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A Short Introduction to Piecewise Deterministic Markov Samplers

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Stochastic Dynamics Out of Equilibrium (IHPStochDyn 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 282))

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Abstract

The use of velocity jump Markov processes in MCMC algorithms have recently drawn attention in various fields, such as statistical physics or Bayesian statistics. The aim of this paper is to introduce these processes and to give a few justifications on their interest.

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References

  1. Bernard, E.P., Krauth, W., Wilson, D.B.: Event-chain Monte Carlo algorithms for hard-sphere systems. Phys. Rev. E 80(5), 056704 (2009)

    Article  Google Scholar 

  2. Beskos, A., Roberts, G.O.: Exact simulation of diffusions. Ann. Appl. Probab. 15(4), 2422–2444 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bierkens, J., Duncan, A.: Limit theorems for the Zig-Zag process. Adv. Appl. Probab. 49(3), 791–825 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bierkens, J., Fearnhead, P., Roberts, G.: The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. arXiv e-prints, arXiv:1607.03188, July 2016

  5. Bierkens, J., Roberts, G.: A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model. Ann. Appl. Probab. 27(2), 846–882 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bierkens, J., Roberts, G., Zitt, P.-A.: Ergodicity of the zigzag process. arXiv e-prints, arXiv:1712.09875 (2017)

  7. Bouchard-Côté, A., Vollmer, S.J., Doucet, A.: The bouncy particle sampler: a nonreversible rejection-free Markov chain Monte Carlo method. J. Am. Stat. Assoc. 113(522), 855–867 (2018)

    Article  MathSciNet  Google Scholar 

  8. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6(4), 399–424 (2004)

    Article  MathSciNet  Google Scholar 

  9. Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7(1), 69–99 (2005)

    Article  MathSciNet  Google Scholar 

  10. Calvez, V., Raoul, G., Schmeiser, C.: Confinement by biased velocity jumps: aggregation of Escherichia coli. Kinet. Relat. Models 8(4), 651–666 (2015)

    Article  MathSciNet  Google Scholar 

  11. Deligiannidis, G., Bouchard-Côté, A., Doucet, A.: Exponential ergodicity of the bouncy particle sampler. Ann. Stat. 47(3), 1268–1287 (2019)

    Article  MathSciNet  Google Scholar 

  12. Diaconis, P., Holmes, S., Neal, R.M.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10(3), 726–752 (2000)

    Article  MathSciNet  Google Scholar 

  13. Diaconis, P., Miclo, L.: On the spectral analysis of second-order Markov chains. Ann. Fac. Sci. Toulouse Math. (6) 22(3), 573–621 (2013)

    Article  MathSciNet  Google Scholar 

  14. Durmus, A., Guillin, A., Monmarché, P.: Geometric ergodicity of the bouncy particle sampler. arXiv e-prints, arXiv:1807.05401 (2018)

  15. Durmus, A., Moulines, É.: Quantitative bounds of convergence for geometrically ergodic Markov chain in the Wasserstein distance with application to the Metropolis adjusted Langevin algorithm. Stat. Comput. 25(1), 5–19 (2015)

    Article  MathSciNet  Google Scholar 

  16. Erban, R., Othmer, H.G.: From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math. 65(2), 361–391 (2004)

    Article  MathSciNet  Google Scholar 

  17. Fétique, N.: Long-time behaviour of generalised Zig-Zag process. arXiv e-prints, arXiv:1710.01087 (2017)

  18. Fontbona, J., Guérin, H., Malrieu, F.: Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process. Adv. Appl. Probab. 44(4), 977–994 (2012)

    Article  MathSciNet  Google Scholar 

  19. Fontbona, J., Guérin, H., Malrieu, F.: Long time behavior of telegraph processes under convex potentials. Stoch. Process. Appl. 126(10), 3077–3101 (2016)

    Article  MathSciNet  Google Scholar 

  20. Goldstein, S.: On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156 (1951)

    Article  MathSciNet  Google Scholar 

  21. Guillin, A., Monmarché, P.: Optimal linear drift for an hypoelliptic diffusion. Electron. Commun. Probab. 21 (2016)

    Google Scholar 

  22. Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J.: Accelerating Gaussian diffusions. Ann. Appl. Probab. 3, 897–913 (1993)

    Article  MathSciNet  Google Scholar 

  23. Kac, M.: A stochastic model related to the telegrapher’s equation. Rocky Mt. J. Math. 4, 497–509 (1974)

    Article  MathSciNet  Google Scholar 

  24. Lelièvre, T.: Two mathematical tools to analyze metastable stochastic processes. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds.) Numerical Mathematics and Advanced Applications 2011, pp. 791–810. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  25. Lelièvre, T., Nier, F., Pavliotis, G.A.: Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Stat. Phys. 152(2), 237–274 (2013)

    Article  MathSciNet  Google Scholar 

  26. Lelièvre, T., Rousset, M., Stoltz, G.: Long-time convergence of an adaptive biasing force method. Nonlinearity 21(6), 1155–1181 (2008)

    Article  MathSciNet  Google Scholar 

  27. Lemaire, V., Thieullen, M., Thomas, N.: Exact simulation of the jump times of a class of piecewise deterministic Markov processes. J. Sci. Comput. 75(3), 1776–1807 (2018)

    Article  MathSciNet  Google Scholar 

  28. Lewis, P.A.W., Shedler, G.S.: Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26(3), 403–413 (1979)

    Article  MathSciNet  Google Scholar 

  29. Meyn, S., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  30. Michel, M., Durmus, A., Sénécal, S.: Forward event-chain Monte Carlo: fast sampling by randomness control in irreversible Markov chains. arXiv e-prints, arXiv:1702.08397 (2017)

  31. Michel, M., Kapfer, S.C., Krauth, W.: Generalized event-chain Monte Carlo: constructing rejection-free global-balance algorithms from infinitesimal steps. J. Chem. Phys. 140(5), 054116 (2014)

    Article  Google Scholar 

  32. Miclo, L., Monmarché, P.: Étude spectrale minutieuse de processus moins indécis que les autres. 2078, 459–481 (2013)

    Google Scholar 

  33. Monmarché, P.: Hypocoercive relaxation to equilibrium for some kinetic models. Kinet. Relat. Models 7(2), 341–360 (2014)

    Article  MathSciNet  Google Scholar 

  34. Monmarché, P.: Piecewise deterministic simulated annealing. ALEA Lat. Am. J. Probab. Math. Stat. 13(1), 357–398 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Neal, R.M.: Improving asymptotic variance of MCMC estimators: non-reversible chains are better. arXiv Mathematics e-prints, math/0407281 (2004)

    Google Scholar 

  36. Peskun, P.H.: Optimum Monte-Carlo sampling using Markov chains. Biometrika 60, 607–612 (1973)

    Article  MathSciNet  Google Scholar 

  37. Peters, E.A.J.F., de With, G.: Rejection-free Monte Carlo sampling for general potentials. Phys. Rev. E 85, 026703 (2012)

    Article  Google Scholar 

  38. Roberts, G.O., Rosenthal, J.S.: Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. B 60, 255–268 (1997)

    Article  MathSciNet  Google Scholar 

  39. Scemama, A., Lelièvre, T., Stoltz, G., Caffarel, M.: An efficient sampling algorithm for variational Monte Carlo. J. Chem. Phys. 125(11), 114105 (2006)

    Article  Google Scholar 

  40. Turitsyn, K.S., Chertkov, M., Vucelja, M.: Irreversible Monte Carlo algorithms for efficient sampling. Physica D 240, 410–414 (2011)

    Article  Google Scholar 

  41. Vanetti, P., Bouchard-Côté, A., Deligiannidis, G., Doucet, A.: Piecewise-deterministic Markov chain Monte Carlo. arXiv e-prints, arXiv:1707.05296 (2017)

  42. Wu, C., Robert, C.P.: Generalized bouncy particle sampler. arXiv e-prints, arXiv:1706.04781 (2017)

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Author information

Authors and Affiliations

  1. Sorbonne Universités, LJLL, Paris, France

    Pierre Monmarché

Authors
  1. Pierre Monmarché
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Correspondence to Pierre Monmarché .

Editor information

Editors and Affiliations

  1. Laboratoire de Probabilités, Statistiques et Modélisation (UMR 8001), Université Paris Diderot, Paris, France

    Giambattista Giacomin

  2. CEREMADE, CNRS UMR 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

  3. CNRS UMR 8145, MAP5, Université Paris Descartes, Paris, France

    Ellen Saada

  4. Zentrum Mathematik, Technische Universität München, Garching, Bayern, Germany

    Herbert Spohn

  5. CERMICS, Ecole des Ponts, Marne-La-Vallée, France

    Gabriel Stoltz

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Monmarché, P. (2019). A Short Introduction to Piecewise Deterministic Markov Samplers. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_11

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