Automatic Zig-Zag sampling in practice

Abstract

Novel Monte Carlo methods to generate samples from a target distribution, such as a posterior from a Bayesian analysis, have rapidly expanded in the past decade. Algorithms based on Piecewise Deterministic Markov Processes (PDMPs), non-reversible continuous-time processes, are developing into their own research branch, thanks their important properties (e.g., super-efficiency). Nevertheless, practice has not caught up with the theory in this field, and the use of PDMPs to solve applied problems is not widespread. This might be due, firstly, to several implementational challenges that PDMP-based samplers present with and, secondly, to the lack of papers that showcase the methods and implementations in applied settings. Here, we address both these issues using one of the most promising PDMPs, the Zig-Zag sampler, as an archetypal example. After an explanation of the key elements of the Zig-Zag sampler, its implementation challenges are exposed and addressed. Specifically, the formulation of an algorithm that draws samples from a target distribution of interest is provided. Notably, the only requirement of the algorithm is a closed-form differentiable function to evaluate the log-target density of interest, and, unlike previous implementations, no further information on the target is needed. The performance of the algorithm is evaluated against canonical Hamiltonian Monte Carlo, and it is proven to be competitive, in simulation and real-data settings. Lastly, we demonstrate that the super-efficiency property, i.e. the ability to draw one independent sample at a lesser cost than evaluating the likelihood of all the data, can be obtained in practice.

Splitting schemes for second order approximations of piecewise-deterministic Markov processes

“…In that case, the random event times have to be approximated even if the ODE can be solved exactly. This question has recently been addressed in [3,38,14] with three different schemes. Here, rather than designing an ad hoc numerical schemes, we work in the general framework of splitting schemes, which are widely used for e.g.…”

“…Moreover, in this work we describe how to remove the bias introduced by our approximation with a non-reversible Metropolis-Hastings step. Two other works ( [38] and [14]) focus on approximate simulation of the Zig-Zag sampler, which is one of our two main examples. In [38] the authors suggest to approximate event times by using numerical approximations of the integral of the rates along the dynamics (2), as well as a root finding algorithm.…”

“…In [38] the authors suggest to approximate event times by using numerical approximations of the integral of the rates along the dynamics (2), as well as a root finding algorithm. In [14], the authors suggest Algorithm 1: Splitting scheme DBD for ZZS Input : Number of iterations N , initial condition (x, v), step size δ. Output: Chain (X tn , V tn ) N n=0 .…”

“…, d do With probability (1 − exp(δλ i (X t n+1 , V t n+1 ))) set V t n+1 = R i V t n+1 ; end Set X t n+1 = X t n+1 + δ 2 V t n+1 ; Set n = n + 1; end using a numerical optimisation algorithm at each iteration to obtain a suitable bound that enables the use of Poisson thinning. The first difference is that we mainly consider our approximations as discrete time Markov chains, whereas the processes of [38] and [14] are interpreted in continuous time, although neither resulting process is a Markov process due to the nature of the numerical algorithms that are used. Naturally, one could interpret our algorithms as continuous time processes, which again would not be Markov processes.…”