Bayesian Parameter Inference for 1D Nonlinear Stochastic Differential Equation Models

Bayesian statistics has become an indispensable tool in many applied sciences, in particular, for the purpose of uncertainty analysis. Inferring parametric uncertainty, for stochastic differential equation (SDE) models, however, is a computationally hard problem due to the high dimensional integrals that have to be calculated. Here, we consider the generic problem of calibrating a one dimensional SDE model to time series and quantifying the ensuing parametric uncertainty. We re-interpret the Bayesian posterior distribution, for model parameters, as the partition function of a statistical mechanical system and employ a Hamiltonian Monte Carlo algorithm to sample from it. Depending on the number of discretization points and the number of measurement points the dynamics of this system happens on very different time scales. Thus, we employ a multiple time scale integration together with a suitable re-parametrization to derive an efficient inference algorithm. While the algorithm is presented by means of a simple SDE model from hydrology, it is readily applicable to a wide range of inference problems. Furthermore the algorithm is highly parallelizable.