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中的一项重要任务。过去十年间，多种集成式采样方法被提出。与传统马尔可夫链蒙特卡洛方法不同，这些新方法部署大量交互样本，样本间的通信对加速收敛至关重要。为验证这些采样策略的有效性，交互粒子的概念自然引出了平均场理论。该理论建立了编码于耦合常微分方程/随机微分方程中的粒子相互作用与刻画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时序收敛证明的偏微分方程理论相衔接。目前已发展出多种数学工具以进行平均场分析，本文展示了两个典型示例：耦合方法以及基于鞅策略构建的紧性论证。前者已被用于证明集成卡尔曼采样器与集成卡尔曼反演的收敛性，而后者将在验证弗拉索夫-玻尔兹曼模拟器的有效性中展现出强大威力。