Measure transport with kernel mean embeddings

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).