Sequential Bayesian inference for drift estimation in nonlinear stochastic differential equations
From a Bayesian perspective, data assimilation is used to estimate a posterior probability distribution over model states and parameters, conditioned on noisy data time series. The model itself must additionally be inferred from data if it is not fully specified, posing a significant challenge in nonlinear systems observed in discrete time at low frequency. In this context, we investigate the use of nonparametric, Bayesian sequential Monte Carlo methods for drift estimation and model selection in nonlinear stochastic differential equation (SDE) systems observed in discrete time, with a Gaussian process prior on drift functions. Computationally, difficulties arise in the nonparametric setting from the associated high-dimensional estimation problem, and additionally nonlinear drift terms make the likelihood difficult to compute and lead to non-Gaussian posterior distributions. We use several numerical examples in one and two dimensions to show under what conditions the drift is recoverable from data, in particular investigating the effects of various errors introduced by likelihood approximations, and varying model noise and observation frequency; the influence of prior covariance and hyperparameter selection on posterior estimates; and the application of localization techniques to GP drift priors to improve computational efficiency and avoid issues of finite sample size in extended systems.
Dr. Paul Joseph Rozdeba University of Potsdam