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

Abstract

An interacting system of Langevin dynamics driven particles has been proposed forsampling from a given posterior density by Garbuno-Inigo, Hoffmann, Li and StuartinInteracting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler(arXiv:1903:08866v2). The proposed formulation is primarily studiedfrom a formalmean-field limit perspective, while the theoretical behaviour under afinite particlesize is left as an open problem. In this note we demonstrate that theparticle-basedcovariance interaction term requires a non-trivial correction. Wealso show that thecorrected dynamics samples exactly from the desired posterior provided that the em-pirical covariance matrix of the particle system remains non-singular and the posteriorlog-density satisfies the standard Bakry– ́Emery criterion.