  An interacting system of Langevin dynamics driven particles has been proposed
for sampling from a given posterior density by Garbuno-Inigo, Hoffmann, Li and
Stuart in Interacting Langevin Diffusions: Gradient Structure and Ensemble
Kalman Sampler (arXiv:1903:08866v2). The proposed formulation is primarily
studied from a formal mean-field limit perspective, while the theoretical
behaviour under a finite particle size is left as an open problem. In this note
we demonstrate that the particle-based covariance interaction term requires a
non-trivial correction. We also show that the corrected dynamics samples
exactly from the desired posterior provided that the empirical covariance
matrix of the particle system remains non-singular and the posterior
log-density satisfies the standard Bakry-Emery criterion.
