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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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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

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

First available in Project Euclid: 28 April 2008

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Mathematical Reviews number (MathSciNet)

Primary: 60G15: Gaussian processes 62G05: Estimation

Bayesian inference rate of convergence

Copyright © 2008, Institute of Mathematical Statistics


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.

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  • [1] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • [2] Cirel'son, B. S. (1975). Density of the distribution of the maximum of a Gaussian process. Teor. Verojatnost. i Primenen. 20 865–873.
  • [3] de Acosta, A. (1983). Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab. 11 78–101.
  • [4] Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500–531.
  • [5] Ghosal, S. and Roy, A. (2006). Posterior consistency in nonparametric regression problem under gaussian process prior. Ann. Statist. 34 2413–2429.
  • [6] Jameson, G. J. O. (1974). Topology and Normed Spaces. Chapman and Hall, London.
  • [7] Kuelbs, J. and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133–157.
  • [8] Kuelbs, J., Li, W. V. and Linde, W. (1994). The Gaussian measure of shifted balls. Probab. Theory Related Fields 98 143–162.
  • [9] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • [10] Li, W. V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for gaussian measures. Ann. Probab. 27 1556–1578.
  • [11] Li, W. V. and Shao, Q.-M. (2001). Gaussian processes: inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods 533–597. Handbook of Statist. 19. North-Holland, Amsterdam.
  • [12] Rudin, W. (1973). Functional Analysis. McGraw-Hill Book Co., New York.
  • [13] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon.
  • [14] Tokdar, S. and Ghosh, J. (2005). Posterior consistency of gaussian process priors in density estimation. J. Statist. Plann. Inference 137 34–42.
  • [15] van der Vaart, A. and van Zanten, J. (2008). Rates of contraction of posterior distributions based on gaussian process priors. Ann. Statist. To appear.
  • [16] van der Vaart, A. W. (1988). Statistical Estimation in Large Parameter Spaces. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam.
  • [17] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.