Piecewise-Deterministic Markov Chain Monte Carlo

TL;DR

This paper introduces new piecewise-deterministic Markov Chain Monte Carlo methods that leverage Hamiltonian dynamics and approximate integrators, enhancing efficiency and applicability over existing approaches.

Contribution

The authors develop novel continuous-time and discrete-time MCMC algorithms that address limitations of previous methods by exploiting target structure and enabling broader application.

Findings

New algorithms outperform existing methods in complex target distributions

Exact Hamiltonian flows improve sampling efficiency

Approximate dynamics from symplectic integrators are effectively utilized

Abstract

A novel class of non-reversible Markov chain Monte Carlo schemes relying on continuous-time piecewise-deterministic Markov Processes has recently emerged. In these algorithms, the state of the Markov process evolves according to a deterministic dynamics which is modified using a Markov transition kernel at random event times. These methods enjoy remarkable features including the ability to update only a subset of the state components while other components implicitly keep evolving and the ability to use an unbiased estimate of the gradient of the log-target while preserving the target as invariant distribution. However, they also suffer from important limitations. The deterministic dynamics used so far do not exploit the structure of the target. Moreover, exact simulation of the event times is feasible for an important yet restricted class of problems and, even when it is, it is application specific. This limits the applicability of these techniques and prevents the development of a generic software implementation of them. We introduce novel MCMC methods addressing these shortcomings. In particular, we introduce novel continuous-time algorithms relying on exact Hamiltonian flows and novel non-reversible discrete-time algorithms which can exploit complex dynamics such as approximate Hamiltonian dynamics arising from symplectic integrators while preserving the attractive features of continuous-time algorithms. We demonstrate the performance of these schemes on a variety of applications.