  Bayesian optimisation (BO) has been a successful approach to optimise
functions which are expensive to evaluate and whose observations are noisy.
Classical BO algorithms, however, do not account for errors about the location
where observations are taken, which is a common issue in problems with physical
components. In these cases, the estimation of the actual query location is also
subject to uncertainty. In this context, we propose an upper confidence bound
(UCB) algorithm for BO problems where both the outcome of a query and the true
query location are uncertain. The algorithm employs a Gaussian process model
that takes probability distributions as inputs. Theoretical results are
provided for both the proposed algorithm and a conventional UCB approach within
the uncertain-inputs setting. Finally, we evaluate each method's performance
experimentally, comparing them to other input noise aware BO approaches on
simulated scenarios involving synthetic and real data.
