where (z) + = max(0, z), q S (x) = m(x) + Φ -1 (β)s(x) is the β-quantile from the current GP posterior (see, for example, Rasmussen and Williams equation 7 for the details on the posterior distribution of a GP) with β ∈ [0.5, 1), x ⋆ = argmin x∈X S q S (x) and Q S+1 (x S+1 ) is the corresponding β-quantile when one additional observation y(x S+1 ) is added to the data set.