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Reproducing kernel Hilbert spaces of Gaussian priors

J. H. van Zanten and A. W. van der Vaart

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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.

Chapter information

Source
Bertrand Clarke and Subhashis Ghosal, eds., Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 200-222

Dates
First available in Project Euclid: 28 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1209398470

Digital Object Identifier
doi:10.1214/074921708000000156

Mathematical Reviews number (MathSciNet)
MR2459226

Subjects
Primary: 60G15: Gaussian processes 62G05: Estimation

Keywords
Bayesian inference rate of convergence

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

van der Vaart, A. W.; van Zanten, J. H. Reproducing kernel Hilbert spaces of Gaussian priors. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 200--222, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000156. https://projecteuclid.org/euclid.imsc/1209398470


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