  In Bayesian optimization, accounting for the importance of the output
relative to the input is a crucial yet challenging exercise, as it can
considerably improve the final result but often involves inaccurate and
cumbersome entropy estimations. We approach the problem from the perspective of
importance-sampling theory, and advocate the use of the likelihood ratio to
guide the search algorithm towards regions of the input space where the
objective function to be minimized assumes abnormally small values. The
likelihood ratio acts as a sampling weight and can be computed at each
iteration without severely deteriorating the overall efficiency of the
algorithm. In particular, it can be approximated in a way that makes the
approach tractable in high dimensions. The "likelihood-weighted" acquisition
functions introduced in this work are found to outperform their unweighted
counterparts in a number of applications.
