Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-30T03:10:09.677Z Has data issue: false hasContentIssue false
  • English
  • Français

PDMP Monte Carlo methods for piecewise smooth densities

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

Published online by Cambridge University Press:  12 March 2024

Augustin Chevallier Open the ORCID record for Augustin Chevallier [Opens in a new window] ,
Sam Power Open the ORCID record for Sam Power [Opens in a new window] ,
Andi Q. Wang  and
Paul Fearnhead Open the ORCID record for Paul Fearnhead [Opens in a new window]
Augustin Chevallier*
Affiliation:
Université de Strasbourg
Sam Power
Affiliation:
University of Bristol
Andi Q. Wang
Affiliation:
University of Warwick
Paul Fearnhead
Affiliation:
Lancaster University
*
Email address: a.chevallier@unistra.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

There has been substantial interest in developing Markov chain Monte Carlo algorithms based on piecewise deterministic Markov processes. However, existing algorithms can only be used if the target distribution of interest is differentiable everywhere. The key to adapting these algorithms so that they can sample from densities with discontinuities is to define appropriate dynamics for the process when it hits a discontinuity. We present a simple condition for the transition of the process at a discontinuity which can be used to extend any existing sampler for smooth densities, and give specific choices for this transition which work with popular algorithms such as the bouncy particle sampler, the coordinate sampler, and the zigzag process. Our theoretical results extend and make rigorous arguments that have been presented previously, for instance constructing samplers for continuous densities restricted to a bounded domain, and we present a version of the zigzag process that can work in such a scenario. Our novel approach to deriving the invariant distribution of a piecewise deterministic Markov process with boundaries may be of independent interest.

Keywords

Bayesian inferencebouncy particle samplerHamiltonian Monte CarloMarkov chain Monte Carlonon-reversible samplerszigzag process

MSC classification

Secondary: 65C05: Monte Carlo methods 60J25: Continuous-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 42
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Afshar, H. M. and Domke, J. (2015). Reflection, refraction, and Hamiltonian Monte Carlo. In Advances in Neural Information Processing Systems 28 (NIPS 2015), Neural Information Processing Systems Foundation, San Diego, CA, pp. 3007–3015.Google Scholar
Bernard, E. P., Krauth, W. and Wilson, D. B. (2009). Event-chain Monte Carlo algorithms for hard-sphere systems. Phys. Rev. E 80, article no. 056704.CrossRefGoogle ScholarPubMed
Bierkens, J. (2016). Non-reversible Metropolis–Hastings. Statist. Comput. 26, 1213–1228.CrossRefGoogle Scholar
Bierkens, J. et al. (2018). Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains. Statist. Prob. Lett. 136, 148–154.CrossRefGoogle Scholar
Bierkens, J., Fearnhead, P. and Roberts, G. (2019). The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. Ann. Statist. 47, 1288–1320.CrossRefGoogle Scholar
Bierkens, J., Grazzi, S., Kamatani, K. and Roberts, G. (2020). The boomerang sampler. In Proc. 37th International Conference on Machine Learning (Proceedings of Machine Learning Research 119), Machine Learning Research Press, pp. 908–918.Google Scholar
Bierkens, J., Grazzi, S., van der Meulen, F. and Schauer, M. (2021). Sticky PDMP samplers for sparse and local inference problems. Preprint. Available at https://arxiv.org/abs/2103.08478.Google Scholar
Bouchard-Côté, A., Vollmer, S. J. and Doucet, A. (2018). The bouncy particle sampler: a nonreversible rejection-free Markov chain Monte Carlo method. J. Amer. Statist. Assoc. 113, 855–867.CrossRefGoogle Scholar
Chen, F., Lovász, L. and Pak, I. (1999). Lifting Markov chains to speed up mixing. In STOC ’99: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, pp. 275–281.CrossRefGoogle Scholar
Chevallier, A., Fearnhead, P. and Sutton, M. (2023). Reversible jump PDMP samplers for variable selection. J. Amer. Statist. Assoc. 118, 2915–2927.CrossRefGoogle Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Routledge, New York.CrossRefGoogle Scholar
Deligiannidis, G., Paulin, D., Bouchard-Côté, A. and Doucet, A. (2021). Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates. Ann. Appl. Prob. 31, 2612–2662.CrossRefGoogle Scholar
Diaconis, P., Holmes, S. and Neal, R. M. (2000). Analysis of a nonreversible Markov chain sampler. Ann. Appl. Prob. 10, 726–752.CrossRefGoogle Scholar
Dinh, V., Bilge, A., Zhang, C. and Matsen, F. A., IV (2017). Probabilistic path Hamiltonian Monte Carlo. In Proc. 34th International Conference on Machine Learning (Proceedings of Machine Learning 70), Machine Learning Research Press, pp. 1009–1018.Google Scholar
Dunson, D. B. and Johndrow, J. (2020). The Hastings algorithm at fifty. Biometrika 107, 1–23.CrossRefGoogle Scholar
Durmus, A., Guillin, A. and Monmarché, P. (2021). Piecewise deterministic Markov processes and their invariant measures. Ann. Inst. H. Poincaré Prob. Statist. 57, 1442–1475.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Fearnhead, P., Bierkens, J., Pollock, M. and Roberts, G. O. (2018). Piecewise deterministic Markov processes for continuous-time Monte Carlo. Statist. Sci. 33, 386–412.CrossRefGoogle Scholar
Grazzi, S. (2023). Design and applications of Monte Carlo methods based on piecewise deterministic Markov processes. Doctoral Thesis, Delft University of Technology.Google Scholar
Harland, J., Michel, M., Kampmann, T. A. and Kierfeld, J. (2017). Event-chain Monte Carlo algorithms for three- and many-particle interactions. Europhys. Lett.) 117, article no. 30001.CrossRefGoogle Scholar
Koskela, J. (2022). Zig-zag sampling for discrete structures and nonreversible phylogenetic MCMC. J. Comput. Graph. Statist. 31, 684–694.CrossRefGoogle Scholar
Krauth, W. (2021). Event-chain Monte Carlo: foundations, applications, and prospects. Frontiers Phys. 9, article no. 663457.CrossRefGoogle Scholar
Michel, M., Durmus, A. and Sénécal, S. (2020). Forward event-chain Monte Carlo: fast sampling by randomness control in irreversible Markov chains. J. Comput. Graph. Statist. 29, 689–702.CrossRefGoogle Scholar
Nakajima, J. and West, M. (2013). Bayesian analysis of latent threshold dynamic models. J. Business Econom. Statist. 31, 151–164.CrossRefGoogle Scholar
Nishimura, A., Dunson, D. B. and Lu, J. (2020). Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods. Biometrika 107, 365–380.CrossRefGoogle Scholar
Pakman, A. (2017). Binary bouncy particle sampler. Preprint. Available at https://arxiv.org/abs/1711.00922v1.Google Scholar
Pakman, A. and Paninski, L. (2014). Exact Hamiltonian Monte Carlo for truncated multivariate Gaussians. J. Comput. Graph. Statist. 23, 518–542.CrossRefGoogle Scholar
Peters, E. A. J. F. and de With, G. (2012). Rejection-free Monte Carlo sampling for general potentials. Phys. Rev. E 85, article no. 026703.CrossRefGoogle ScholarPubMed
Raftery, A. E. and Akman, V. (1986). Bayesian analysis of a Poisson process with a change-point. Biometrika 73, 85–89.CrossRefGoogle Scholar
Terenin, A. and Thorngren, D. (2018). A piecewise deterministic Markov process via $(r,\theta)$ swaps in hyperspherical coordinates. Preprint. Available at http://arxiv.org/abs/1807.00420.Google Scholar
Vanetti, P., Bouchard-Côté, A., Deligiannidis, G. and Doucet, A. (2017). Piecewise-deterministic Markov chain Monte Carlo. Preprint. Available at http://arxiv.org/abs/1707.05296.Google Scholar
Wang, A. Q., Pollock, M., Roberts, G. O. and Steinsaltz, D. (2021). Regeneration-enriched Markov processes with application to Monte Carlo. Ann. Appl. Prob. 31, 703–735.CrossRefGoogle Scholar
Wu, C. and Robert, C. P. (2020). Coordinate sampler: a non-reversible Gibbs-like MCMC sampler. Statist. Comput. 30, 721–730.CrossRefGoogle Scholar
Zhou, G. (2020). Mixed Hamiltonian Monte Carlo for mixed discrete and continuous variables. In NIPS ’20: Proceedings of the 34th Conference on Neural Information Processing Systems, Neural Information Processing Systems Foundation, San Diego, CA, pp. 17094–17104.Google Scholar