Measure transport with kernel mean embeddings

Understand

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 reproducing kernel Hilbert spaces.
• Focusing on the continuous-time setting, we develop a family of interacting particle systems (termed $\textit{KME-dynamics}$) that bridge between prior and posterior, and that include the Kalman-Bucy filter as a special case.
• KME-dynamics does not require the score of the target, but rather estimates the score implicitly and intrinsically, and we develop links to score-based generative modeling and importance reweighting.