Variational inference via Wasserstein gradient flows
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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 π, VI aims at producing a simple but effective approximation π̂ to π for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, VI is still poorly understood and dominated by heuristics. In this work, we propose principled methods for VI, in which π̂ 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 π is log-concave.
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