Subject of the book

The book is about systems of random variables, that is, sets of random vari-ables ordered in two ways: by their contents (the questions the variables answer) and by their contexts (conditions under which they are recorded). Such a sys-tem can be contextual or noncontextual. The meaning of contextuality in this book generalizes that in quantum physics, where its special forms are known as nonlocality and Kochen–Specker contextuality. Another related notion is that of selectiveness of influences in cognitive science. However, the treatment in the book is general and abstract, applying to all systems of random variables, whether they describe phenomena in quantum physics, computer science, or psychology.

The mathematical essence of contextuality lies in the similarity of random variables having the same content in different contexts. There is a principled way of measuring how similar two such variables are: by computing the maxi-mal probability with which they could coincide if they were jointly distributed (de facto, no two variables are if they are in different contexts). A system is noncontextual if these maximal probabilities of coincidence for all same-content variables are compatible with the joint distributions of the variables in their re-spective contexts. In the special case when any two same-content variables have the same distribution (consistently connected systems), this definition translates into the following: a system is noncontextual if any two same-content variables can be considered identical. This is the case of nonlocality or Kochen–Specker contextuality.

In contextual systems, the difference in the similarity of two random vari-ables considered within and without their contexts is not due to any physical action of contexts upon the variables. Contextuality neither requires nor admits explanations in terms of physical causes and effects. It is a purely mathematical property, based on the fact that a random variable is a function whose domain is determined by all other variables it is jointly distributed with. Contextuality cannot be pinned to specific variables (as it would have to be if we dealt with causal influences). When a system is contextual, we know that some of the same-content variables are more dissimilar within their contexts than they are without – but one can never say what this difference in similarity is for any particular pair. Whether a system is or is not contextual is a system-wise and system-wide property of the system.

Character of the book

The book systematically and comprehensively covers the basics of contextuality on an abstract mathematical level. This excludes many specialized and domain-specific topics, those that may be of interest in, or even central to, a substantive area of research, such as computer science, psychology, or even quantum physics, in which most of the contextuality research has been conducted.

No previous knowledge of the material is required. The book is written to be accessible to readers with only modest knowledge of mathematics: basic set-theoretic notions and notation, elementary aspects of probability theory, and, in places, basics of linear algebra. Each chapter is followed by exercises with com-plete solutions. Some of them provide additional examples and clarifications, and even offer additional theoretical material. The book is mostly confined to systems with a finite number of random variables, and to variables with a finite number of possible values. This allows the reader to focus on conceptual issues without getting into technical details of measure-theoretic constructs (although these are introduced too, as optional reading).

However, in spite of its technical accessibility, the material presented in this book is advanced enough to be of interest to scholars from a broad spectrum of disciplines: from mathematics to philosophy to quantum physics to computer science to behavioral and social sciences.

Contents of the book’s 11 chapters

Chapter 1 opens with an example of a system of random variables, and it shows how four very different empirical situations, taken from quantum physics, cog-nitive science, and psychometrics, are all described by this system. If random variables are treated as epistemic ones, the same system also describes a vari-ant of the Liar paradox in logic. The chapter introduces the double-indexation of random variables, critical for the contextuality theory, and the notion of a special value of a random variable, interpreted as “undefined ”.

The notions of overt influences and contextuality as components of context-dependence are introduced in Chapter 2, using the example of the smallest pos-sible nontrivial system (a cyclic system of rank 2, describing, e.g., the question-order effect in cognitive science). Overt influences are causal and variable-specific, while contextuality is non-causal and system-wide. The chapter intro-duces the notion of a probabilistic coupling, the main mathematical tool of the contextuality theory.

After reviewing the basics of the theory of random variables in Chapter 3, the book proceeds to elaborate the theory of couplings in Chapter 4. This chapter introduces the notion of maximal and multimaximal couplings, central for the contextuality theory, and studies their existence and uniqueness properties.

The general and systematic theory of contextuality is presented in Chapter 5. It shows that the contextuality status of a system (i.e., whether it is con-textual or noncontextual) is invariant under some but may change under other transformations of the system. The chapter emphasizes multiple ways in which one and the same empirical situation can be parsed into contents and contexts, and the dependence of the contextuality status on the choice of these parsings. Chapter 6 specializes the theory to systems of dichotomous random variables, and shows how any system can be dichotomized. Depending on the structure imposed on the set of values of a variable, its dichotomization may involve all possible or only some of the possible splits of this set into two subsets. The redundancy of the set of all possible splits is irrelevant here: the contextuality status of a system generally changes after one adds redundant splits. It is shown, however, that the contextuality status of the system with all possible splits is determined by a certain, special (and itself redundant) subset thereof.

Chapter 7 presents a complete theory of cyclic systems, which played a central historical role in contextuality research. The chapter establishes two closed-form criteria (necessary and sufficient conditions) for contextuality.

Most of the traditional contextuality research has been confined to con-sistently connected and undisturbed systems, those in which distributions of random variables are context-independent. The specialization of the theory to such systems is presented in Chapter 8. It is shown that any system, without changing its contextuality status, can be consistified, that is, reformulated as an undisturbed system of a special form. The chapter also shows that the existence of undisturbed contextual systems rules out the ensemble interpretation of ran-domness in the systems’ random variables in favor of fundamental stochasticity. The chapter also addresses the problem of counterfactual definiteness (whether a measurement of a group of random variables would have remained the same had they been measured in another context). This notion is often confused with that of noncontextuality. It is shown, however, that any undisturbed system satisfies counterfactual definiteness.

Chapter 9 is about hidden variable models (HVMs). Any system can be described by an HVM in which the distribution of hidden variables is context-independent but their mapping into observable variables is context-dependent; or by an HVM in which this mapping is context-independent but the distribution of the hidden variables varies with context. It is shown that these two types of models are equivalent and intertranslatable. As a mathematical language, HVMs generally do not distinguish between overt influences and contextuality.

Chapter 10 is about measures of contextuality. Its main development is a hierarchical measure of contextuality, one that establishes on what level of joint distributions (the level of pairs, triples, or generally n-tuples) the system becomes contextual, and to what degree. The latter is defined as a “city-block” distance from a point representing the system to the noncontextuality polytope. The chapter also discusses the contextual fraction measure and a measure based on signed probability measures (“negative probabilities”). Finally, Chapter 11 specializes the discussion of the measures to cyclic sys-tems. Here, the level of contextuality is always 2, and the geometry of the noncontextuality polytopes is especially transparent.

Title: Contextuality in Random Variables

Authors: Ehtibar N. Dzhafarov, Janne V. Kujala and Víctor H. Cervantes

ISBN: 9781009671927