On representations of mean-field variational inference

Abstract Variational Bayes has allowed the analysis of Bayes’ rule in terms of gradient flows, partial differential equations (PDE), and diffusion processes. Mean-field variational inference (MFVI) is a version of approximate Bayesian inference that optimizes over product distributions. In the spirit of variational Bayes, we represent the MFVI problem in three different manners: a gradient flow in a Wasserstein space, a system of quasilinear PDE, and a McKean–Vlasov diffusion process. Furthermore, we show that a time-discretized coordinate ascent variational inference algorithm in the product Wasserstein space of measures yields a gradient flow in the small-time-step limit. A similar result is obtained for their associated densities, with the limit given by a system of quasilinear PDEs. We illustrate how the tools provided here can be used to guarantee convergence of algorithms, and can be extended to a variety of approaches, old and new, to solve MFVI problems.