Variational Filtering and Inference

We present a variational treatment of dynamic models that furnishes the time-dependent conditional densities of a system's states and the time-independent densities of its parameters. These obtain by maximising the variational free energy of the system with respect to the conditional densities. The ensuing free energy represents a lower-bound approximation to the models marginal likelihood or log-evidence required for model selection and averaging. This approach rests on formulating the optimization of free energy dynamically, in generalised coordinates of motion. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and eschews some limitations of existing approaches, such as Kalman and particle filtering. We refer to this approach as dynamic expectation maximisation (DEM).