Stochastic Unrelatedness, Couplings, and Contextuality

TL;DR

This paper explores the concepts of stochastic unrelatedness and couplings, emphasizing their fundamental role in probability theory and the foundational understanding of stochastic independence and contextuality.

Contribution

It clarifies the distinction between stochastic unrelatedness and independence, highlighting the importance of couplings in the foundation of probability and contextuality theory.

Findings

Stochastic independence is a special case of stochastic dependence.

Stochastic unrelatedness occurs when variables have no joint distribution.

Couplings can impose joint distributions on unrelated variables in many ways.

Abstract

R. Duncan Luce once mentioned in a conversation that he did not consider Kolmogorov's probability theory well-constructed because it treats stochastic independence as a "numerical accident," while it should be treated as a fundamental relation, more basic than the assignment of numerical probabilities. I argue here that stochastic independence is indeed a "numerical accident," a special form of stochastic dependence between random variables (most broadly defined). The idea that it is fundamental may owe its attractiveness to the confusion of stochastic independence with stochastic unrelatedness, the situation when two or more random variables have no joint distribution, "have nothing to do with each other." Kolmogorov's probability theory cannot be consistently constructed without allowing for stochastic unrelatedness, in fact making it a default situation: any two random variables recorded under mutually incompatible conditions are stochastically unrelated. However, stochastically unrelated random variables can always be probabilistically coupled, i.e., imposed a joint distribution upon, and this generally can be done in an infinity of ways, independent coupling being merely one of them. The notions of stochastic unrelatedness and all possible couplings play a central role in the foundation of probability theory and, especially, in the theory of probabilistic contextuality.