Inference With Aggregate Data in Probabilistic Graphical Models: An Optimal Transport Approach

We consider inference (filtering) problems over probabilistic graphical models with aggregate data generated by a large population of individuals. We propose a new efficient belief propagation type algorithm over tree graphs with polynomial computational complexity as well as a global convergence guarantee. This is in contrast to previous methods that either exhibit prohibitive complexity as the population grows or do not guarantee convergence. Our method is based on optimal transport, or more specifically, multimarginal optimal transport theory. In particular, we consider an inference problem with aggregate observations, that can be seen as a structured multimarginal optimal transport problem where the cost function decomposes according to the underlying graph. Consequently, the celebrated Sinkhorn/iterative scaling algorithm for multi-marginal optimal transport can be leveraged together with the standard belief propagation algorithm to establish an efficient inference scheme which we call Sinkhorn belief propagation (SBP). We further specialize the SBP algorithm to cases associated with hidden Markov models due to their significance in control and estimation. We demonstrate the performance of our algorithm on applications such as inferring population flow from aggregate observations. We also show that in the special case where the aggregate observations are in Dirac form, our algorithm naturally reduces to the standard belief propagation algorithm.