Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <s> The primary criterion of adequacy of a probabilistic causal analysis is that the causal variable should render the simultaneous phenomenological data conditionally independent. The intuition back of this idea is that the common cause of the phenomena should factor out the observed correlations. So we label the principle the common cause criterion. If we find that the barometric pressure and temperature are both dropping at the same time, we do not think of one as the cause of the other but look for a common dynamical cause within the physical theory of meteorology. If we find fever and headaches positively correlated, we look for a common disease as the source and do not consider one the cause of the other. But we do not want to suggest that satisfaction of this criterion is the end of the search for causes or probabilistic explanations. It does represent a significant and important milestone in any particular investigation. <s> BIB001 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB002 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <s> Abstract The analysis of reaction time (RT) additivity and interactions via factorial manipulation is a widely used and potentially powerful tool for elucidating mental processing; however, current implementation is limited by the scope of systems for which predictions are available, as well as statistical weakness. Predictions for an expanded set of system dimensions (serial/parallel, independent/dependent, and selective/nonselective influence) are given. The theorems are not limited to particular families of distributions. Statistical considerations are examined in the context of an exemplary nonadditive parallel system which adequately fits “additive” data but will only be rejected as additive when power is increased beyond the traditional criteria. <s> BIB003 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <s> What is the meaning of saying that random variables {X(1), em leader, X(n)} (such as aptitude scores or hypothetical response time components), not necessarily stochastically independent, are selectively influenced respectively by subsets {Gamma(1), em leader, Gamma(n)} of a factor set Phi upon which the joint distribution of {X(1), em leader, X(n)} is known to depend? One possible meaning of this statement, termed conditionally selective influence, is completely characterized in Dzhafarov (1999, Journal of Mathematical Psychology, 43, 123-157). The present paper focuses on another meaning, termed unconditionally selective influence. It occurs when two requirements are met. First, for i=1, em leader, n, the factor subset Gamma(i) is the set of all factors that effectively change the marginal distribution of X(i). Second, if {X(1), em leader, X(n)} are transformed so that all marginal distributions become the same (e.g., standard uniform or standard normal), the transformed variables are representable as well-behaved functions of the corresponding factor subsets {Gamma(1), em leader, Gamma(n)} and of some common set of sources of randomness whose distribution does not depend on any factors. Under the constraint that the factor subsets {Gamma(1), em leader, Gamma(n)} are disjoint, this paper establishes the necessary and sufficient structure of the joint distribution of {X(1), em leader, X(n)} under which they are unconditionally selectively influenced by {Gamma(1), em leader, Gamma(n)}. The unconditionally selective influence has two desirable properties, uniqueness and nestedness: {X(1), em leader, X(n)} cannot be influenced selectively by more than one partition {Gamma(1), em leader, Gamma(n)} of the factor set Phi, and the components of any subvector of {X(1), em leader, X(n)} are selectively influenced by the components of the corresponding subpartition of {Gamma(1), em leader, Gamma(n)}. Copyright 2001 Academic Press. <s> BIB004 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <s> Let each of several (generally interdependent) random vectors, taken separately, be influenced by a particular set of external factors. Under what kind of the joint dependence of these vectors on the union of these factor sets can one say that each vector is selectively influenced by “its own” factor set? The answer proposed and elaborated in this paper is: One can say this if and only if one can find a factor-independent random vector given whose value the vectors in question are conditionally independent, with their conditional distributions selectively influenced by the corresponding factor sets. Equivalently, the random vectors should be representable as deterministic functions of “their” factor sets and of some mutually independent and factor-independent random variables, some of which may be shared by several of the functions. <s> BIB005 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB006 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB007 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB008 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB009 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> A Historical Note <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> BIB010
The issue of selective influences was introduced to psychology in Sternberg's influential paper  , in the context of studying consecutive "stages" of information processing. Sternberg acknowledged that selective influences can hold even if the durations of the stages are not stochastically independent, but he lacked mathematical apparatus for dealing with this possibility. Townsend BIB003 proposed to formalize the notion of selectively influenced and stochastically interdependent random variables by the concept of "indirect nonselectiveness": the conditional distribution of the variable A 1 given any value a 2 of the variable A 2 , depends on α 1 only, and, by symmetry, the conditional distribution of A 2 at any A 1 = a 1 depends on α 2 only. Under the name of "conditionally selective influence" this notion was mathematically characterized and generalized in BIB004 A different approach to selective influences, reviewed below, is based on BIB004 BIB005 BIB006 BIB008 BIB009 BIB010 BIB007 . As it turns out 1 this approach parallels the development in QM of the issue of whether an EPR experiment can have a "classical" explanation (in terms of non-contextual local variables). The Joint Distribution Criterion which is at the heart of this development (see below) was indirectly introduced in the celebrated work of Bell  , and explicitly in BIB002  BIB001 .