Variational filtering in generated coordinates of motion
Karl Friston, University College London, United Kingdom
This presentation reviews 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 obtain by maximizing 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 optimization in generalized co-ordinates of motion. The resulting scheme can be used for on-line 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 inference on a system’s states, parameters and hyperparameters using exactly the same principles. Free-form (Variational filtering) and fixed form (Dynamic Expectation Maximization) variants of the scheme will be demonstrated using simulated (bird-song) and real data (from hemodynamic systems studied in neuroimaging).
