Rates of contraction of posterior distributions based on Gaussian process priors

Details Rates of contraction of posterior distributions based on Gaussian process priors Rates of contraction of posterior distributions based on Gaussian process priors

2008-06-18

arxiv.org/abs/0806.3024v1

Annals of Statistics 2008, Vol. 36, No. 3, 1435-1463

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/009053607000000613 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000613

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.

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