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'.

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.

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 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 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 direct influences of contexts (closed-open arrangements of the slits) upon the marginal distributions of the random variables involved. The analysis presented is entirely within the framework of abstract classical probability theory (with contextually labeled random variables). The only physical constraint used in the analysis is that a particle cannot pass through a closed slit. The noncontextuality of the double-slit system does not generalize to systems describing experiments with more than two slits: in an abstract triple-slit system, almost any set of observable detection probabilities is compatible with both a contextual scenario and a noncontextual scenario of the particle passing though various combinations of open and closed slits (although the issue of physical realizability of these scenarios remains open).

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'.

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.

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-Binicioglu-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.

Beyond quantum probability: another formalism shared by quantum physics and psychology.

There is another meeting place for quantum physics and psychology, both within and outside of cognitive modeling. In physics it is known as the issue of classical (probabilistic) determinism, and in psychology it is known as the issue of selective influences. The formalisms independently developed in the two areas for dealing with these issues turn out to be identical, opening ways for mutually beneficial interactions.

The Fechnerian idea.

From the principle that subjective dissimilarity between 2 stimuli is determined by their ratio, Fechner derives his logarithmic law in 2 ways. In one derivation, ignored and forgotten in modern accounts of Fechner's theory, he formulates the principle in question as a functional equation and reduces it to one with a known solution. In the other derivation, well known and often criticized, he solves the same functional equation by differentiation. Both derivations are mathematically valid (the much-derided "expedient principle" mentioned by Fechner can be viewed as merely an inept way of pointing at a certain property of the differentiation he uses). Neither derivation uses the notion of just-noticeable differences. But if Weber's law is accepted in addition to the principle in question, then the dissimilarity between 2 stimuli is approximately proportional to the number of just-noticeable differences that fit between these stimuli: The smaller Weber's fraction the better the approximation, and Weber's fraction can always be made arbitrarily small by an appropriate convention. We argue, however, that neither the 2 derivations of Fechner's law nor the relation of this law to thresholds constitutes the essence of Fechner's approach. We see this essence in the idea of additive cumulation of sensitivity values. Fechner's work contains a surprisingly modern definition of sensitivity at a given stimulus: the rate of growth of the probability-of-greater function, with this stimulus serving as a standard. The idea of additive cumulation of sensitivity values lends itself to sweeping generalizations of Fechnerian scaling.

True contextuality beats direct influences in human decision making.

In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default theory allows one to separate true contextuality from direct influences in the overall context-dependence. The Contextuality-by-Default analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowd-sourcing experiments that exhibit true contextuality in simple decision making. The design of these experiments is an elaboration of one introduced in the Snow Queen experiment (Decision 5, 193-204, 2018), in which contextuality was for the first time demonstrated unequivocally. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Corrigendum: Noncontextuality with marginal selectivity in reconstructing mental architectures.

[This corrects the article on p. 735 in vol. 6, PMID: 26136694.].

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.

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.