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In Bayesian optimization, accounting for the importance of the outputrelative to the input is a crucial yet challenging exercise, as it canconsiderably improve the final result but often involves inaccurate andcumbersome entropy estimations. We approach the problem from the perspective ofimportance-sampling theory, and advocate the use of the likelihood ratio toguide the search algorithm towards regions of the input space where theobjective function to be minimized assumes abnormally small values. Thelikelihood ratio acts as a sampling weight and can be computed at eachiteration without severely deteriorating the overall efficiency of thealgorithm. In particular, it can be approximated in a way that makes theapproach tractable in high dimensions. The "likelihood-weighted" acquisitionfunctions introduced in this work are found to outperform their unweightedcounterparts in a number of applications.

Bayesian Optimization with Output-Weighted Optimal Sampling