Rates of Contraction of Posterior Distributions Based on Gaussian Process Priors

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 bal...

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Bibliographic Details
Published in:The Annals of statistics 2008-06, Vol.36 (3), p.1435-1463
Main Authors: van der Vaart, A. W., van Zanten, J. H.
Format: Article
Language:English
Subjects:
60G15
62G05
Banach space
Bayesian inference
Brownian motion
classification
Density
Density estimation
Distribution theory
Exact sciences and technology
Expansion & contraction
Functional analysis
General topics
Hilbert spaces
Integers
Mathematical analysis
Mathematical functions
Mathematical independent variables
Mathematics
nonparametric density estimation
nonparametric regression
Normal distribution
Parameter estimation
Polynomials
Probability and statistics
Probability theory and stochastic processes
Rate of convergence
Regression analysis
Sciences and techniques of general use
Series expansion
Statistics
Stochastic processes
Studies
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/009053607000000613