Bayesian Inference for Optimal Transport with Stochastic Cost

Abstract

In machine learning and computer vision, optimal transport has had significant success inlearning generative models and defining metric distances between structured and stochasticdata objects, that can be cast as probability measures. The key element of optimal trans-port is the so called lifting of anexactcost (distance) function, defined on the sample space,to a cost (distance) between probability measures over the sample space. However, in manyreal life applications the cost isstochastic: e.g., the unpredictable traffic flow affects the costof transportation between a factory and an outlet. To take this stochasticity into account,we introduce a Bayesian framework for inferring the optimal transport plan distributioninduced by the stochastic cost, allowing for a principled way to include prior informationand to model the induced stochasticity on the transport plans. Additionally, we tailor anHMC method to sample from the resulting transport plan posterior distribution.