Limit theorems for the Zig-Zag process

TL;DR

This paper analyzes the Zig-Zag process, a piecewise deterministic Markov process for sampling, establishing conditions for a CLT, characterizing its asymptotic variance, and comparing its performance to traditional MCMC methods.

Contribution

It provides the first detailed theoretical analysis of the Zig-Zag sampler's asymptotic properties and performance in the one-dimensional case, including CLT conditions and diffusion limits.

Findings

Conditions for CLT to hold for the Zig-Zag process.

Characterization of the asymptotic variance.

Comparison of Zig-Zag performance with existing Monte Carlo methods.

Abstract

Markov chain Monte Carlo methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis- Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the Zig-Zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime (Bierkens, Fearnhead and Roberts (2016)). In this paper we study the performance of the Zig-Zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a Central limit theorem (CLT) holds and characterize the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the Zig-Zag process by identifying a diffusion limit as the switching rate tends to infinity. Based on our results we compare the performance of the Zig-Zag sampler to existing Monte Carlo methods, both analytically and through simulations.