Expectation propagation for diffusion processes by moment closure approximations

We consider the inverse problem of reconstructing the trajectory of a diffusion process from discrete and continuous time (soft) observations. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of state space marginals using variational approximate inference. The resulting optimisation algorithm is a hybrid expectation propagation/variational message passing algorithm. We then show how the approximation can be extended to a wide class of discrete-state Markovian jump processes by making use of the chemical Langevin equation. Our empirical results show that this is a computationally feasible and accurate method to approach these intractable classes of inverse problems.