Description
Gaussian processes (GP) provide nonparametric approaches for inferring latent functions from data. In this talk I will discuss application of GPs to the case where the function models the drift (i.e. the deterministic driving force) in stochastic differential equations (SDE). I will discuss different scenarios: If the path of the SDE is observed densely in time, drift estimation reduces to simple GP regression. If observations are sparse in time, the GP likelihood can not be computed in an efficient way and we resort to an approximate EM algorithm where the unobserved paths of the SDE are treated as latent random variables. Since the complete data likelihood contains a continuum of infinite latent data we use a sparse Gaussian variational approximation to obtain a tractable MAP estimator. Finally, we discuss cases, where GPs can be used to estimate the drift using the empirical density of observations alone.