Contextuality is About Identity of Random Variables

TL;DR

This paper introduces a new approach to contextuality in probability theory, treating random variables recorded under different conditions as inherently different, and uses couplings to analyze non-contextuality without implying causality.

Contribution

It proposes the Contextuality-by-Default framework, redefining contextuality through probabilistic couplings, avoiding assumptions of variable identity across conditions.

Findings

Contextuality is characterized by the non-existence of certain couplings.

Random variable identity is determined by recording conditions, not causality.

The approach clarifies the nature of contextuality without violating physical laws.

Abstract

Contextual situations are those in which seemingly "the same" random variable changes its identity depending on the conditions under which it is recorded. Such a change of identity is observed whenever the assumption that the variable is one and the same under different conditions leads to contradictions when one considers its joint distribution with other random variables (this is the essence of all Bell-type theorems). In our Contextuality-by-Default approach, instead of asking why or how the conditions force "one and the same" random variable to change "its" identity, any two random variables recorded under different conditions are considered different "automatically". They are never the same, nor are they jointly distributed, but one can always impose on them a joint distribution (probabilistic coupling). The special situations when there is a coupling in which these random variables are equal with probability 1 are considered non-contextual. Contextuality means that such couplings do not exist. We argue that the determination of the identity of random variables by conditions under which they are recorded is not a causal relationship and cannot violate laws of physics.