Oliver Bodemer
IBM J. Res. Dev
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.
Oliver Bodemer
IBM J. Res. Dev
Shashanka Ubaru, Lior Horesh, et al.
Journal of Biomedical Informatics
Hiroki Yanagisawa
ICML 2023
Raúl Fernández Díaz, Lam Thanh Hoang, et al.
ACS Fall 2024