# Contextuality in Canonical Systems of Random Variables

## TL;DR

This paper introduces a canonical framework for analyzing contextuality in systems of binary random variables, providing a criterion for determining when such systems are contextual based on their maximal couplings.

## Contribution

It proposes a canonical representation for measurement systems and establishes a criterion for contextuality, especially for dichotomizations of categorical variables.

## Key findings

- Canonical representation simplifies analysis of measurement systems.
- A criterion for contextuality in systems of dichotomized variables.
- Illustration with a system of categorical random variables.

## Abstract

Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables.