Mean-field Variational Inference via Wasserstein Gradient Flow

One of the core problems of modern Bayesian inference is to compute the posterior distribution, a joint probability measure over unknown quantities, such as model parameters and unobserved latent variables, obtained by combining data information with prior knowledge in a principled manner. Modern statistics often rely on complex models for which the posterior distribution is analytically intractable and requires approximate computation. As a common alternative strategy to conventional Markov chain Monte Carlo (MCMC) sampling approach for approximating the posterior, variational inference (VI, [10]), or variational Bayes [27], finds the closest member in a user specified class of analytically tractable distributions, referred to as the variational (distribution) family, to approximate the target posterior. Although MCMC is asymptotically exact, VI is usually orders of magnitude faster [12, 62] since it turns the sampling or integration into an optimization problem. VI has successfully demonstrated its power in a wide variety of applications, including clustering [11, 23], semi-supervised learning [38], neural-network training [5, 52], and probabilistic modeling [13, 36]. Among various approximating schemes, the mean-field (MF) approximation, which originates from statistical mechanics and uses the approximating family consisting of all fully factorized density functions over (blocks of) the unknown quantities, is the most widely used and representative instance of VI that is conceptually simple yet practically powerful. On the downside, VI still requires certain conditional conjugacy structure to facilitate efficient computation (c.f.