Optimal Transport, Ricci Curvature and Mean Field Variational Bayesian Approxima
We start this short course by introducing some of the geometric aspects of the theory of optimal transport that have been developed by Lott, Villani and Sturm. The theory which establishes a relation between statistical mechanics and geometry leads to a notion for a measured length space to have Ricci curvature bounded below. (about 5 sessions) For this part we start by Monge Kantarovich theory and then review the relevant role of Ricci curvature in differential geometry and continue by discussing the stablility properties of the notion of lower Ricci curvature bound and if time permits the smoothness of optimal transport in curved spaces.
Next for about two sessions we try to cover some standard mean field equations in random energy models and talk about its generalization in the framework of graphical models and variational Bayesian inference (2 sessions).
At the end we will talk about our results on the theory of mean field variational Bayesian approximation which paves the way for a rigorous treatment of this area of Bayesian inference and its applications.
 Information Physics and Computation, Marc Mezard and Andrea Montanari.
 Optimal transport, Ricci curvature and mean field variational Bayesian approximation, my preprint.