DEM: a variational treatment of dynamic systems

This paper presents a variational treatment of dynamic models that furnishes time-dependent conditional densities on the path or trajectory of a system's states and the time-independent densities of its parameters. These are obtained by maximising a variational action with respect to conditional densities, under a fixed-form assumption about their form. The action or path-integral of free-energy represents a lower bound on the model's log-evidence or marginal likelihood required for model selection and averaging. This approach rests on formulating the optimisation dynamically, in generalised coordinates of motion. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for dual and triple inferences on a system's states, parameters and hyperparameters using exactly the same principles. We refer to this approach as dynamic expectation maximisation (DEM).