  In this work we introduce a new framework for multi-objective Bayesian
optimisation where the multi-objective functions can only be accessed via
choice judgements, such as ``I pick options A,B,C among this set of five
options A,B,C,D,E''. The fact that the option D is rejected means that there is
at least one option among the selected ones A,B,C that I strictly prefer over D
(but I do not have to specify which one). We assume that there is a latent
vector function f for some dimension $n_e$ which embeds the options into the
real vector space of dimension n, so that the choice set can be represented
through a Pareto set of non-dominated options. By placing a Gaussian process
prior on f and deriving a novel likelihood model for choice data, we propose a
Bayesian framework for choice functions learning. We then apply this surrogate
model to solve a novel multi-objective Bayesian optimisation from choice data
problem.
