Fokker-Planck particle systems for Bayesian inference: Computational approaches

TL;DR

This paper explores particle-based methods for Bayesian inference using Fokker-Planck dynamics, introducing preconditioning and localized gradient-free approaches to improve convergence and applicability to multimodal distributions.

Contribution

It develops new interacting particle systems with preconditioning and localization, extending gradient-free Bayesian inference to multimodal measures.

Findings

Affine invariant formulations accelerate convergence.

Preconditioning enables gradient-free implementations.

Localized approximations extend applicability to multimodal distributions.

Abstract

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.