Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no‐hidden‐variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem. <s> BIB001 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> Abstract The way external factors influence distribution functions for the overall time required to perform a mental task (such as responding to a stimulus, or solving a problem) may be informative as to the underlying mental architecture, the hypothetical network of interconnected processes some of which are selectively influenced by some of the external factors. Under the assumption that all processes contributing to the overall performance time are stochastically independent, several basic results have been previously established. These results relate patterns of response time distribution functions produced by manipulating external factors to such questions as whether the hypothetical constituent processes in the mental architecture enter AND gates or OR gates, and whether pairs of processes are sequential or concurrent. The present study shows that all these results are also valid for stochastically interdependent component times, provided the selective dependence of these components upon external factors is understood within the framework of a recently proposed theory of selective influence. According to this theory each component is representable as a function of three arguments: the factor set selectively influencing it, a component-specific source of randomness, and a source of randomness shared by all the components. <s> BIB002 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> The paper provides conceptual clarifications for the issues related to the dependence of jointly distributed systems of random entities on external factors. This includes the theory of selective influence as proposed in Dzhafarov [(2003a). Selective influence through conditional independence. Psychometrika, 68, 7–26] and generalized versions of the notions of probabilistic causality [Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199] and dimensionality in the latent variable models [Levine, M. V. (2003). Dimension in latent variable models. Journal of Mathematical Psychology, 47, 450–466]. One of the basic observations is that any system of random entities whose joint distribution depends on a factor set can be represented by functions of two arguments: a single factor-independent source of randomness and the factor set itself. In the case of random variables (i.e., real-valued random entities endowed with Borel sigma-algebras) the single source of randomness can be chosen to be any random variable with a continuous distribution (e.g., uniformly distributed between 0 and 1). <s> BIB003 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> 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,y)(x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x′x,x′ and y,y′y,y′, ::: ::: sxy≤sxy′+sx′y′+sx′y,sxy≤sxy′+sx′y′+sx′y, ::: ::: where sxy=E[|f(Axy,x)−g(Bxy,y)|p]1/p with arbitrary f,gf,g, and p≥1p≥1, and (Axy,Bxy)(Axy,Bxy) denoting (A,B)(A,B) at specific values of x,yx,y. For p=2(f(Axy,x),g(Bxy,y))p=2(f(Axy,x),g(Bxy,y)) this condition is superseded by a more restrictive one: ::: ::: |ρxyρxy′−ρx′yρx′y′|≤1−ρxy21−ρxy′2+1−ρx′y21−ρx′y′2, ::: ::: where ρxyρxy is the correlation between f(Axy,x)f(Axy,x) and g(Bxy,y)g(Bxy,y). For bivariate normally distributed (f(Axy,x),g(Bxy,y))(f(Axy,x),g(Bxy,y)) this condition, if satisfied on a 2×2 subset {x,x′}×{y,y′}{x,x′}×{y,y′}, is also sufficient for a selective dependence of (A,B)(A,B) on (x,y)(x,y) confined to this subset. <s> BIB004 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> 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 rules. It has also been shown, however, that a broad class of Thurstonian-type models, called “well-behaved” (and including, e.g., models with multivariate normal perceptual representations whose parameters are smooth functions of stimuli) cannot simultaneously account for two empirically plausible properties of same–different comparisons, Regular Minimality (which essentially says that “being least discriminable from” is a symmetric relation) and nonconstancy of the minima of discrimination probabilities (the fact that different pairs of least discriminable stimuli are discriminated with different probabilities). These results have been obtained for stimulus spaces represented by regions of Euclidean spaces. In this paper, the impossibility for well-behaved Thurstonian-type models to simultaneously account for Regular Minimality and nonconstancy of minima is established for a much broader notion of well-behavedness applied to a much broader class of stimulus spaces (any Hausdorff arc-connected ones). The universality of Thurstonian-type models with independent perceptual images and deterministic decision rules is shown (by a simpler proof than before) to hold for arbitrary stimulus spaces. <s> BIB005 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> 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 corresponds to a treatment is distributed as the original variables at this treatment. The distance tests (necessary conditions) for selective influence previously formulated for two random variables in a two-by-two factorial design (Kujala & Dzhafarov, 2008, J. Math. Psychol., 52, 128–144) are extended to arbitrary sets of factors and random variables. The generalization turns out to be the simplest possible one: the distance tests should be applied to all two-by-two designs extractable from a given set of factors. <s> BIB006 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> 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 distribution for a certain set of random variables. For inputs and outputs with finite sets of values this criterion translates into a test of consistency of a certain system of linear equations and inequalities (Linear Feasibility Test) which can be performed by means of linear programming. While new in the behavioral context, both this test and the Joint Distribution Criterion on which it is based have been previously proposed in quantum physics, in dealing with generalizations of Bell inequalities for the quantum entanglement problem. The parallels between this problem and that of selective influences in behavioral sciences is established by observing that noncommuting measurements in quantum physics are mutually exclusive and can therefore be treated as different levels of one and the same factor. <s> BIB007 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Basic Notions <s> 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 an especially versatile and widely applicable pseudo-quasi-metric called an order-distance and its special case called a classification distance. <s> BIB008
Aimed at providing a broad overview of concepts and results, the content of this paper partially overlaps with that of several previous publications, especially BIB007 BIB008 BIB004 . Random variables are understood in the broadest sense, as measurable functions X : V s → V , with no restrictions on the sample spaces (V s , Σ s , µ s ) and the induced probability spaces (distributions) (V, Σ, µ). In particular, any set X of jointly distributed random variables (functions on the same sample space) is a random variable, and its distribution (V, Σ, µ) is referred to as the joint distribution of its elements. We use symbol ∼ in the meaning of "has the same distribution as." A random variable in the narrow sense is a special case of a random entity, with V a finite product of countable sets and intervals of reals, and Σ the smallest sigma-algebra containing the corresponding product of power sets and Lebesgue sigma-algebras. Note that a vector of random variables in the narrow sense is a random variable in the narrow sense. Consider an indexed set α = α λ : λ ∈ Λ , with each α λ being a set referred to as a (deterministic) input, with the elements of {λ} × α λ called input points. Input points therefore are pairs of the form x = (λ, w), with w ∈ α λ , and should not be confused with input values w. A nonempty set Φ ⊂ λ∈Λ α λ is called a set of (allowable) treatments. A treatment therefore is a function φ : Λ → λ∈Λ α λ such that φ (λ) ∈ α λ for any λ ∈ Λ. Let there be a collection of sets of random variables be a random variable with a known distribution (the joint distribution of all A λ φ in A φ ) for every treatment φ ∈ Φ. We define with the understanding that A λ is not generally a random variable, i.e., A λ φ for different φ are not necessarily jointly distributed. The definition of the relation interpreted as "for each λ ∈ Λ, A λ may be influenced by α λ only," can be given in three equivalent forms: (SI 1 ) there are independent random variables C, S λ : λ ∈ Λ , and functions such that, for any treatment φ ∈ Φ, (SI 2 ) there is a random variable C and functions such that, for any treatment φ ∈ Φ, (JDC) there is a set of jointly distributed random variables (one random variable for every value of every input), such that, for any treatment φ ∈ Φ, The latter statement constitutes the Joint Distribution Criterion (JDC) for selective influences, and H is called the JDC (indexed) set. The proof of the equivalence BIB006 obtains essentially by the definition of a joint distribution, which seems to have been overlooked in the earlier derivations BIB001  . If Λ = {1, . . . , n} and all outputs A λ are random variables in the narrow sense, then C in SI 2 and C, S 1 , . . . , S n in SI 1 can also be chosen to be random variables in the narrow sense; moreover, their distribution functions can be chosen arbitrarily, provided they are continuous and strictly increasing on their domains, e.g., unit uniform BIB007 . Two important consequences of (7) are as follows: (invariance with respect to input-value-specific transformations) for any set of measurable functions where These properties should be viewed as desiderata for any reasonable definition of selective influences. In QM, SI 1 corresponds to the existence of a "classical" probabilistic explanation. In psychology, statement SI 1 combined with auxiliary assumptions was used in  and BIB005 to analyze the representability of same-different pairwise discrimination probabilities by means of Thurstonian-type models in which two stimuli being compared are mapped into random entities (distributed in some hypothetical space of mental images) that in turn are mapped (deterministically or probabilistically) into a response, "same" or "different." Statement SI 1 was also used to analyze the response time distributions for parallel-serial networks of mental operations with selectively influenced components BIB002 . Note that the representation of the outputs A λ as functions of the corresponding inputs α i and unobservable sources of randomness, A λ -specific (S λ ) and common (C), includes as special cases all conceivable generalizations and combinations of regression and factor analyses, with our term "input" corresponding to the traditional "regressor," and the term "source of randomness" to the factor-analytic "factor." This observation alone shows the potentially unlimited sphere of applicability of SI 1 . Statement SI 2 (corresponding in QM to "classical" determinsitic explanation) and JDC turn out to be more convenient in dealing with certain foundational probabilistic issues BIB003 and for the construction of the working tests (necessary conditions) for selective influences BIB006 BIB007 BIB008 BIB004 . The tests are discussed below. The following is a table of correspondences between the general terminology used in dealing with the issue of selective influences, and that of QM in dealing with EPR. Selective Probabilistic Causality (general) Quantum Entanglement Problem observed random output outcome of a given measurement on a given particle input (factor) set of noncommuting measurements on a given particle input value one of noncommuting measurements on a given particle joint distribution criterion joint distribution criterion diagram of selective influences "classical" explanation representation in the form SI1 probabilistic "classical" explanation representation in the form SI2 deterministic "classical" explanation