Posterior consistency of Gaussian process prior for nonparametric binary regression

Details Posterior consistency of Gaussian process prior for nonparametric binary regression Posterior consistency of Gaussian process prior for nonparametric binary regression

2007-02-23

arxiv.org/abs/math/0702686v1

Annals of Statistics 2006, Vol. 34, No. 5, 2413-2429

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000795 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000795

Consider binary observations whose response probability is an unknown smooth function of a set of covariates. Suppose that a prior on the response probability function is induced by a Gaussian process mapped to the unit interval through a link function. In this paper we study consistency of the resulting posterior distribution. If the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values, we show that the posterior distribution is consistent in the $L_1$-distance. As an auxiliary result to our proofs, we show that, under certain conditions, a Gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function. This result may be of independent interest in the literature for small ball probabilities of Gaussian processes.

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