Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo

TL;DR

This paper explores the unifying framework of piecewise deterministic Markov processes for continuous-time Monte Carlo algorithms, highlighting their advantages in big-data Bayesian analysis and introducing new implementation insights.

Contribution

It demonstrates the connection between continuous-time MCMC and SMC methods through PDMPs and broadens the class of possible algorithms for efficient Bayesian sampling.

Findings

Unified framework for continuous-time Monte Carlo methods

Sub-sampling enhances efficiency in big-data Bayesian inference

Guidelines for implementing and improving these algorithms

Abstract

Recently there have been exciting developments in Monte Carlo methods, with the development of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has led to some fundamentally new Monte Carlo algorithms which can be used to sample from, say, a posterior distribution. Interestingly, continuous-time algorithms seem particularly well suited to Bayesian analysis in big-data settings as they need only access a small sub-set of data points at each iteration, and yet are still guaranteed to target the true posterior distribution. Whilst continuous-time MCMC and SMC methods have been developed independently we show here that they are related by the fact that both involve simulating a piecewise deterministic Markov process. Furthermore we show that the methods developed to date are just specific cases of a potentially much wider class of continuous-time Monte Carlo algorithms. We give an informal introduction to piecewise deterministic Markov processes, covering the aspects relevant to these new Monte Carlo algorithms, with a view to making the development of new continuous-time Monte Carlo more accessible. We focus on how and why sub-sampling ideas can be used with these algorithms, and aim to give insight into how these new algorithms can be implemented, and what are some of the issues that affect their efficiency.