  Variational inference, such as the mean-field (MF) approximation, requires
certain conjugacy structures for efficient computation. These can impose
unnecessary restrictions on the viable prior distribution family and further
constraints on the variational approximation family. In this work, we introduce
a general computational framework to implement MF variational inference for
Bayesian models, with or without latent variables, using the Wasserstein
gradient flow (WGF), a modern mathematical technique for realizing a gradient
flow over the space of probability measures. Theoretically, we analyze the
algorithmic convergence of the proposed approaches, providing an explicit
expression for the contraction factor. We also strengthen existing results on
MF variational posterior concentration from a polynomial to an exponential
contraction, by utilizing the fixed point equation of the time-discretized WGF.
Computationally, we propose a new constraint-free function approximation method
using neural networks to numerically realize our algorithm. This method is
shown to be more precise and efficient than traditional particle approximation
methods based on Langevin dynamics.
