Keywords: Bures-Wasserstein, Kalman filter, mixture of Gaussians, variational inference, Wasserstein gradient flow
TL;DR: We leverage the theory of Wasserstein gradient flows to propose new algorithms (with convergence guarantees) for approximating a posterior distribution by Gaussians or mixtures of Gaussians.
Abstract: Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior $\pi$, VI aims at producing a simple but effective approximation $\hat \pi$ to $\pi$ for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, algorithmic guarantees for VI are still relatively less well-understood. In this work, we propose principled methods for VI, in which $\hat \pi$ is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures--Wasserstein space of Gaussian measures. Akin to MCMC, it comes with strong theoretical guarantees when $\pi$ is log-concave.
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