Rates of contraction of posterior distributions based on Gaussian process priors

2008/05/26 by Aad van der Vaart, A. W. van der Vaart, J. H. van Zanten · 16 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Gaussian Processes and Bayesian Inference #Statistical Methods and Inference #math.ST #msc:60G15 #msc:62G05 #stat.TH

paper · pdf · doi:10.1214/009053607000000613

published as Annals of Statistics 2008, Vol. 36, No. 3, 1435-1463 · Published in at http://dx.doi.org/10.1214/009053607000000613 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2008/05/26 · arxiv created 2008/06/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors.

Citations

• Fractional Brownian Motions, Fractional Noises and Applications
• Bayesian Analysis of Binary and Polychotomous Response Data

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