ISBA 2012 Lecture on Bayesian Foundations
Title: Confidence in nonparametric credible sets?
Aad van der Vaart (University of Leiden, Netherlands)
In nonparametric statistics the posterior distribution is used in exactly the same way as in any Bayesian analysis. It supposedly gives us the likelihood of various parameter values given the data. A difference with parametric analysis is that it is often difficult to have an intuitive understanding of the prior, which affects the believability of the posterior distribution as a quantification of uncertainty. A second difference is that the posterior distribution is much more sensitive to the prior: its “fine properties” matter. This is true even in the asymptotic situation when the informativeness of the data increases indefinitely. In this talk, we start by reviewing frequentist asymptotic results and insights on posterior distributions in the semi- and nonparametric setting obtained in the last decade. These results show that posterior distributions can be effective in recovering a true parameter provided some care is taken when choosing a prior. We next go on to ask whether posterior distributions are also capable in giving a correct idea of error in the reconstructions. Are credible sets in any way comparable to confidence regions? We shall not present an answer to this question, but show by example that it will be delicate.