Kalman filters constitute a scalable and robust methodology for approximate Bayesian inference, matching first and second order moments of the target posterior. To improve the accuracy in nonlinear and non-Gaussian settings, we extend this principle to include more or different characteristics, based on kernel mean embeddings (KMEs) of probability measures into their corresponding Hilbert spaces. Focusing on the continuous-time setting, we develop a family of interacting particle systems (termed $\textit{KME-dynamics}$) that bridge between the prior and the posterior, and that include the Kalman-Bucy filter as a special case. A variant of KME-dynamics has recently been derived from an optimal transport perspective by Maurais and Marzouk, and we expose further connections to (kernelised) diffusion maps, leading to a variational formulation of regression type. Finally, we conduct numerical experiments on toy examples and the Lorenz-63 model, the latter of which show particular promise for a hybrid modification (called Kalman-adjusted KME-dynamics).

翻译：卡尔曼滤波器是一种可扩展且稳健的近似贝叶斯推断方法，通过匹配目标后验分布的一阶和二阶矩进行近似。为提升在非线性和非高斯场景下的精度，我们将该原理扩展至基于概率测度核均值嵌入（KME）的希尔伯特空间，以纳入更多或不同特征。聚焦连续时间设定，我们发展了一族连接先验与后验的交互粒子系统（称为$\textit{KME-动力学}$），其中卡尔曼-布西滤波器作为其特例。Maurais与Marzouk近期从最优传输视角推导出KME-动力学的一个变体，我们进一步揭示其与（核化）扩散映射的关联，从而引出回归类型的变分公式。最后，我们在玩具示例与Lorenz-63模型上进行数值实验，后者对一种称为卡尔曼调整KME-动力学的混合修正方案展现出特殊前景。