  Preferential Bayesian optimisation (PBO) deals with optimisation problems
where the objective function can only be accessed via preference judgments,
such as "this is better than that" between two candidate solutions (like in A/B
tests or recommender systems). The state-of-the-art approach to PBO uses a
Gaussian process to model the preference function and a Bernoulli likelihood to
model the observed pairwise comparisons. Laplace's method is then employed to
compute posterior inferences and, in particular, to build an appropriate
acquisition function. In this paper, we prove that the true posterior
distribution of the preference function is a Skew Gaussian Process (SkewGP),
with highly skewed pairwise marginals and, thus, show that Laplace's method
usually provides a very poor approximation. We then derive an efficient method
to compute the exact SkewGP posterior and use it as surrogate model for PBO
employing standard acquisition functions (Upper Credible Bound, etc.). We
illustrate the benefits of our exact PBO-SkewGP in a variety of experiments, by
showing that it consistently outperforms PBO based on Laplace's approximation
both in terms of convergence speed and computational time. We also show that
our framework can be extended to deal with mixed preferential-categorical BO,
where binary judgments (valid or non-valid) together with preference judgments
are available.
