Continuous-time belief propagation

Many temporal processes can be naturally modeled as a stochastic system that evolves con-tinuously over time. The representation language of continuous-time Bayesian networks allows to succinctly describe multi-component continuous-time stochastic processes. A crucial element in applications of such models is inference. Here we introduce a variational approximation scheme, which is a natural extension of Belief Propagation for continuous-time processes. In this scheme, we view messages as inhomogeneous Markov processes over individual components. This leads to a relatively simple procedure that allows to easily incorporate adaptive ordinary differential equation (ODE) solvers to perform individual steps. We provide the theoretical foundations for the approximation, and show how it performs on a range of networks. Our results demonstrate that our method is quite accurate on singly connected networks, and provides close approximations in more complex ones.