Sampling using Piecewise Deterministic Markov Processes

On this website (continuously work in progress) 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. I will highlight papers which in my opinion reflect the current state of the art. Inevitably some of this page will reflect some of my own judgements and prejudice. 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.

Main idea

Bayesian models provide a generic approach to problems in statistics and machine learning (and are therefore potentially useful in all quantitative scientific fields). However it is a real challenge in Bayesian models to perform efficient computations. The main computational tool is Markov Chain Monte Carlo (MCMC). 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).

First, a few images

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.

Mathematics of PDMPs

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.

Methodology using PDMPs

Here papers are listed which describe sampling methods based on PDMPs. A concise introduction to the topic may be found in the bold paper.

Statistics literature

Physics literature

As usual many ideas concerning sampling methods originated from physics; these are a few key papers in this respect.

Machine learning literature

Key references on non-reversible MCMC sampling

The use of PDMPs for sampling originated from the attempts to design non-reversible MCMC algorithms. Here I list several key references from the physics, mathematics and statistics literature.

Last update: May 2018