  We present a novel approach to approximate Gaussian and mixture-of-Gaussians
filtering. Our method relies on a variational approximation via a gradient-flow
representation. The gradient flow is derived from a Kullback--Leibler
discrepancy minimization on the space of probability distributions equipped
with the Wasserstein metric. We outline the general method and show its
competitiveness in posterior representation and parameter estimation on two
state-space models for which Gaussian approximations typically fail: systems
with multiplicative noise and multi-modal state distributions.
