Monte Carlo sampling with integrator snippets

Abstract

Assume interest is in sampling from a probability distribution $\mu$ defined on $(\mathsf{Z},\mathscr{Z})$. We develop a framework for sampling algorithms which takes full advantage of ODE numerical integrators, say $\psi\colon\mathsf{Z}\rightarrow\mathsf{Z}$ for one integration step, to explore $\mu$ efficiently and robustly. The popular Hybrid Monte Carlo (HMC) algorithm \cite{duane1987hybrid,neal2011mcmc} and its derivatives are examples of such a use of numerical integrators. A key idea developed here is that of sampling integrator snippets, that is fragments of the orbit of an ODE numerical integrator $\psi$, and the definition of an associated probability distribution $\bar{\mu}$ such that expectations with respect to $\mu$ can be estimated from integrator snippets distributed according to $\bar{\mu}$. The integrator snippet target distribution $\bar{\mu}$ takes the form of a mixture of pushforward distributions which suggests numerous generalisations beyond mappings arising from numerical integrators, e.g. normalising flows. Very importantly this structure also suggests new principled and robust strategies to tune the parameters of integrators, such as the discretisation stepsize, effective integration time, or number of integration steps, in a Leapfrog integrator. We focus here primarily on Sequential Monte Carlo (SMC) algorithms, but the approach can be used in the context of Markov chain Monte Carlo algorithms. We illustrate performance and, in particular, robustness through numerical experiments and provide preliminary theoretical results supporting observed performance.