  Bayesian inference can be embedded into an appropriately defined dynamics in
the space of probability measures. In this paper, we take Brownian motion and
its associated Fokker--Planck equation as a starting point for such embeddings
and explore several interacting particle approximations. More specifically, we
consider both deterministic and stochastic interacting particle systems and
combine them with the idea of preconditioning by the empirical covariance
matrix. In addition to leading to affine invariant formulations which
asymptotically speed up convergence, preconditioning allows for gradient-free
implementations in the spirit of the ensemble Kalman filter. While such
gradient-free implementations have been demonstrated to work well for posterior
measures that are nearly Gaussian, we extend their scope of applicability to
multimodal measures by introducing localised gradient-free approximations.
Numerical results demonstrate the effectiveness of the considered
methodologies.
