Contextuality In Random Variables – Ehtibar N Dzhafarov (1)

It can be directly verified that in the former case S2 q ≽S1 q (equivalently, q ≽R1 q) and in the latter case S1 q ≽S2 q (equivalently, R1 q ≽R2 q). Indeed, in the first matrix, α1 ≤β1 because α1 = p11 ≤p11 + p41 = β1; and we show analogously that α2 ≤β2, α3 ≤β3, α5 ≤β5, α6 ≤β6. The only exception is α4 ≥β4 because α4 = p41 +p42 +· · ·+p46 ≥p44 = β4.

This means R2 q ≽R1 q. That R1 q ≽R2 q in the second case is shown similarly. Let us now demonstrate the “if” part of the proposition: nominal domi- nance ⇒noncontextuality. Using again the case with k = 6 as an example, let R1 q ≽R2 q.

Without loss of generality, assume that αi ≥βi for all i except for i = 2. Then the matrix q = 1 q = 1 α1 α2 α3 α4 α5 α6 β1 β2 β3 β4 β5 β6 can be filled in as follows: q = 1 q = 1 β1 α1 −β1 α1 α2 α2 α3 −β3 β3 α3 α4 −β4 β4 α4 α5 −β5 β5 α5 α6 −β6 β6 α6 β1 β2 β3 β4 β5 β6 . Contextuality II: Dichotomizations and Criteria of Contextuality One can verify that the probabilities in each row and each column sum to the corresponding marginals.

In particular, (α1 −β1) + α2 + (α3 −β3) + (α4 −β4) + (α5 −β5) + (α6 −β6) = 1 −β1 −β3 −β4 −β5 −β6 = β2. This proves that if R1 q ≽R2 q, the coupling q, S2 q

of the form implying noncontextuality does exist. The case R2 q ≽R1 q is symmetrical. One consequence of the proposition just proved is this: Contextuality of Systems That Fail the Nominal Dominance Test: If a system of categorical random variables contains a pair of content-sharing variables  Rc q, Rc′ q  neither of which nominally dominates the other, then the complete dichotomization of the system is contextual.

This proposition follows from the fact that a system is contextual whenever any of its subsystems is contextual. This is a useful proposition because any system of variables, possibly very large, can be analyzed by considering all pairs of its content-sharing variables: And if at least one of these pairs fails the nominal dominance test, the dichotomization of the system is contextual.

The mathematical essence of contextuality lies in the similarity of random variables answering the same question in different contexts: Contextuality means they are less similar when considered within their respective contexts than when isolated from them. This book presents a principled way of measuring this similarity and distin- guishing two forms of context-dependence: contextuality and disturbance. While applicable across a broad range of disciplines, the concept of contextuality in this book is closest to that in quantum physics, where its special forms – in the absence of disturbance – are known as Bell nonlocality and Kochen–Specker contextuality.

This systematic introduction requires no prior familiarity with the subject and a very modest mathematical background. Structured as a textbook, complete with exercises and solutions, it is accessible to a broad readership and suitable for teach- ing. It will be useful to researchers and students in quantum mechanics, philosophy of science, psychology, computer science, linguistics, and probability theory. E H T I B A R N.

D Z H A FA ROV is Professor Emeritus at Purdue University, USA. He has published over 170 papers in psychology, mathematics, philosophy, and foundations of quantum mechanics, and he has edited six books and four special journal issues. He served as President of the Society for Mathematical Psychology and has received a Humboldt Research Award. JA N N E V. K U JA L A is Associate Professor at the University of Turku, Finland.

He has published over 60 papers in computational statistics, applied probability, foundations of quantum mechanics, mathematical psychology, and learning ana- lytics, and has edited one special journal issue. He received the William K. Estes Early Career award in mathematical psychology. V Í C TO R H. C E RVA N T E S is Assistant Professor at the University of Illinois Urbana-Champaign, USA. He was awarded a Fulbright Scholarship in 2014, and he received his doctoral degree in mathematical and computational cognitive science from Purdue University.

CONTEXTUALITY IN RANDOM VARIABLES A Systematic Introduction EHTIBAR N. DZHAFAROV Purdue University, USA JANNE V. KUJALA University of Turku, Finland VÍCTOR H.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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