Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo

Probabilistic modeling provides the capability to represent and manipulate uncertainty in data, models, decisions and predictions. 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 closedform 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 very practical approximations. This is a Monte Carlo based method, where the so-called 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 method is that it is guaranteed to converge to the “true solution” under mild assumptions, despite being based on a practical implementation of a particle filter (i.e., using a finite number of particles). We will also provide a motivating numerical example illustrating the method which we have implemented in an in-house developed modeling language, serving the purpose of abstracting away the underlying mathematics of the Monte Carlo approximations from the user. This modeling language will 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.