Nonparametric Bayesian uncertainty quantification
Aad van der Vaart, Leiden University
In Bayesian nonparametrics a functional parameter (density, regression function) is equipped with a prior distribution, and a posterior distribution is obtained in the standard manner. The center of the posterior distribution can be used as a point estimator. It has been documented in fair generality that reasonable priors give good reconstructions of an unknown function. In particular, priors that come with a bandwidth or sparsity parameter that is tuned to the data by a hierarchical or empirical Bayes method typically lead to posterior distributions that contract at optimal rates to the true function. However, at the core of the Bayesian method is uncertainty quantification through the full spread of the posterior distribution. For instance, one would hope that the area covered by a plot of a sample of draws (functions) from the posterior distribution can be interpreted as a confidence set. The purpose of the three talks is to investigate to what extent this is justified. A full and general answer is currently not available, but we discuss results for special models, which are thought to extend to other models as well. The talks will assume no prior knowledge of Bayesian nonparametrics; we shall start with examples of priors and contraction rates.