Testing for selectivity in the dependence of random variables on external factors

Random variables AA and BB, whose joint distribution depends on factors (x,y)(x,y), are selectively influenced by xx and yy, respectively, if AA and BB can be represented as functions of, respectively, (x,SA,C)(x,SA,C) and (y,SB,C)(y,SB,C), where SA,SB,CSA,SB,C are stochastically independent and do not depend on (x,y)(x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of AA may depend on yy. But parameters characterizing stochastic interdependence of AA and BB, such as their mixed moments, are generally functions of both xx and yy. We derive two simple necessary conditions for selective dependence of (A,B)(A,B) on (x…

On minima of discrimination functions

Abstract A discrimination function ψ ( x , y ) assigns a measure of discriminability to stimulus pairs x , y (e.g., the probability with which they are judged to be different in a same-different judgment scheme). If for every x there is a single y least discriminable from x , then this y is called the point of subjective equality (PSE) for x , and the dependence h ( x ) of the PSE for x on x is called a PSE function. The PSE function g ( y ) is defined in a symmetrically opposite way. If the graphs of the two PSE functions coincide (i.e.,  g ≡ h − 1 ), the function is said to satisfy the Regular Minimality law. The minimum level functions are restrictions of ψ to the graphs of the PSE funct…

Regular Minimality and Thurstonian-type modeling

Abstract A Thurstonian-type model for pairwise comparisons is any model in which the response (e.g., “they are the same” or “they are different”) to two stimuli being compared depends, deterministically or probabilistically, on the realizations of two randomly varying representations (perceptual images) of these stimuli. The two perceptual images in such a model may be stochastically interdependent but each has to be selectively dependent on its stimulus. It has been previously shown that all possible discrimination probability functions for same–different comparisons can be generated by Thurstonian-type models of the simplest variety, with independent percepts and deterministic decision ru…

On Contextuality in Behavioral Data

Dzhafarov, Zhang, and Kujala (Phil. Trans. Roy. Soc. A 374, 20150099) reviewed several behavioral data sets imitating the formal design of the quantum-mechanical contextuality experiments. The conclusion was that none of these data sets exhibited contextuality if understood in the generalized sense proposed in Dzhafarov, Kujala, and Larsson (Found. Phys. 7, 762-782, 2015), while the traditional definition of contextuality does not apply to these data because they violate the condition of consistent connectedness (also known as marginal selectivity, no-signaling condition, no-disturbance principle, etc.). In this paper we clarify the relationship between (in)consistent connectedness and (non…

Proof of a Conjecture on Contextuality in Cyclic Systems with Binary Variables

We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary and sufficient condition) for contextuality in a broad class of quantum systems. This class includes Leggett-Garg, EPR/Bell, and Klyachko-Can-Binicioglu-Shumovsky type systems as special cases. In a system of this class certain physical properties $q_{1},...,q_{n}$ are measured in pairs $(q_{i},q_{j})$; every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a proper…

Contextuality-by-Default 2.0: Systems with Binary Random Variables

The paper outlines a new development in the Contextuality-by-Default theory as applied to finite systems of binary random variables. The logic and principles of the original theory remain unchanged, but the definition of contextuality of a system of random variables is now based on multimaximal rather than maximal couplings of the variables that measure the same property in different contexts: a system is considered noncontextual if these multimaximal couplings are compatible with the distributions of the random variables sharing contexts. A multimaximal coupling is one that is a maximal coupling of any subset (equivalently, of any pair) of the random variables being coupled. Arguments are …

No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov's probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice-Bob EPR paradigm, non-contextuality means that the identity of Alice's spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis \alphai chosen by Alice, irrespective of Bob's axis \betaj (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice's and Bob's spins are identified as Aij and Bij, even though their distributions are determined by, respectively, \alphai alone and \betaj alone, in accordance with the no-signaling requir…

Probabilistic foundations of contextuality

Contextuality is usually defined as absence of a joint distribution for a set of measurements (random variables) with known joint distributions of some of its subsets. However, if these subsets of measurements are not disjoint, contextuality is mathematically impossible even if one generally allows (as one must) for random variables not to be jointly distributed. To avoid contradictions one has to adopt the Contextuality-by-Default approach: measurements made in different contexts are always distinct and stochastically unrelated to each other. Contextuality is reformulated then in terms of the (im)possibility of imposing on all the measurements in a system a joint distribution of a particul…

Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology

This paper is a brief overview of the concepts involved in measuring the degree of contextuality and detecting contextuality in systems of binary measurements of a finite number of objects. We discuss and clarify the main concepts and terminology of the theory called "contextuality-by-default," and then discuss a possible generalization of the theory from binary to arbitrary measurements.

Contextuality Analysis of the Double Slit Experiment (With a Glimpse Into Three Slits)

The Contextuality-by-Default theory is illustrated on contextuality analysis of the idealized double-slit experiment. The experiment is described by a system of contextually labeled binary random variables each of which answers the question: has the particle hit the detector, having passed through a given slit (left or right) in a given state (open or closed)? This system of random variables is a cyclic system of rank 4, formally the same as the system describing the EPR/Bell paradigm with signaling. Unlike the latter, however, the system describing the double-slit experiment is always noncontextual, i.e., the context-dependence in it is entirely explainable in terms of direct influences of…

Erratum to “Testing for selectivity in the dependence of random variables on external factors” [J. Math. Psych. 52 (2008) 128–144]

Is there contextuality in behavioural and social systems?

Most behavioral and social experiments aimed at revealing contextuality are confined to cyclic systems with binary outcomes. In quantum physics, this broad class of systems includes as special cases Klyachko-Can-Binicioglu-Shumovsky-type, Einstein-Podolsky-Rosen-Bell-type, and Suppes-Zanotti-Leggett-Garg-type systems. The theory of contextuality known as Contextuality-by-Default allows one to define and measure contextuality in all such system, even if there are context-dependent errors in measurements, or if something in the contexts directly interacts with the measurements. This makes the theory especially suitable for behavioral and social systems, where direct interactions of "everythin…

Probabilistic Contextuality in EPR/Bohm-type Systems with Signaling Allowed

In this chapter, we review a principled way of defining and measuring contextuality in systems with deterministic inputs and random outputs, recently proposed and developed in \citep{KujalaDzhafarovLarsson2015,DKL2015FooP}.

Contextuality in canonical systems of random variables

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 o…

Order-distance and other metric-like functions on jointly distributed random variables

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). These functions are useful in dealing with the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality or its consequences by one of our functions when applied to such a set rules out the existence of this joint distribution. We focus on…

Context–content systems of random variables : The Contextuality-by-Default theory

Abstract This paper provides a systematic yet accessible presentation of the Contextuality-by-Default theory. The consideration is confined to finite systems of categorical random variables, which allows us to focus on the basics of the theory without using full-scale measure-theoretic language. Contextuality-by-Default is a theory of random variables identified by their contents and their contexts, so that two variables have a joint distribution if and only if they share a context. Intuitively, the content of a random variable is the entity the random variable measures or responds to, while the context is formed by the conditions under which these measurements or responses are obtained. A …

Selectivity in Probabilistic Causality: Drawing Arrows from Inputs to Stochastic Outputs

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one determine, for each of the outputs, which of the inputs it is influenced by? The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of "what influences what" is equivalent to that of the existence of a joint distr…

The Joint Distribution Criterion and the Distance Tests for Selective Probabilistic Causality

A general definition and a criterion (a necessary and sufficient condition) are formulated for an arbitrary set of external factors to selectively influence a corresponding set of random entities (generalized random variables, with values in arbitrary observation spaces), jointly distributed at every treatment (a set of factor values containing precisely one value of each factor). The random entities are selectively influenced by the corresponding factors if and only if the following condition, called the joint distribution criterion, is satisfied : there is a jointly distributed set of random entities, one entity for every value of every factor, such that every subset of this set that corr…

A Qualified Kolmogorovian Account of Probabilistic Contextuality

We describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central principle contextuality-by-default is that the outputs indexed by mutually incompatible values of inputs are stochastically unrelated; but they can be coupled imposed a joint distribution on in a variety of ways. A system is characterized by a pattern of which outputs can be "directly influenced" by which inputs a primitive relation, hypothetical or normative, and by certain constraints imposed on the outputs such as Bell-type inequalities or their quantum analogues. The set of couplings compatible with these …

Measuring Observable Quantum Contextuality

Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two …

Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology

This paper is a brief overview of the concepts involved in measuring the degree of contextuality and detecting contextuality in systems of binary measurements of a finite number of objects. We discuss and clarify the main concepts and terminology of the theory called “contextuality-by-default,” and then discuss generalizations of the theory to arbitrary systems of arbitrary random variables.

Contextuality-by-Default 2.0: Systems with Binary Random Variables

The paper outlines a new development in the Contextuality-by-Default theory as applied to finite systems of binary random variables. The logic and principles of the original theory remain unchanged, but the definition of contextuality of a system of random variables is now based on multimaximal rather than maximal couplings of the variables that measure the same property in different contexts: a system is considered noncontextual if these multimaximal couplings are compatible with the distributions of the random variables sharing contexts. A multimaximal coupling is one that is a maximal coupling of any subset (equivalently, of any pair) of the random variables being coupled. Arguments are …

All-Possible-Couplings Approach to Measuring Probabilistic Context.

From behavioral sciences to biology to quantum mechanics, one encounters situations where (i) a system outputs several random variables in response to several inputs, (ii) for each of these responses only some of the inputs may "directly" influence them, but (iii) other inputs provide a "context" for this response by influencing its probabilistic relations to other responses. These contextual influences are very different, say, in classical kinetic theory and in the entanglement paradigm of quantum mechanics, which are traditionally interpreted as representing different forms of physical determinism. One can mathematically construct systems with other types of contextuality, whether or not …

Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview

Similar formalisms have been independently developed in psychology, to deal with the issue of selective influences (deciding which of several experimental manipulations selectively influences each of several, generally non-independent, response variables), and in quantum mechanics (QM), to deal with the EPR entanglement phenomena (deciding whether an EPR experiment allows for a "classical" account). The parallels between these problems are established by observing that any two noncommuting measurements in QM are mutually exclusive and can therefore be treated as analogs of different values of one and the same input. Both problems reduce to that of the existence of a jointly distributed syst…

A Supplementary Text to “Contextuality in Canonical Systems of Random Variables” by Ehtibar N. Dzhafarov, Víctor H. Cervantes, and Janne V. Kujala (Phil. Trans. Roy. Soc. A xxx, 10.1098/rsta.xxxx.xxxx) from Contextuality in canonical systems of random variables

Mathematical Proofs

All-Possible-Couplings Approach to Measuring Probabilistic Context

From behavioral sciences to biology to quantum mechanics, one encounters situations where (i) a system outputs several random variables in response to several inputs, (ii) for each of these responses only some of the inputs may ‘‘directly’’ influence them, but (iii) other inputs provide a ‘‘context’’ for this response by influencing its probabilistic relations to other responses. These contextual influences are very different, say, in classical kinetic theory and in the entanglement paradigm of quantum mechanics, which are traditionally interpreted as representing different forms of physical determinism. One can mathematically construct systems with other types of contextuality, whether or …

Contextuality Analysis of the Double Slit Experiment (with a Glimpse into Three Slits)

The Contextuality-by-Default theory is illustrated on contextuality analysis of the idealized double-slit experiment. The experiment is described by a system of contextually labeled binary random variables each of which answers the question: Has the particle hit the detector, having passed through a given slit (left or right) in a given state (open or closed)? This system of random variables is a cyclic system of rank 4, formally the same as the system describing the Einsten-Podolsky-Rosen-Bell paradigm with signaling. Unlike the latter, however, the system describing the double-slit experiment is always noncontextual, i.e., the context-dependence in it is entirely explainable in terms of d…

Selectivity in Probabilistic Causality: Where Psychology Runs Into Quantum Physics

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one determine, for each of the outputs, which of the inputs it is influenced by? The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of "what influences what" is equivalent to that of the existence of a joint distr…

Contextuality is About Identity of Random Variables

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…

Necessary and Sufficient Conditions for an Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems

The notion of (non)contextuality pertains to sets of properties measured one subset (context) at a time. We extend this notion to include so-called inconsistently connected systems, in which the measurements of a given property in different contexts may have different distributions, due to contextual biases in experimental design or physical interactions (signaling): a system of measurements has a maximally noncontextual description if they can be imposed a joint distribution on in which the measurements of any one property in different contexts are equal to each other with the maximal probability allowed by their different distributions. We derive necessary and sufficient conditions for th…

A new definition of well-behaved discrimination functions

Abstract A discrimination function shows the probability or degree with which stimuli are discriminated from each other when presented in pairs. In a previous publication [Kujala, J.V., & Dzhafarov, E.N. (2008). On minima of discrimination functions. Journal of Mathematical Psychology , 52 , 116–127] we introduced a condition under which the conformity of a discrimination function with the law of Regular Minimality (which says, essentially, that “being least discriminable from” is a symmetric relation) implies the constancy of the function’s minima (i.e., the same level of discriminability of every stimulus from the stimulus least discriminable from it). This condition, referred to as “well…

Context-Content Systems of Random Variables: The Contextuality-by-Default Theory

This paper provides a systematic yet accessible presentation of the Contextuality-by-Default theory. The consideration is confined to finite systems of categorical random variables, which allows us to focus on the basics of the theory without using full-scale measure-theoretic language. Contextuality-by-Default is a theory of random variables identified by their contents and their contexts, so that two variables have a joint distribution if and only if they share a context. Intuitively, the content of a random variable is the entity the random variable measures or responds to, while the context is formed by the conditions under which these measurements or responses are obtained. A system of…

Random Variables Recorded Under Mutually Exclusive Conditions: Contextuality-by-Default

We present general principles underlying analysis of the dependence of random variables (outputs) on deterministic conditions (inputs). Random outputs recorded under mutually exclusive input values are labeled by these values and considered stochastically unrelated, possessing no joint distribution. An input that does not directly influence an output creates a context for the latter. Any constraint imposed on the dependence of random outputs on inputs can be characterized by considering all possible couplings (joint distributions) imposed on stochastically unrelated outputs. The target application of these principles is a quantum mechanical system of entangled particles, with directions of …

Embedding Quantum into Classical: Contextualization vs Conditionalization

We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability theory. In the contextualization approach each random variable is "automatically" labeled by all conditions under which it is recorded, and the random variables across a set of mutually exclusive conditions are probabilistically coupled (imposed a joint distribution upon). Analysis of all possible probabilistic couplings for a given set of random variables allows one to characterize various relations between their separate distributions (such as Bell-type ine…