A scalable framework to transform samples from one continuous distribution to another

We present a framework to transform a sample from one continuous distribution P to another ℚ. Our previous work considered the special case of Bayesian inference where P is the prior and ℚ is the posterior, showing that this can be solved with convex optimization under appropriate conditions. Here, our contribution is two fold: (i) we consider the more general case of arbitrary P and ℚ and show using optimal transport theory and KL divergence minimization that convexity holds provided that ℚ has a log-concave density; (ii) we develop a largescale distributed solver. With this general framework finding the optimal Bayesian map is done through a series of MAP estimation problems. Interesting applications are also presented.