  In order to improve the performance of Bayesian optimisation, we develop a
modified Gaussian process upper confidence bound (GP-UCB) acquisition function.
This is done by sampling the exploration-exploitation trade-off parameter from
a distribution. We prove that this allows the expected trade-off parameter to
be altered to better suit the problem without compromising a bound on the
function's Bayesian regret. We also provide results showing that our method
achieves better performance than GP-UCB in a range of real-world and synthetic
problems.
