On this website I aim to collect references to papers on the use and analysis of Piecewise Deterministic Markov Processes (PDMPs) for use in MCMC. I will try to list all papers that I find important, will put years of first appearance on arXiv or elsewhere, and list in reverse chronological order. Inevitably this page will reflect some of my own judgements. If you want to bring some paper to my attention, or feel I should correct something, please write me at `joris.bierkens AT tudelft.nl`

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Markov Chain Monte Carlo (MCMC) is a probabilistic computational method which is of vital importance in data science (statistics, machine learning) as well as in the physical sciences (physics, chemistry, biology). For example, MCMC is applied in Bayesian statistics, for the training of (deep) neural networks, as well as for the simulation of many-particle systems.

In recent years it has been established that PDMPs may play a very useful role in designing new efficient MCMC methods which have good convergence properties and allow various clever tricks in dealing with large data sets (subsampling of the data, for example).

The following images are generated using the R package `RZigZag`

, which can be installed via the command `install.packages("RZigZag")`

. The plots can then be produced using this R script.

What we see are two-dimensional projections of higher dimensional trajectories of two important PDMP samplers: the Zig-Zag Sampler and the Bouncy Particle Sampler (BPS). The trajectories can be seen to be continuous (in contrast to most MCMC algorithms). Also shown are discrete samples drawn along the trajectories, which can be interpreted in the classical MCMC sense. Both plots in this example correspond to a 5-dimensional standard normal distribution. The BPS trajectory is generated using a refreshment rate of `0.01`

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Here papers are listed which concern the mathematical analysis of process underlying sampling by means of PDMPs. I have chosen not to list papers concerning the analysis of PDMPs under assumptions which are not immediately useful for MCMC.

- C. Andrieu, A. Durmus, N. NÃ¼sken, J. Roussel, Hypercoercivity of Piecewise Deterministic Markov Process-Monte Carlo (2018).
*This important paper gives results in a very general context for scaling with dimension of various PDMP samplers. The obtained results seem to be sharp, for example in the sense that they agree with the results obtained for the specialized case of a factorized distribution in Bierkens, Kamatani and Roberts (2018) discussed below. No doubt the title will soon to change to "Hypocoercivity ... "* - G. Deligiannidis, D. Paulin, A. Doucet, Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates (2018).
*For a refreshment rate decaying to zero as O(d^{-1/2}) the first coordinate of the BPS has a scaling limit as the dimension d goes to infinity, implying that the mixing time of the first coordinate process is dimension-independent. Note however that in Bierkens, Kamatani and Roberts (2018), discussed below, we show that the radial component will not mix if the switching intensity decays to zero.* - Joris Bierkens, Kengo Kamatani, Gareth Roberts, High-dimensional scaling limits of piecewise deterministic sampling algorithms (2018).
*In the case of a standard normal distribution on R^d, the computational efficiency of Zig-Zag and BPS (with constant positive refreshment) are compared by means of scaling limits for certain summary statistics.* - Alain Durmus, Arnaud Guillin, Pierre Monmarché, Geometric ergodicity of the bouncy particle sampler (2018).
*Compared to Deligiannidis et al. (2017) the conditions for exponential ergodicity of BPS are weakened, based on a coupling argument.* - Joris Bierkens, Gareth Roberts, Pierre-André Zitt, Ergodicity of the zigzag process (2017).
*In this paper it is shown that under mild conditions the zigzag process is (exponentially) ergodic, even without refreshments of the velocity.* - George Deligiannidis, Alexandre Bouchard-Côté, Arnaud Doucet, Exponential Ergodicity of the Bouncy Particle Sampler (2017)
- Joris Bierkens, Andrew Duncan, Limit theorems for the Zig-Zag process (2016)
- Joris Bierkens, Gareth Roberts, A piecewise deterministic scaling limit of Lifted Metropolis-Hastings in the Curie-Weiss model (2015).
*In this work we discovered PDMPs as a scaling limit of a discrete MCMC algorithm (see the paper by Turitsyn et al. (2008) listed below). We obtain exponential ergodicity of the Zig-Zag process in the one-dimensional case.* - Pierre Monmarché, Piecewise deterministic simulated annealing (2014).
*Most of the building blocks for sampling in the one-dimensional case are here, but the focus is on using PDMPs for optimization by means of simulated annealing. So in a sense, this paper is immediately taking the next step and skipping a discussion of MCMC.*

Here papers are listed which describe sampling methods based on PDMPs.

- Paul Vanetti, Alexandre Bouchard-Côté, George Deligiannidis, Arnaud Doucet, Piecewise Deterministic Markov Chain Monte Carlo (2017)
- Changye Wu, Christian P. Robert, Generalized Bouncy Particle Sampler (2017)
- Joris Bierkens, Alexandre Bouchard-Côté, Arnaud Doucet, Andrew B. Duncan, Paul Fearnhead, Thibaut Lienart, Gareth Roberts, Sebastian J. Vollmer, Piecewise Deterministic Markov Processes for Scalable Monte Carlo on Restricted Domains (2017)
- Paul Fearnhead, Joris Bierkens, Murray Pollock, Gareth O Roberts, Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo (2016)
- Joris Bierkens, Paul Fearnhead, Gareth Roberts, The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data (2016)
- Alexandre Bouchard-Côté, Sebastian J. Vollmer, Arnaud Doucet, The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method (2015)

- Manon Michel, Sebastian C. Kapfer, Werner Krauth, Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps (2013)
- E. A. J. F. Peters, G. de With, Rejection-free Monte Carlo sampling for general potentials (2011)
- Konstantin S. Turitsyn, Michael Chertkov, Marija Vucelja, Irreversible Monte Carlo Algorithms for Efficient Sampling (2008)

- Ari Pakman, Binary Bouncy Particle Sampler (2017)
- Ari Pakman, Dar Gilboa, David Carlson, Liam Paninski, Stochastic Bouncy Particle Sampler (2016)

*Last update: September 2018*