Dynamical Systems, Stochastic Processes and Bayesian Inference

The modelling of continuous-time dynamical systems from uncertain observations is an important task that comes up in a wide range of applications ranging from numerical weather prediction over finance to genetic networks and motion capture in video. Often, we may assume that the dynamical models are formulated by systems of differential equations. In a Bayesian approach, we may then incorporate a priori knowledge about the dynamics by providing probability distributions over the unknown functions, which correspond for example to driving forces and appear as coefficients or parameters in the differential equations. Hence, such functions become stochastic processes in a probabilistic Bayesian framework. Gaussian processes (GPs) provide a natural and flexible framework in such circumstances. The use of GPs in the learning of functions from data is now a well-established technique in Machine Learning. Nevertheless, their application to dynamical systems becomes highly nontrivial when the dynamics is nonlinear in the (Gaussian) parameter functions as closed form analytical posterior predictions (even in the case of Gaussian observation noise) are no longer possible. Moreover, their computation requires the entire underlying Gaussian latent process at all times (not just at the discrete observation times). Hence, inference of the dynamics would require nontrivial sampling methods or approximation techniques. The aim of this workshop is to provide a forum for discussing open problems related to stochastic dynamical systems, their links to Bayesian inference and their relevance to Machine Learning.