That is, no sampling method can achieve more collisions than formula_67 on one pair without achieving fewer collisions than formula_67 on another pair, where the reduced pair is more similar under formula_67 than the increased pair. This theorem is true for the Jaccard Index of sets (if interpreted as uniform distributions) and the probability Jaccard, but not of the weighted Jaccard. (The theorem uses the word "sampling method" to describe a joint distribution over all distributions on a space, because it derives from the use of weighted minhashing algorithms that achieve this as their collision probability.)