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arXiv:2511.01783 · Cambridge University Press eBooks

Preface to Contextuality in Random Variables: A Systematic Introduction, by E. N. Dzhafarov, J. V. Kujala, and V. H. Cervantes

Ehtibar N. Dzhafarov

math.PRquant-phSubmitted Nov 3, 2025• Announced Nov 4, 2025• 1 citation• Revised Nov 4, 2025 (v2)
v1 · Nov 3, 2025v2 · Nov 4, 2025
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Abstract

This is the preface for the book by E. N. Dzhafarov, J. V. Kujala, and V. H. Cervantes, titled Contextuality in Random Variables: A Systematic Introduction. It is to be published by Cambridge University Press in 2026.
DOI
10.1017/9781009742221

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We introduce a new notion, that of a contextuality profile of a system of random variables. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating degree of contextuality to level at which the system is considered, level1⋯n−1n>1n+1⋯Ndegree0⋯0dn>0dn+1≥dn⋯dN≥dN−1 \begin{array}{c|c|c|c|c|c|c|c} \textnormal{level} & 1 & \cdots & n-1 & n>1 & n+1 & \cdots & N\\ \hline \textnormal{degree} & 0 & \cdots & 0 & d_{n}>0 & d_{n+1}\geq d_{n} & \cdots & d_{N}\geq d_{N-1} \end{array} leveldegree​10​⋯⋯​n−10​n>1dn​>0​n+1dn+1​≥dn​​⋯⋯​NdN​≥dN−1​​​, where N is the maximum number of variables per system's context. A system is represented at level n if one only considers the joint distributions with k≤n k\leq n k≤n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.
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The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets (“linkedness” being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point.