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Limit theorems for the zig-zag process

Part of: Markov processes Limit theorems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  08 September 2017

Joris Bierkens  and
Andrew Duncan
Joris Bierkens*
Affiliation:
University of Warwick
Andrew Duncan*
Affiliation:
Imperial College London
*
* Current address: Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD, Delft, The Netherlands. Email address: joris.bierkens@tudelft.nl
** Current address: School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK.
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Abstract

Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

Keywords

MCMCnonreversible Markov processpiecewise deterministic Markov processcontinuous-time Markov processcentral limit theoremfunctional central limit theorem

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 791 - 825
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Azaïs, R., Bardet, J.-B., Génadot, A., Krell, N. and Zitt, P.-A. (2014). Piecewise deterministic Markov process—recent results. ESAIM: Proc. 44, 276–290. CrossRefGoogle Scholar
[2] Bierkens, J. and Roberts, G. (2016). A piecewise deterministic scaling limit of lifted Metropolis–Hastings in the Curie–Weiss model. Ann. Appl. Prob. 27, 846–882. Google Scholar
[3] Bierkens, J., Fearnhead, P. and Roberts, G. (2016). The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. Preprint. Available at https://arxiv.org/abs/1607.03188. Google Scholar
[4] Bouchard-Côté, A., Vollmer, S. J. and Doucet, A. (2017). The bouncy particle sampler: a non-reversible rejection-free Markov chain Monte Carlo method. To appear in J. Amer. Statist. Assoc. Available at http://dx.doi.org/10.1080/01621459.2017.1294075. CrossRefGoogle Scholar
[5] Cattiaux, P., Chafaï, D. and Guillin, A. (2012). Central limit theorems for additive functionals of ergodic Markov diffusions processes. ALEA Lat. Am. J. Prob. Math. Statist. 9, 1–39. Google Scholar
[6] Chen, T.-L. and Hwang, C.-R. (2013). Accelerating reversible Markov chains. Statist. Prob. Lett. 83, 1956–1962. CrossRefGoogle Scholar
[7] Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. With discussion. J. Roy. Statist. Soc. B 46, 353–388. Google Scholar
[8] Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987). Hybrid Monte Carlo. Phys. Lett. B 195, 216–222. CrossRefGoogle Scholar
[9] Duncan, A. B., Lelièvre, T. and Pavliotis, G. A. (2016). Variance reduction using nonreversible Langevin samplers. J. Statist. Phys. 163, 457–491. CrossRefGoogle ScholarPubMed
[10] Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA. Google Scholar
[11] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes. John Wiley, New York. Google Scholar
[12] Fontbona, J., Guérin, H. and Malrieu, F. (2012). Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process. Adv. Appl. Prob. 44, 977–994. CrossRefGoogle Scholar
[13] Fontbona, J., Guérin, H. and Malrieu, F. (2016). Long time behavior of telegraph processes under convex potentials. Stoch. Process. Appl. 126, 3077–3101. CrossRefGoogle Scholar
[14] Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Prob. 24, 916–931. CrossRefGoogle Scholar
[15] Hastings, W. K. (1970). Monte carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109. CrossRefGoogle Scholar
[16] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (1993). Accelerating gaussian diffusions. Ann. Appl. Prob. 3, 897–913. CrossRefGoogle Scholar
[17] Jarner, S. F. and Roberts, G. O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Statist. 34, 781–815. CrossRefGoogle Scholar
[18] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. CrossRefGoogle Scholar
[19] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104, 1–19. CrossRefGoogle Scholar
[20] Kolmogorov, A. N. and Fomin, S. V. (1975). Introductory Real Analysis. Dover, New York. Google Scholar
[21] Komorowski, T., Landim, C. and Olla, S. (2012). Fluctuations in Markov Processes (Fund. Principles Math. Sci. 345). Springer, Heidelberg. 491pp. CrossRefGoogle Scholar
[22] Lelièvre, T., Nier, F. and Pavliotis, G. A. (2013). Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Statist. Phys. 152, 237–274. CrossRefGoogle Scholar
[23] Lewis, P. A. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26, 403–413. CrossRefGoogle Scholar
[24] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092. CrossRefGoogle Scholar
[25] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517. CrossRefGoogle Scholar
[26] Monmarché, P. (2015). On H 1 and entropic convergence for contractive PDMP. Electron. J. Prob. 20, 1–30. CrossRefGoogle Scholar
[27] Monmarché, P. (2016). Piecewise deterministic simulated annealing. ALEA Lat. Am. J. Prob. Math. Statist. 13, 357–398. CrossRefGoogle Scholar
[28] Neal, R. M. (2011). MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo, Chapman and Hall, Boca Raton, FL, pp. 113–162. CrossRefGoogle Scholar
[29] Ottobre, M., Pillai, N. S., Pinski, F. J. and Stuart, A. M. (2016). A function space HMC algorithm with second order Langevin diffusion limit. Bernoulli 22, 60–106. CrossRefGoogle Scholar
[30] Peters, E. A. J. F. and de With, G. (2012). Rejection-free Monte Carlo sampling for general potentials. Phys. Rev. E 85, 026703. CrossRefGoogle ScholarPubMed
[31] Rey-Bellet, L. and Spiliopoulos, K. (2015). Irreversible Langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28, 2081–2103. CrossRefGoogle Scholar
[32] Rey-Bellet, L. and Spiliopoulos, K. (2015). Variance reduction for irreversible Langevin samplers and diffusion on graphs. Electron. Commun. Prob. 20, 16pp. CrossRefGoogle Scholar
[33] Rey-Bellet, L. and Spiliopoulos, K. (2016). Improving the convergence of reversible samplers. J. Statist. Phys. 164, 472–494. CrossRefGoogle Scholar
[34] Sun, Y., Schmidhuber, J. and Gomez, F. J. (2010). Improving the asymptotic performance of Markov chain Monte–Carlo by inserting vortices. Adv. Neural Inf. Process. Syst. 23, 2235–2243. Google Scholar
[35] Welling, M. and Teh, Y. W. (2011). Bayesian learning via stochastic gradient Langevin dynamics. Proc. 28th Int. Conf. Machine Learning, ICML-11, 681–688. Google Scholar