Piecewise-Deterministic Markov Chain Monte Carlo

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.

arXiv comment42 pages, 4 figures

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• Regeneration-enriched Markov processes with application to Monte Carlo
• Accelerated Sampling on Discrete Spaces with Non-Reversible Markov Processes
• A general perspective on the Metropolis-Hastings kernel
• Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents
• Cores for Piecewise-Deterministic Markov Processes used in Markov Chain Monte Carlo
• Binary Bouncy Particle Sampler
• Complexity of zigzag sampling algorithm for strongly log-concave distributions
• The Coordinate Sampler: A Non-Reversible Gibbs-like MCMC Sampler
• Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets
• Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions
• Parallelising MCMC via Random Forests
• Constructing sampling schemes via coupling: Markov semigroups and optimal transport
• Velocity Jumps for Molecular Dynamics
• Analysis of high-dimensional Continuous Time Markov Chains using the Local Bouncy Particle Sampler
• A note on the jump locations of Markov processes
• Efficient posterior sampling for high-dimensional imbalanced logistic regression
• Bouncy Hybrid Sampler as a Unifying Device
• Zigzag path connects two Monte Carlo samplers: Hamiltonian counterpart to a piecewise deterministic Markov process
• Piecewise deterministic sampling with splitting schemes
• The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo
• Generalizing Parallel Replica Dynamics: Trajectory Fragments, Asynchronous Computing, and PDMPs
• Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and Application to hard dimers