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Posterior consistency of Gaussian process prior for nonparametric binary regression. (English) Zbl 1106.62039
Summary: 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. 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.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
46N30 Applications of functional analysis in probability theory and statistics
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