Efficient Bayesian inference methods via convex optimization and optimal transport

In this paper, we consider many problems in Bayesian inference - from drawing samples to posteriors, to calculating confidence intervals, to implementing posterior matching algorithms, by finding maps that push one distribution to another. We show that for a large class of problems (with log-concave likelihoods and log-concave priors), these problems can be efficiently solved using convex optimization. We provide example applications within the context of dynamic statistical signal processing.