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An interacting system of Langevin dynamics driven particles has been proposedfor sampling from a given posterior density by Garbuno-Inigo, Hoffmann, Li andStuart in Interacting Langevin Diffusions: Gradient Structure and EnsembleKalman Sampler (arXiv:1903:08866v2). The proposed formulation is primarilystudied from a formal mean-field limit perspective, while the theoreticalbehaviour under a finite particle size is left as an open problem. In this notewe demonstrate that the particle-based covariance interaction term requires anon-trivial correction. We also show that the corrected dynamics samplesexactly from the desired posterior provided that the empirical covariancematrix of the particle system remains non-singular and the posteriorlog-density satisfies the standard Bakry-Emery criterion.

Note on Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler by Garbuno-Inigo, Hoffmann, Li and Stuart