A fundamental task of intelligent and autonomous robots is to infer from observations the state of the world. This inference is generally achieved by employing a filter, which consists of a model and filtering law. Learning this model and filtering law from observations is another fundamental part of robotics, and is generally referred to as system identification. Neuroscientist K.J. Friston has developed a relatively novel theory on biologically plausible human brain inference called the Free-Energy Principle. One of the theories within the Free-Energy Principle, namely that Dynamic Expectation Maximization (DEM), has been suggested as a novel method for filtering and system identification. This method is expected to outperform standard Expectation Maximization (EM) in terms of hidden state and parameter estimation in settings where noise is correlated. However, in order for this neuroscientific theory to be properly used for robot inference, two problems must first be solved. The first of these problems is the fact that the theory is defined in the continuous-time domain, whereas data available for system identification is always discrete. In this thesis I will suggest three discrete-time interpretations for DEM-based system identification. The major difference between the three methods is the information that is embedded in the generalized signals: predictions, derivatives and past data. The second problem is that the filtering method corresponding to the Free-Energy Principle depends on data which is not available: the derivative signals of measured in- and outputs. I introduce two fundamentally different solutions to this feasibility issue: a numerical differentiator and a stable filter. Both of these solutions are shown to find an estimate for the unavailable data. However, the former is shown to significantly outperform the latter. Furthermore, the theory described in this thesis is implemented into a novel python toolbox for system identification. This toolbox can be used as a basis for further research and be approved along with it, until at some point it is ready to be used for real applications. Using the toolbox, the DEM-based identification and filtering methods are tested though various numerical simulations and the results are compared with the EM method. Results show that with the implemented settings none of the suggested discrete-time filtering methods outperforms the conventional Kalman filter. The main cause of this inferior performance is shown to be instability in the filtering method. I make some suggestions for overcoming this problem. As a result of the inferior performance, the joint-performance of the suggested DEM-based parameter- and state- estimation methods also proves to be inferior in terms of parameter estimation accuracy. However, results show that the theoretical parameter optima of the Free-Energy as determined from known hidden states are in fact close on the real parameters, and furthermore show to be invariant to noise correlation. This suggests that should the instability issue as some point be solved and a better means to approximate the theoretical optimum be found, the DEM-based methods might in fact outperform EM both in terms of hidden-state and parameter accuracy in settings with correlated noise.