Reproducing kernel Hilbert spaces of Gaussian priors

2008/05/21 by A. W. van der Vaart, J. H. van Zanten · 1 citation
Mathematics · #math.FA #math.OA #math.ST #stat.TH #msc:60G15 #msc:62G05

paper · pdf · doi:10.1214/074921708000000156

published as IMS Collections 2008, Vol. 3, 200-222 · Published in at http://dx.doi.org/10.1214/074921708000000156 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2008/05/21 · arxiv updated 2009/12/01

Abstract

We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reproducing kernel Hilbert spaces under linear maps are useful tools to compute the concentration function.

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