Universiteit Leiden

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Credible sets in nonparametric regression

Promotor: Prof.dr. A.W. van der Vaart

S. Sniekers
01 december 2015
Thesis in Leiden Repository

A common problem in Bayesian statistics is to determine whether a quantity obtained from a Bayesian posterior distribution is also meaningful in a frequentist context. In this thesis, we try to answer this question for credible sets in the so-called fixed design model. Taking a specific prior distribution, we study whether credible sets based on this prior can also be used as confidence intervals. In particular, our aim is to construct a credible set for a parameter function. Under certain assumptions on the smoothness of the function, it turns out that we can obtain meaningful results about both the frequentist coverage and the width of the credible set. We consider several different classes of functions in this thesis, each with a different set of assumptions. In the first chapter, we assume that we know rather a lot about the structure of the function, and use this to obtain a useful method. In chapter 2, we extend this to a method that adapts to the structure of the function. Finally, in the last chapter, we extend this to so-called credible bands, that describe the behaviour of the entire function, rather than at a specific point.