Approximate Bayes learning of stochastic differential equations

Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process regression. We also develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation. I. INTRODUCTION Dynamical systems in the physical world evolve in continuous time and often the (noisy) dynamics is described naturally in terms of (stochastic) differential equations [1]. However, due to missing information and/or the complexity of a system it may be difficult to derive such a model from first principles. Instead, the goal often is to fit it to observations of the state at discrete points in time [2]. So far most inference approaches for these systems have dealt with the estimation of parameters contained in the drift function (e.g. Assumptions for the stochastic part were often simple: additive noise with the diffusion constant as the only parameter to estimate. But as both drift and diffusion can be nonlinear functions of the state vector, a nonparametric estimation would be a natural generalization, when a large number of data points is available. Previous nonparametric approaches were based on solving the adjoint Fokker-Planck equation [5] and on kernel estimators [6] and are effectively restricted to one-dimensional models. An alternative would be a Bayesian nonparametric approach, where prior knowledge on the unknown functions--such as smoothness, variability, or periodicity--can be encoded in a probability distribution. A recent result by [7, 8] presented an important step in this direction. The authors have shown that Gaussian processes (GPs) provide a natural family of prior probability measures over drift functions. If a path of the stochastic dynamics is observed densely, the posterior process over the drift is also a GP. Unfortunately, this simplicity is lost, when observations are not dense, but separated by larger time intervals. In [7] the case of sparse observations has been treated by a Monte Carlo approach, which alternates between sampling complete diffusion paths of the stochastic differential equation (SDE) and sampling from GP for the drift given a philipp.batz@tu-berlin.de