Title: Sampling Bayesian Posteriors via Optimal Transport and Gradient Flows

Abstract: This talk presents two complementary perspectives on modern Bayesian computation through a geometric lens on the space of probability measures. The first perspective is a static optimal transport (OT) framework, where a deterministic map is learned from a reference distribution to the posterior. This map-based approach avoids common MCMC limitations, including intractable normalizing constants and sampling inefficiency, while supporting mixed parameter types and enabling OT-based exploratory analysis. The second perspective adopts a dynamic optimization viewpoint, using gradient flows with respect to the Kullback–Leibler (KL) divergence to minimize general objectives. We introduce the Implicit KL Proximal Descent (IKLPD) algorithm, a discretization of the KL gradient flow that guarantees global convergence for convex functionals and achieves exponential convergence rates in Bayesian inference.