This presentation reviews variational treatments of dynamic models that furnish 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. 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 coordinates of motion. The resulting scheme can be used for on-line Bayesian inversion of nonlinear hierarchical dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for inference on a models 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) data.
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