  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.
