Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo

•Representing uncertainty in learning of dynamical systems is important.•Self-contained tutorial to the use of the particle filter inside Metropolis-Hastings.•The method converges to the true solution also for nonlinear models.•A new modeling language makes the new algorithms accessible. Probabilistic modeling provides the capability to represent and manipulate uncertainty in data, models, predictions and decisions. We are concerned with the problem of learning probabilistic models of dynamical systems from measured data. Specifically, we consider learning of probabilistic nonlinear state-space models. There is no closed-form solution available for this problem, implying that we are forced to use approximations. In this tutorial we will provide a self-contained introduction to one of the state-of-the-art methods—the particle Metropolis-Hastings algorithm—which has proven to offer a practical approximation. This is a Monte Carlo based method, where the particle filter is used to guide a Markov chain Monte Carlo method through the parameter space. One of the key merits of the particle Metropolis-Hastings algorithm is that it is guaranteed to converge to the “true solution” under mild assumptions, despite being based on a particle filter with only a finite number of particles. We will also provide a motivating numerical example illustrating the method using a modeling language tailored for sequential Monte Carlo methods. The intention of modeling languages of this kind is to open up the power of sophisticated Monte Carlo methods—including particle Metropolis-Hastings—to a large group of users without requiring them to know all the underlying mathematical details.