Bayesian sampling is an important task in statistics and machine learning. Over the past decade, many ensemble-type sampling methods have been proposed. In contrast to the classical Markov chain Monte Carlo methods, these new methods deploy a large number of interactive samples, and the communication between these samples is crucial in speeding up the convergence. To justify the validity of these sampling strategies, the concept of interacting particles naturally calls for the mean-field theory. The theory establishes a correspondence between particle interactions encoded in a set of coupled ODEs/SDEs and a PDE that characterizes the evolution of the underlying distribution. This bridges numerical algorithms with the PDE theory used to show convergence in time. Many mathematical machineries are developed to provide the mean-field analysis, and we showcase two such examples: The coupling method and the compactness argument built upon the martingale strategy. The former has been deployed to show the convergence of ensemble Kalman sampler and ensemble Kalman inversion, and the latter will be shown to be immensely powerful in proving the validity of the Vlasov-Boltzmann simulator.

翻译：贝叶斯采样是统计学与机器学习领域的一项重要任务。过去十年间，学界提出了多种集成式采样方法。与经典马尔可夫链蒙特卡罗方法不同，这些新方法部署了大量交互样本，样本间的信息交互对加速收敛至关重要。为验证此类采样策略的有效性，交互粒子的概念自然引出平均场理论。该理论建立了由耦合常/随机微分方程刻画的粒子相互作用与描述底层分布演化的偏微分方程之间的对应关系，从而将数值算法与用于证明时间收敛性的偏微分方程理论联系起来。目前已有多种数学工具被开发用于提供平均场分析，本文重点展示两个典型案例：耦合方法及基于鞅策略的紧性论证。前者已成功证明集成卡尔曼采样器与集成卡尔曼反演的收敛性，后者将展示其在证明弗拉索夫-玻尔兹曼模拟器有效性方面的强大能力。