60100

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

Keywords
Markov chain Monte Carlo uniformly ergodic Markov chain discrepancy theory probabilistic method

Citation

Dick, Josef; Rudolf, Daniel; Zhu, Houying. Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. Ann. Appl. Probab. 26 (2016), no. 5, 3178--3205. doi:10.1214/16-AAP1173. https://projecteuclid.org/euclid.aoap/1476884315


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References

  • [1] Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 1000–1034.
  • [2] Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 337–404.
  • [3] Beck, J. (1984). Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31 33–41.
  • [4] Brauchart, J. S. and Dick, J. (2013). A characterization of Sobolev spaces on the sphere and an extension of Stolarsky’s invariance principle to arbitrary smoothness. Constr. Approx. 38 397–445.
  • [5] Chen, S. (2011). Consistency and convergence rate of Markov chain quasi-Monte Carlo with examples. Ph.D. thesis, Stanford Univ., Stanford, CA.
  • [6] Chen, S., Dick, J. and Owen, A. B. (2011). Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. Ann. Statist. 39 673–701.
  • [7] Chen, S., Matsumoto, M., Nishimura, T. and Owen, A. B. (2012). New inputs and methods for Markov chain quasi-Monte Carlo. In Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proc. Math. Stat. 23 313–327. Springer, Heidelberg.
  • [8] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Applications of Mathematics (New York) 31. Springer, New York.
  • [9] Dick, J. and Pillichshammer, F. (2010). Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge Univ. Press, Cambridge.
  • [10] Fang, K.-T. and Wang, Y. (1994). Number-Theoretic Methods in Statistics. Chapman & Hall, London.
  • [11] Glynn, P. W. and Ormoneit, D. (2002). Hoeffding’s inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143–146.
  • [12] Gnewuch, M. (2008). Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy. J. Complexity 24 154–172.
  • [13] Haussler, D. (1995). Sphere packing numbers for subsets of the Boolean $n$-cube with bounded Vapnik–Chervonenkis dimension. J. Combin. Theory Ser. A 69 217–232.
  • [14] Heinrich, S., Novak, E., Wasilkowski, G. W. and Woźniakowski, H. (2001). The inverse of the star-discrepancy depends linearly on the dimension. Acta Arith. 96 279–302.
  • [15] L’Ecuyer, P., Lecot, C. and Tuffin, B. (2008). A randomized quasi-Monte Carlo simulation method for Markov chains. Oper. Res. 56 958–975.
  • [16] Lemieux, C. and Sidorsky, P. (2006). Exact sampling with highly uniform point sets. Math. Comput. Modelling 43 339–349.
  • [17] Leopardi, P. (2009). Diameter bounds for equal area partitions of the unit sphere. Electron. Trans. Numer. Anal. 35 1–16.
  • [18] Liao, L. (1998). Variance reduction in Gibbs sampler using quasi random numbers. J. Comput. Graph. Statist. 7 253–266.
  • [19] Mathé, P. and Novak, E. (2007). Simple Monte Carlo and the Metropolis algorithm. J. Complexity 23 673–696.
  • [20] Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
  • [21] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [22] Miasojedow, B. (2014). Hoeffding’s inequalities for geometrically ergodic Markov chains on general state space. Statist. Probab. Lett. 87 115–120.
  • [23] Mira, A. and Tierney, L. (2002). Efficiency and convergence properties of slice samplers. Scand. J. Stat. 29 1–12.
  • [24] Neal, R. M. (2003). Slice sampling. Ann. Statist. 31 705–767.
  • [25] Owen, A. B. and Tribble, S. D. (2005). A quasi-Monte Carlo Metropolis algorithm. Proc. Natl. Acad. Sci. USA 102 8844–8849 (electronic).
  • [26] Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Probab. 20 no. 79, 32.
  • [27] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [28] Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2 13–25 (electronic).
  • [29] Roberts, G. O. and Rosenthal, J. S. (1998). On convergence rates of Gibbs samplers for uniform distributions. Ann. Appl. Probab. 8 1291–1302.
  • [30] Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20–71.
  • [31] Rudolf, D. (2012). Explicit error bounds for Markov chain Monte Carlo. Dissertationes Math. (Rozprawy Mat.) 485 1–93.
  • [32] Smith, R. L. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Oper. Res. 32 1296–1308.
  • [33] Sobol, I. M. (1974). Pseudorandom numbers for the construction of discrete Markov chains by a Monte Carlo method. USSR Compat. Math. Math. Phys. 14 36–45.
  • [34] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28–76.
  • [35] Tribble, S. D. (2007). Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. Ph.D. thesis, Stanford Univ.
  • [36] Tribble, S. D. and Owen, A. B. (2008). Construction of weakly CUD sequences for MCMC sampling. Electron. J. Stat. 2 634–660.