Order-distance and other metric-like functions on jointly distributed random variables

Abstract

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables. These functions satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). We apply these functions to the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality by one of our functions when applied to such a set rules out the existence of the joint distribution. We focus on an especially versatile and widely applicable class of pseudo-quasi-metrics called order-distances. We show, in particular, that the Bell-CHSH-Fine inequalities of quantum physics follow from the triangle inequalities for appropriately defined order-distances.