Dynamic expectation maximisation (DEM) is a variational treatment of hierarchical, nonlinear dynamic or static models. It uses a fixed-form Laplace assumption to approximate the conditional, variational or ensemble density of unknown states and parameters. This is an approximation to the density that would obtain from Variational Filtering (VF) in generalized coordinates of motion. The first demonstration with VF uses a simple convolution model and allows one to compare DEM and VF. We also demonstrate the inversion of increasingly complicated models; ranging from a simple General Linear Model to a Lorenz attractor. It is anticipated that the reader will examine the routines called to fully understand the nature of the scheme.
 
DEM presents a variational treatment of dynamic models that furnishes time-dependent conditional densities on the 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 logevidence 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. DEM can be regarded as the fixed-form homologue of variational filtering (which is covered in the demonstrations): Variational filtering represents a simple Bayesian filtering scheme, using variational calculus, for inference on the hidden states of dynamic systems. Variational filtering is a stochastic scheme that propagates particles over a changing variational energy landscape, such that their sample density approximates the conditional density of hidden states and inputs. Again, he key innovation, on which variational filtering rests, is a formulation in generalised coordinates of motion. This renders the scheme much simpler and more versatile than existing approaches, such as those based on particle filtering. We demonstrate variational filtering using simulated and real data from hemodynamic systems studied in neuroimaging and provide comparative evaluations using particle filtering and the fixed-form homologue of variational filtering, namely dynamic expectation maximisation.
 
K.J. Friston. Variational filtering. NeuroImage, 41(3):747-766, 2008.
K.J. Friston, N. Trujillo-Bareto, and J. Daunizeau. DEM: A variational treatment of dynamic systems. NeuroImage, 41(3):849-885, 2008.
 
These descriptions of the new features are taken from the SPM8 Release Notes
 
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