Hidden variables, free choice, context-independence and all that

This paper provides a systematic account of the hidden variable models (HVMs) formulated to describe systems of random variables with mutually exclusive contexts. Any such system can be described either by a model with free choice but generally context-dependent mapping of the hidden variables into observable ones, or by a model with context-independent mapping but generally compromised free choice. These two types of HVMs are equivalent, one can always be translated into another. They are also unfalsifiable, applicable to all possible systems. These facts, the equivalence and unfalsifiability, imply that freedom of choice and context-independent mapping are no assumptions at all, and they tell us nothing about freedom of choice or physical influences exerted by contexts as these notions would be understood in science and philosophy. The conjunction of these two notions, however, defines a falsifiable HVM that describes non-contextuality when applied to systems with no disturbance or to consistifications of arbitrary systems. This HVM is most adequately captured by the term 'context-irrelevance', meaning that no distribution in the model changes with context. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.

Quantum Mechanics Is Compatible with Counterfactual Definiteness

Counterfactual definiteness (CFD) means that if some property is measured in some context, then the outcome of the measurement would have been the same had this property been measured in a different context. A context includes all other measurements made together with the one in question, and the spatiotemporal relations among them. The proviso for CFD is non-disturbance: any physical influence of the contexts on the property being measured is excluded by the laws of nature, so that no one measuring this property has a way of ascertaining its context. It is usually claimed that in quantum mechanics CFD does not hold, because if one assigns the same value to a property in all contexts it is measured in, one runs into a logical contradiction, or at least contravenes quantum theory and experimental evidence. We show that this claim is not substantiated if one takes into account that only one of the possible contexts can be a factual context, all other contexts being counterfactual. With this in mind, any system of random variables can be viewed as satisfying CFD. The concept of CFD is closely related to but distinct from that of noncontextuality, and it is the latter property that may or may not hold for a system, in particular being contravened by some quantum systems.

Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies

Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems' couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory T is contextually equivalent to a theory T' in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in T and T' such that the corresponding systems in T and T' are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together.

Contextuality and Informational Redundancy

A noncontextual system of random variables may become contextual if one adds to it a set of new variables, even if each of them is obtained by the same context-wise function of the old variables. This fact follows from the definition of contextuality, and its demonstration is trivial for inconsistently connected systems (i.e., systems with disturbance). However, it also holds for consistently connected (and even strongly consistently connected) systems, provided one acknowledges that if a given property was not measured in a given context, this information can be used in defining functions among the random variables. Moreover, every inconsistently connected system can be presented as a (strongly) consistently connected system with essentially the same contextuality characteristics.

Assumption-Free Derivation of the Bell-Type Criteria of Contextuality/Nonlocality

Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of contextuality in the contextuality-by-default approach, and then specializing these criteria to consistently connected systems.

Contextuality Analysis of Impossible Figures

This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense.

On joint distributions, counterfactual values and hidden variables in understanding contextuality

This paper deals with three traditional ways of defining contextuality: (C1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts; (C2) in terms of relationship between factual measurements in a given context and counterfactual measurements that could be made if one used other contexts; and (C3) in terms of (non)existence of 'hidden variables' that determine the outcomes of all factually performed measurements. It is generally believed that the three meanings are equivalent, but the issues involved are not entirely transparent. Thus, arguments have been offered that C2 may have nothing to do with C1, and the traditional formulation of C1 itself encounters difficulties when measurement outcomes in a contextual system are treated as random variables. I show that if C1 is formulated within the framework of the Contextuality-by-Default (CbD) theory, the notion of a probabilistic coupling, the core mathematical tool of CbD, subsumes both counterfactual values and 'hidden variables'. In the latter case, a coupling itself can be viewed as a maximally parsimonious choice of a hidden variable. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Measures of contextuality and non-contextuality

We discuss three measures of the degree of contextuality in contextual systems of dichotomous random variables. These measures are developed within the framework of the Contextuality-by-Default (CbD) theory, and apply to inconsistently connected systems (those with 'disturbance' allowed). For one of these measures of contextuality, presented here for the first time, we construct a corresponding measure of the degree of non-contextuality in non-contextual systems. The other two CbD-based measures do not suggest ways in which degree of non-contextuality of a non-contextual system can be quantified. We find the same to be true for the contextual fraction measure developed by Abramsky, Barbosa and Mansfield. This measure of contextuality is confined to consistently connected systems, but CbD allows one to generalize it to arbitrary systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

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 of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

Mathematical Theorizing versus Mathematical Metaphorizing: A Commentary on Rudolph

Science may benefit from aesthetically pleasing and intellectually stimulating mathematical metaphors. To make them into mathematical theories, however, one has to complement them by links to well-defined theoretical primitives in turn linked to well-defined empirical procedures and observable phenomena. Rudolph’s (2006) mathematical metaphors for psychological time are fascinating, but the mathematical rigor with which they can be described does not compensate for the conspicuous lack of both theoretical and operational clarity in the notions these metaphors are supposed to pertain to, including the very notion of psychological time

On contextuality in behavioural data

Dzhafarovet al.(Dzhafarovet al.2016Phil. Trans. R. Soc. A374, 20150099. (doi:10.1098/rsta.2015.0099)) reviewed several behavioural datasets imitating the formal design of the quantum-mechanical contextuality experiments. The conclusion was that none of these datasets exhibited contextuality if understood in the generalized sense proposed by Dzhafarovet al.(2015Found. Phys.7, 762-782. (doi:10.1007/s10701-015-9882-9)), 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-signalling condition, no-disturbance principle, etc.). In this paper, we clarify the relationship between (in)consistent connectedness and (non)contextuality, as well as between the traditional and extended definitions of (non)contextuality, using as an example the Clauser-Horn-Shimony-Holt inequalities originally designed for detecting contextuality in entangled particles.

Is there contextuality in behavioural and social systems?

Most behavioural 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 systems, 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 behavioural and social systems, where direct interactions of 'everything with everything' are ubiquitous. For cyclic systems with binary outcomes, the theory provides necessary and sufficient conditions for non-contextuality, and these conditions are known to be breached in certain quantum systems. We review several behavioural and social datasets (from polls of public opinion to visual illusions to conjoint choices to word combinations to psychophysical matching), and none of these data provides any evidence for contextuality. Our working hypothesis is that this may be a broadly applicable rule: behavioural and social systems are non-contextual, i.e. all 'contextual effects' in them result from the ubiquitous dependence of response distributions on the elements of contexts other than the ones to which the response is presumably or normatively directed.

Analyzability, ad hoc restrictions, and excessive flexibility of evidence-accumulation models: reply to two critical commentaries

Jones and Dzhafarov (2014) proved the linear ballistic accumulator (LBA) and diffusion model (DM) of speeded choice become unfalsifiable if 2 assumptions are removed: that growth rate variability between trials follows a Gaussian distribution and that this distribution is invariant under certain experimental manipulations. The former assumption is purely technical and has never been claimed as a theoretical commitment, and the latter is logically and empirically suspect. Heathcote, Wagenmakers, and Brown (2014) questioned the distinction between theoretical and technical assumptions and argued that only the predictions of the whole model matter. We respond that it is valuable to understand how a model's predictions depend on each of its assumptions to know what is critical to an explanation and to generalize principles across phenomena or domains. Smith, Ratcliff, and McKoon (2014) claimed unfalsifiability of the generalized DM relies on parameterizations with negligible diffusion and proposed a theoretical commitment to simple growth-rate distributions. We respond that a lower bound on diffusion would be a new, ad hoc assumption, and restrictions on growth-rate distributions are only theoretically justified if one supplies a model of what determines growth-rate variability. Finally, we summarize a simulation of the DM that retains the growth-rate invariance assumption, requires the growth-rate distribution to be unimodal, and maintains a contribution of diffusion as large as in past fits of the standard model. The simulation demonstrates mimicry between models with different theoretical assumptions, showing the problems of excess flexibility are not limited to the cases to which Smith et al. objected. (PsycINFO Database Record (c) 2014 APA, all rights reserved).

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 the existence of such a description in a broad class of systems including Klyachko-Can-Binicioğlu-Shumvosky-type (KCBS), EPR-Bell-type, and Leggett-Garg-type systems. Because these conditions allow for inconsistent connectedness, they are applicable to real experiments. We illustrate this by analyzing an experiment by Lapkiewicz and colleagues aimed at testing contextuality in a KCBS-type system.

Noncontextuality with marginal selectivity in reconstructing mental architectures

We present a general theory of series-parallel mental architectures with selectively influenced stochastically non-independent components. A mental architecture is a hypothetical network of processes aimed at performing a task, of which we only observe the overall time it takes under variable parameters of the task. It is usually assumed that the network contains several processes selectively influenced by different experimental factors, and then the question is asked as to how these processes are arranged within the network, e.g., whether they are concurrent or sequential. One way of doing this is to consider the distribution functions for the overall processing time and compute certain linear combinations thereof (interaction contrasts). The theory of selective influences in psychology can be viewed as a special application of the interdisciplinary theory of (non)contextuality having its origins and main applications in quantum theory. In particular, lack of contextuality is equivalent to the existence of a "hidden" random entity of which all the random variables in play are functions. Consequently, for any given value of this common random entity, the processing times and their compositions (minima, maxima, or sums) become deterministic quantities. These quantities, in turn, can be treated as random variables with (shifted) Heaviside distribution functions, for which one can easily compute various linear combinations across different treatments, including interaction contrasts. This mathematical fact leads to a simple method, more general than the previously used ones, to investigate and characterize the interaction contrast for different types of series-parallel architectures.

Quantum models for psychological measurements: an unsolved problem

There has been a strong recent interest in applying quantum theory (QT) outside physics, including in cognitive science. We analyze the applicability of QT to two basic properties in opinion polling. The first property (response replicability) is that, for a large class of questions, a response to a given question is expected to be repeated if the question is posed again, irrespective of whether another question is asked and answered in between. The second property (question order effect) is that the response probabilities frequently depend on the order in which the questions are asked. Whenever these two properties occur together, it poses a problem for QT. The conventional QT with Hermitian operators can handle response replicability, but only in the way incompatible with the question order effect. In the generalization of QT known as theory of positive-operator-valued measures (POVMs), in order to account for response replicability, the POVMs involved must be conventional operators. Although these problems are not unique to QT and also challenge conventional cognitive theories, they stand out as important unresolved problems for the application of QT to cognition. Either some new principles are needed to determine the bounds of applicability of QT to cognition, or quantum formalisms more general than POVMs are needed.

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 inequalities or quantum-mechanical constraints). In the conditionalization approach one considers the conditions under which the random variables are recorded as if they were values of another random variable, so that the observed distributions are interpreted as conditional ones. This approach is uninformative with respect to relations between the distributions observed under different conditions because any set of such distributions is compatible with any distribution assigned to the conditions.

On selective influences, marginal selectivity, and bell/CHSH inequalities

The Bell/CHSH inequalities of quantum physics are identical with the inequalities derived in mathematical psychology for the problem of selective influences in cases involving two binary experimental factors and two binary random variables recorded in response to them. The following points are made regarding cognitive science applications: (1) compliance of data with these inequalities is informative only if the data satisfy the requirement known as marginal selectivity; (2) both violations of marginal selectivity and violations of the Bell/CHSH inequalities are interpretable as indicating that at least one of the two responses is influenced by both experimental factors.

Unfalsifiability and mutual translatability of major modeling schemes for choice reaction time

[Correction Notice: An Erratum for this article was reported in Vol 121(1) of Psychological Review (see record 2014-03591-005). The link to supplemental material was missing. All versions of this article have been corrected.] Much current research on speeded choice utilizes models in which the response is triggered by a stochastic process crossing a deterministic threshold. This article focuses on 2 such model classes, 1 based on continuous-time diffusion and the other on linear ballistic accumulation (LBA). Both models assume random variability in growth rates and in other model components across trials. We show that if the form of this variability is unconstrained, the models can exactly match any possible pattern of response probabilities and response time distributions. Thus, the explanatory or predictive content of these models is determined not by their structural assumptions but, rather, by distributional assumptions (e.g., Gaussian distributions) that are traditionally regarded as implementation details. Selective influence assumptions (i.e., which experimental manipulations affect which model parameters) are shown to have no restrictive effect, except for the theoretically questionable assumption that speed-accuracy instructions do not affect growth rates. The 2nd contribution of this article concerns translation of falsifiable models between universal modeling languages. Specifically, we translate the predictions of the diffusion and LBA models (with their parametric and selective influence assumptions intact) into the Grice modeling framework, in which accumulation processes are deterministic and thresholds are random variables. The Grice framework is also known to reproduce any possible pattern of response probabilities and times, and hence it can be used as a common language for comparing models. It is found that only a few simple properties of empirical data are necessary predictions of the diffusion and LBA models.

Matrices satisfying regular minimality

A matrix of discrimination measures (discrimination probabilities, numerical estimates of dissimilarity, etc.) satisfies Regular Minimality (RM) if every row and every column of the matrix contains a single minimal entry, and an entry minimal in its row is minimal in its column. We derive a formula for the proportion of RM-compliant matrices among all square matrices of a given size and with no tied entries. Under a certain "meta-probabilistic" model this proportion can be interpreted as the probability with which a randomly chosen matrix turns out to be RM-compliant.

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 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 and 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.

Reconstructing Distances among Objects from Their Discriminability

We describe a principled way of imposing a metric representing dissimilarities on any discrete set of stimuli (symbols, handwritings, consumer products, X-ray films, etc.), given the probabilities with which they are discriminated from each other by a perceiving system, such as an organism, person, group of experts, neuronal structure, technical device, or even an abstract computational algorithm. In this procedure one does not have to assume that discrimination probabilities are monotonically related to distances, or that the distances belong to a predefined class of metrics, such as Minkowski. Discrimination probabilities do not have to be symmetric, the probability of discriminating an object from itself need not be a constant, and discrimination probabilities are allowed to be 0's and 1's. The only requirement that has to be satisfied is Regular Minimality, a principle we consider the defining property of discrimination: for ordered stimulus pairs (a,b), b is least frequently discriminated from a if and only if a is least frequently discriminated from b. Regular Minimality generalizes one of the weak consequences of the assumption that discrimination probabilities are monotonically related to distances: the probability of discriminating a from a should be less than that of discriminating a from any other object. This special form of Regular Minimality also underlies such traditional analyses of discrimination probabilities as Multidimensional Scaling and Cluster Analysis.

Multidimensional Fechnerian Scaling: Basics

Fechnerian scaling is a theory of how a certain (Fechnerian) metric can be computed in a continuous stimulus space of arbitrary dimensionality from the shapes of psychometric (discrimination probability) functions taken in small vicinities of stimuli at which these functions reach their minima. This theory is rigorously derived in this paper from three assumptions about psychometric functions: (1) that they are continuous and have single minima around which they increase in all directions; (2) that any two stimulus differences from these minimum points that correspond to equal rises in discrimination probabilities are comeasurable in the small (i.e., asymptotically proportional), with a continuous coefficient of proportionality; and (3) that oppositely directed stimulus differences from a minimum point that correspond to equal rises in discrimination probabilities are equal in the small. A Fechnerian metric derived from these assumptions is an internal (or generalized Finsler) metric whose indicatrices are asymptotically similar to the horizontal cross-sections of the psychometric functions made just above their minima. Copyright 2001 Academic Press.

Unconditionally Selective Dependence of Random Variables on External Factors

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.

Double Skew-Dual Scaling: A Conjoint Scaling of Two Sets of Objects Related by a Dominance Matrix

Consider two sets of objects, {alpha(1), em leader, alpha(n)} and {beta(1), em leader, beta(m), such as n subjects solving m tasks, or n stimuli presented first and m stimuli presented second in a pairwise comparison experiment. Let any pair (alpha(i), beta(j)) be associated with a real number a(ij), interpreted as the degree of dominance of alpha(i) over beta(j) (e.g., the probability of alpha(i) relating in a certain way to beta(j)). Intuitively, the problem addressed in this paper is how to conjointly, in a "naturally" coordinated fashion, characterize the alpha-objects and beta-objects in terms of their overall tendency to dominate or be dominated. The gist of the solution is as follows. Let A denote the nxm matrix of a(ij) values, and let there be a class of monotonic transformations straight phi with nonnegative codomains. For a given straight phi, a complementary matrix B is defined so that straight phi(a(ij))+straight phi(b(ij))=const, and one computes vectors D(alpha) and D(beta) (the dominance values for alpha-objects and beta-objects) by solving the equations straight phi(A) straight phi(D(beta))/Sigma;straight phi(D(beta))=straight phi(D(alpha)) and straight phi(B(T)) straight phi(D(alpha))/Sigmastraight phi(D(alpha))=straight phi(D(beta)), where (T) is transposition, Sigma is the sum of elements, and straight phi applies elementwise. One also computes vectors S(alpha) and S(beta) (the subdominance values for alpha-objects and beta-objects) by solving the equations straight phi(B) straight phi(S(beta))/Sigmastraight phi(S(beta))=straight phi(S(alpha)) and straight phi(A(T)) straight phi(S(alpha))/Sigmastraight phi(S(alpha))=straight phi(S(beta)). The relationship between S-vectors and D-vectors is complex: intuitively, D(alpha) characterizes the tendency of an alpha-object to dominate beta-objects with large dominance values, whereas S(alpha) characterizes the tendency of an alpha-objects to fail to dominate beta-objects with large subdominance values. For classes containing more than one straight phi-transformation, one can choose an optimal straight phi as the one maximizing some measure of discrimination between individual elements of vectors straight phi(D(alpha)), straight phi(D(beta)), straight phi(S(alpha)), and straight phi(S(beta)), such as the product or minimum of these vectors' variances. The proposed analysis of dominance matrices has only superficial similarities with the classical dual scaling (Nishisato, 1980). Copyright 1999 Academic Press.

Fechnerian metrics in unidimensional and multidimensional stimulus spaces

A new theory is proposed for subjective (Fechnerian) distances among stimuli in a continuous stimulus space of arbitrary dimensionality. Each stimulus in such a space is associated with a psychometric function that determines probabilities with which it is discriminated from other stimuli, and a certain measure of its discriminability from its infinitesimally close neighboring stimuli is computed from the shape of this psychometric function in the vicinity of its minimum. This measure of discriminability can be integrated along any path connecting any two points in the stimulus space, yielding the psychometric length of this path. The Fechnerian distance between two stimuli is defined as the infimum of the psychometric lengths of all paths connecting the two stimuli. For a broad class of models defining the dichotomy of response bias versus discriminability, the Fechnerian distances are invariant under response bias changes. In the case in which physically multidimensional stimuli are discriminated along some unidimensional subjective attribute, a systematic construction of the Fechnerian metric leads to a resolution of the long-standing controversy related to the numbers of just-noticeable differences between isosensitivity curves. It is argued that for unidimensional stimulus continua, the proposed theory is close to the intended meaning of Fechner's original theory.

Conditionally Selective Dependence of Random Variables on External Factors

Selective influence of experimental factors upon observable or hypothetical random variables is a key concept in the analysis of processing architectures and response time decompositions. This paper deals with the notion of conditionally selective influence, defined as follows. Let {X1, em leader, Xn} be stochastically interdependent random variables (e.g., hypothetical components of response time), and let Phi be a set of external factors affecting the joint distribution of {X1, em leader, Xn}. A subset of factors &Lambdai conditionally selectively influences Xi if at any fixed values of the remaining random variables the conditional distribution of Xi only depends on factors inside &Lambdai. The notion of conditional selectivity generalizes the relationship between factors and random variables described in Townsend (1984) as "indirect nonselectivity." This paper establishes the structure of the joint distribution of {X1, em leader, Xn} that is necessary and sufficient for {X1, em leader, Xn} to be conditionally selectively influenced by (not necessarily disjoint) factor subsets {&Lambda1, em leader, Gamman}, respectively. The notion of conditional selectivity is compared to that of unconditional selectivity, defined as follows. A subset of factors &Gammai unconditionally selectively influences Xi if the latter can be presented as a deterministic function of &Gammai and of some random variables (the same for all Xi, i=1, em leader, n) whose joint distribution does not depend on any factors from Phi. The two forms of selective influence are generally incompatible. Copyright 1999 Academic Press.

Empirical Recovery of Response Time Decomposition Rules I. Sample-Level Decomposition Tests

E. N. Dzhafarov and R. Schweickert (1995, Journal of Mathematical Psychology, 39, 285-314) developed a mathematical theory for the decomposability of response time (RT) into two component times that are selectively influenced by different factors and are either stochastically independent or perfectly positively stochastically interdependent (in which case they are increasing functions of a common random variable). In this theory, RT is obtained from its component times by means of an associative and commutative operation. For any such operation, there is a decomposition test, a relationship between observable RT distributions that holds if and (under mild constraints) only if the RTs are decomposable by means of this operation. In this paper, we construct a sample-level version of these decomposition tests that serve to determine whether RTs that are represented by finite samples are decomposable by means of a given operation (under a given form of stochastic relationship between component times, independence or perfect positive interdependence). The decision is based on the asymptotic p-values associated with the maximal distance between empirical distribution functions computed by combining in a certain way the RT samples corresponding to different treatments.

Empirical Recovery of Response Time Decomposition Rules II. Discriminability of Serial and Parallel Architectures

Among the possible response time (RT) decomposition rules, three are of a traditional interest: addition (serial RT architecture), minimum (parallel-OR architecture), and maximum (parallel-AND architecture). Given RT samples, one can decide which of these three operation is the true decomposition rule by choosing the operation producing the smallest Smirnov distance between the RT samples combined in a certain way, as described by E. N. Dzhafarov and J. M. Cortese (1996, Journal of Mathematical Psychology 40, 185-202). By means of Monte-Carlo simulations, we determine at what sample sizes this decision identifies the true decomposition rule reliably. The results indicate that for a broad class of RT distribution functions the sample sizes required are by an order of magnitude larger when the component times are stochastically independent than when they are perfectly positively stochastically interdependent. In both cases, however, the required sample sizes are realistically achievable in an experiment, provided the experimental factors selectively influencing component times are sufficiently effective. Addition and maximum are generally more difficult to discriminate than addition and minimum, which in turn are more difficult to discriminate than maximum and minimum.

Detection of changes in speed and direction of motion: reaction time analysis

Observers reacted to the change in the movement of a random-dot field whose initial velocity, V0, was constant for a random period and then switched abruptly to another value, V1. The two movements, both horizontally oriented, were either in the same direction (speed increments or decrements), or in the opposite direction but equal in speed (direction reversals). One of the two velocities, V0 or V1, could be zero (motion onset and offset, respectively). In the range of speeds used, 0-16 deg/sec (dps), the mean reaction time (MRT) for a given value of V0 depended on magnitude of V1-V0 only: MRT approximately r+c(V0)/magnitude of V1-V0 beta, where beta = 2/3, r is a velocity-independent component of MRT, and c(V0) is a parameter whose value is constant for low values of V0 (0-4 dps), and increases beginning with some value of V0 between 4 and 8 dps. These and other data reviewed in the paper are accounted for by a model in which the time-position function of a moving target is encoded by mass activation of a network of Reichardt-type encoders. Motion-onset detection (V0 = 0) is achieved by weighted temporal summation of the outputs of this network, the weights assigned to activated encoders being proportional to their squared spatial spans. By means of a "subtractive normalization," the visual system effectively reduces the detection of velocity changes (a change from V0 to V1) to the detection of motion onset (a change from 0 to V1-V0). Subtractive normalization operates by readjustment of weights: the weights of all encoders are amplified or attenuated depending on their spatial spans, temporal spans, and the initial velocity V0. Assignment of weights and weighted temporal summation are thought of as special-purpose computations performed on the dynamic array of activations in the motion-encoding network, without affecting the activations themselves.

Pitch motion with random chord sequences

Perception of global pitch motion was studied through psychoacoustic experiments with random chord sequences. Chords contained either six or eight (fixed) tone elements, being sinusoidal, sawtooth-like, or Shepard tones, which were either on or off according to a probability controlled by the experimenter. Sequences of 2, 4, 5, or 8 chords were used. Identification by subjects of the perceived direction of overall pitch motion (up or down) was found to be well accounted for by a model in which the ultimate pitch motion percept is given by a sum of contributions from selected element transitions--that is, transitions between adjoining tone elements in successive time frames only. In its simplest form, this dipole contribution model has only one free parameter, the perceptual noise for an element transition, which was estimated for various acoustic tone representations and chord arrangements. Results of two experiments, which were carried out independently in two different laboratories, are reported.

Motion direction identification in random cinematograms: a general model

The cinematograms of 12 two-state elements arranged in the clock positions in space and in a sequence of adjacent 100-ms frames in time were used as stimuli. Some positions in each frame (or all 12 of them) could be labeled as "domain" ones, and every element that was T frames and S positions (clockwise or counterclockwise) apart from a domain element could repeat the latter's state with probability P. The probability of the rotation direction identification was obtained as a function of T, S, P, number of frames, and the domain positions selection scheme. A generalized version of the reversed phi phenomenon was obtained: if P less than .5, then the psychometric value lies below .5 level. All the data can be accounted for by a simple model according to which the choice of direction is based on the counts of the different types of dipoles, each type being characterized by the probability and the weight of its count: In most situations all dipoles but the shortest ones (connecting the neighboring elements of successive frames) can be ignored.

Reaction time to motion onset: local dispersion model analysis

Data on the simple reaction time to motion onset presented in Ball and Sekuler [Psychol. Rev. 87, 435-469 (1980)] and Tynan and Sekuler [Vision Res. 20, 709-715 (1982)] are re-analysed on the basis of local dispersion model of motion detectability. According to this model the detectability at the moment t is determined by the mean value of the local dispersion function LD (t) within the interval (t-T, t), LD(t) being some measure of scattering (namely, running variance) of spatial positions passed through during the period (t-tau, t). Reaction time is assumed to be equal to the time that takes the detectability to reach some critical level plus constant execution time. Theoretical predictions fit the experimental data perfectly when two main parameters of the model, T/tau and tau are the same as were found appropriate in other, independent experiments on motion detection.