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Wednesday, 18.07.2018, 10:00 (WIAS-ESH)
Forschungsseminar Mathematische Statistik
Dr. J. van Waaij, Humboldt Universität zu Berlin:
Adaptive nonparametric Bayesian methods for SDEs
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

We consider continuous time observations (Xt_0le tle T of the SDE dXt = Theta(Xt)dt + dWt: Our goal is estimating the unknown drift function Theta. Due to their numerical advantages, Bayesian methods are often used for this. In this talk I discuss optimal rates of convergence for these methods. I will start with an introduction to Bayesian methods for SDEs and what sufficient con- ditions are for posterior convergence. I show that the sufficient conditions are satisfied for the Gaussian process prior, which leads to optimal convergence rates, provided the smoothness of Theta matches the smootness of the Gaussian process. Adaptivity can be obtained by equipping the hyperparameter(s) of the Gaussian process prior with an additional prior. We discuss several choices for such hyperparameters. A new promising approach to obtain adaptivity is empirical Bayes. Here the optimal hyperparameters for the Gaussian process prior are first esti- mated from the data and then the prior with those hyperparameters plugged- in is used for the inference. This talk is based on van Waaij, 2018 and joint work with Frank van der Meulen (TU Delft), Moritz Schauer (Leiden University) and Harry van Zanten (University of Amsterdam).

Humboldt-Universität zu Berlin
Universität Potsdam
WIAS Berlin