Inference with GPs. In BO, we exploit GPs for regression, with uncertainty quantification, to build a surrogate model for f (x). We have observations {(x i , y i )} n i=1 from previous experiments (previous iterations of BO), with y i the measured property of NPM x i . Under the Bayesian view, y i is a noise-free observation of the random variable f (x i ). We wish to know the distribution of the random variable f (x) for an unevaluated NPM x, to determine the utility of evaluating it in the next experiment. Imposing the assumption in eqn. for a specific collection of points on the domain of f (x) composed of (i) the n evaluated NPMs from the past experiments {x 1 , x 2 , ..., x n } and (ii) the unevaluated NPM, x: