The Probability Jaccard Index has a geometric interpretation as the area of an intersection of simplices. Every point on a unit formula_44-simplex corresponds to a probability distribution on formula_45 elements, because the unit formula_44-simplex is the set of points in formula_45 dimensions that sum to 1. To derive the Probability Jaccard Index geometrically, represent a probability distribution as the unit simplex divided into sub simplices according to the mass of each item. If you overlay two distributions represented in this way on top of each other, and intersect the simplices corresponding to each item, the area that remains is equal to the Probability Jaccard Index of the distributions.