Adaptive posterior contraction rates for diffusions

Diffusions have many applications in science and can be described with a stochastic differential equation (SDE). We consider the following SDE, which was for example used in moleculair dynamics (see e.g. Papaspiliopoulos et al. (2012)), dXt=θ(Xt)dt+dWt, where θ is measurable, one-periodic and ∫01θ(x)2dx<∞. We are interested in estimating θ from an observation (Xt:t∈[0,T]) of the SDE. We study the posterior rates of contraction for several nonparametric Bayesian methods for diffusions. For Gaussian process priors we derive optimal posterior contraction rates, when the smoothness of the Gaussian process coincides with the smoothness of the target drift function. Adaptivity to the unknown smoothness is achieved by random scaling of the Gaussian process prior, or by equipping the baseline smoothness hyperparameter with a hyperprior. We derive good adaptive posterior contraction results for priors defined as randomly truncated series priors. We consider expansions in orthonormal bases and in the Faber-Schauder basis, both with inverse gamma scaling. We also study the empirical Bayes approach to selecting the scaling parameter of the Gaussian process prior. Here the parameter is estimated from the data and plugged into the prior. Adaptive optimal contraction rates for the associated posterior are derived