Belief Propagation and Beyond for Particle Tracking
  We describe a novel approach to statistical learning from particles tracked while moving in a random environment. The problem consists in inferring properties of the environment from recorded snapshots. We consider here the case of a fluid seeded with identical passive particles that diffuse and are advected by a flow. Our approach rests on efficient algorithms to estimate the weighted number of possible matchings among particles in two consecutive snapshots, the partition function of the underlying graphical model. The partition function is then maximized over the model parameters, namely diffusivity and velocity gradient. A Belief Propagation (BP) scheme is the backbone of our algorithm, providing accurate results for the flow parameters we want to learn. The BP estimate is additionally improved by incorporating Loop Series (LS) contributions. For the weighted matching problem, LS is compactly expressed as a Cauchy integral, accurately estimated by a saddle point approximation. Numerical experiments show that the quality of our improved BP algorithm is comparable to the one of a fully polynomial randomized approximation scheme, based on the Markov Chain Monte Carlo (MCMC) method, while the BP-based scheme is substantially faster than the MCMC scheme. 
Michael Chertkov, Lukas Kroc, Massimo Vergassola