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Bayesian inference can be embedded into an appropriately defined dynamics inthe space of probability measures. In this paper, we take Brownian motion andits associated Fokker--Planck equation as a starting point for such embeddingsand explore several interacting particle approximations. More specifically, weconsider both deterministic and stochastic interacting particle systems andcombine them with the idea of preconditioning by the empirical covariancematrix. In addition to leading to affine invariant formulations whichasymptotically speed up convergence, preconditioning allows for gradient-freeimplementations in the spirit of the ensemble Kalman filter. While suchgradient-free implementations have been demonstrated to work well for posteriormeasures that are nearly Gaussian, we extend their scope of applicability tomultimodal measures by introducing localised gradient-free approximations.Numerical results demonstrate the effectiveness of the consideredmethodologies.

Fokker-Planck particle systems for Bayesian inference: Computational approaches