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

TL;DR

This paper introduces a class of metric-like functions on jointly distributed random variables, useful for analyzing probabilistic causality and joint distribution existence in behavioral sciences and quantum physics.

Contribution

It constructs and studies order-distance and classification distance functions that serve as tools for causality analysis and joint distribution verification.

Findings

Order-distance satisfies the triangle inequality.

Violation of the triangle inequality indicates no joint distribution exists.

These functions are applicable in behavioral sciences and quantum physics.

Abstract

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). These functions are useful in dealing with 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 or its consequences by one of our functions when applied to such a set rules out the existence of this joint distribution. We focus on an especially versatile and widely applicable pseudo-quasi-metric called an order-distance and its special case called a classification distance.