The Upper Confidence Bound in Disguise

Probability of improvement fixes a threshold $\tau$ and reads off a probability: how much of the predictive mass sits above $\tau$? Expected improvement integrates the gap above $\tau$. The upper confidence bound 1 fixes a probability $q$ and reads off a threshold: the value that only $1 - q$ of the mass exceeds. All three are views of the same conditional CDF. Three views of one conditional CDF.…