English: The probability jaccard index is optimal in the following sense. If a sampling method has more collisions than the probability jaccard index on two distributions, x and y, then there must be a pair of distributions that are more similar under the probability jaccard index than x and y are on which that sampling method has fewer collisions. This diagram shows the proof mechanism graphically on 3 element distributions. For x and y to collide more on the green element, they have to give up collisions with z on either red or blue, and can’t gain any more on green to compensate. As a result, the collisions with z must decrease, but z is more similar to both x and y than they are to each other.