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.

翻译：假设我们关注从定义在$(\mathsf{Z},\mathscr{Z})$上的概率分布$\mu$中采样。本文发展了一个采样算法框架，该框架充分利用常微分方程数值积分器（记单步积分映射为$\psi\colon\mathsf{Z}\rightarrow\mathsf{Z}$）来高效且稳健地探索$\mu$。流行的混合蒙特卡洛算法及其衍生方法正是此类数值积分器应用的实例。本文发展的一个核心思想是采样积分器片段，即常微分方程数值积分器$\psi$轨道中的片段，并定义一个关联的概率分布$\bar{\mu}$，使得关于$\mu$的期望可以通过服从$\bar{\mu}$分布的积分器片段进行估计。积分器片段目标分布$\bar{\mu}$表现为一系列前推分布的混合形式，这启发了超越数值积分器映射的多种推广（例如标准化流）。尤为重要的是，这种结构也为调整积分器参数（如蛙跳积分器中的离散化步长、有效积分时间或积分步数）提供了新的原则性稳健策略。本文主要聚焦于序贯蒙特卡洛算法，但该方法同样适用于马尔可夫链蒙特卡洛算法。我们通过数值实验展示了算法性能（特别是稳健性），并提供了支持观测性能的初步理论结果。