  The main challenges that arise when adopting Gaussian Process priors in
probabilistic modeling are how to carry out exact Bayesian inference and how to
account for uncertainty on model parameters when making model-based predictions
on out-of-sample data. Using probit regression as an illustrative working
example, this paper presents a general and effective methodology based on the
pseudo-marginal approach to Markov chain Monte Carlo that efficiently addresses
both of these issues. The results presented in this paper show improvements
over existing sampling methods to simulate from the posterior distribution over
the parameters defining the covariance function of the Gaussian Process prior.
This is particularly important as it offers a powerful tool to carry out full
Bayesian inference of Gaussian Process based hierarchic statistical models in
general. The results also demonstrate that Monte Carlo based integration of all
model parameters is actually feasible in this class of models providing a
superior quantification of uncertainty in predictions. Extensive comparisons
with respect to state-of-the-art probabilistic classifiers confirm this
assertion.
