Relating Causal and Probabilistic Approaches to Contextuality
Matt Jones
TL;DR
This paper bridges causal and probabilistic frameworks to better understand contextuality, demonstrating their equivalence and formalizing the concept through hidden influences and signaling in probabilistic models.
Contribution
It introduces a causal perspective on contextuality, proving the equivalence between model-based and CbD frameworks, and formalizes hidden influences as a core aspect of contextuality.
Findings
M-contextuality aligns with standard contextuality in consistent systems
M-contextuality is equivalent to CbD-contextuality in general systems
Hidden influences or signaling are necessary for non-contextual models
Abstract
A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabilistic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here we take a causal view of direct influences, defining direct influence within any causal model as the probability of all latent states of the system in which a change of context changes the outcome of a measurement. Model-based contextuality (M-contextuality) is then defined as the necessity of stronger direct influences to model a full…
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\subject
physics, psychology, mathematics
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Matt Jones
Relating Causal and Probabilistic Approaches to Contextuality
Matt Jones1
1University of Colorado Boulder
Abstract
A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabilistic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here we take a causal view of direct influences, defining direct influence within any causal model as the probability of all latent states of the system in which a change of context changes the outcome of a measurement. Model-based contextuality (M-contextuality) is then defined as the necessity of stronger direct influences to model a full system than when considered individually. For consistently connected systems, M-contextuality agrees with standard contextuality. For general systems, it is proved that M-contextuality is equivalent to the property that any model of a system must contain “hidden influences”, meaning direct influences that go in opposite directions for different latent states, or equivalently signaling between observers that carries no information. This criterion can be taken as formalizing the “no-conspiracy” principle that has been proposed in connection with CbD. M-contextuality is then proved to be equivalent to CbD-contextuality, thus providing a new interpretation of CbD-contextuality as the non-existence of a model for a system without hidden direct influences.
keywords:
contextuality, probabilistic causal models, contextuality-by-default, direct influence
1 Introduction
Probabilistic contextuality describes an empirical system of measurements wherein a set of observables can be measured in different subsets in different contexts, and even though each observable has the same distribution in all contexts in which it is measured, the joint distributions of measurements within each context cannot be pieced together into a global joint distribution. The original significance lies in Bell’s theorem and related results [5, 9], which state that certain contextual systems cannot be explained by any local hidden-variables theory, although such systems are predicted by quantum mechanics and have been experimentally confirmed [3, 22].
The standard formulation of contextuality applies only to cases where the distribution of each observable is identical across contexts. This property of a system of measurements is called consistent connectedness [25], marginal selectivity [14, 31], or no-disturbance [29]. Dzhafarov and Kujala have made persuasive arguments for extending contextuality to inconsistently connected systems, including the fact that real experiments never eliminate all sources of contamination, and that sample frequencies in finite datasets will generally not be equal even if the true generating probabilities are [12, 15, 16]. Beyond these pragmatic considerations, it is of interest to know whether contextuality can be usefully defined in cases where the distributions of observables are truly different across contexts. This might enable contextuality analysis to be applied to other domains, such as human cognition or behavior [1, 2, 8, 18].
The contextuality-by-default theory (CbD) offers one approach for defining contextuality for inconsistently connected systems [16, 17, 25]. CbD treats measurements of each observable in different contexts as different random variables, by default. It then asks whether the distributions of random variables in each context are compatible with a global distribution in which all variables for each observable are made as equal as possible, in a rigorous sense based on probabilistic couplings [30]. If not, then the system is CbD-contextual (we use this term to distinguish from standard contextuality and the model-based M-contextuality introduced below). For consistently connected systems, CbD-contextuality agrees with standard contextuality.
In addition to applying beyond consistently connected systems, CbD departs from previous approaches to contextuality in that it is a purely probabilistic theory of random variables, not grounded in theories of the physical system generating the measurements. As such, it is unclear what CbD-contextuality indicates about that system. Does establishing that an inconsistently connected set of measurements is CbD-contextual imply anything about viable theories of the physical system, similar to how standard contextuality implies a system cannot be described by any local hidden-variable theory?
The present article proves an affirmative answer to this question, based on a characterization of contextuality recently advanced by Cavalcanti [7] in terms of probabilistic causal models. Probabilistic causal models are widely used in statistics, computer science, machine learning, and psychology and are well suited for situations involving stochastic latent structure [24, 27]. They provide a useful generalization of hidden-variable theories in physics, enabling a physical system to be described by context variables controlled by the experimenter, unobservable variables representing theoretical latent (hidden) states of the system, and measured observables. Within this framework, we propose a definition of model-based contextuality (M-contextuality) that applies to both consistently and inconsistently connected systems, and we prove that it is equivalent both to CbD-contextuality and to the non-existence of a certain type of model for the system under investigation.
Our approach builds on two of the main principles that have motivated CbD: the distinction between contextuality and direct influence [8, 14, 16], and Cervantes and Dzhafarov’s no-conspiracy principle prohibiting “hidden” direct influences [8]. Early work on CbD showed that contextuality can be defined as a context-dependence of the identity of random variables over and above the dependence due to direct influences. Although the notion of direct influence was founded on the theory of selective influences in probabilistic causal models [11], subsequent developments of CbD have defined direct influence in purely probabilistic terms, to refer to the difference in an observable’s distribution across different contexts [8, 16]. Here we refer to such distributional differences as inconsistent connectedness, reserving direct influence to refer to a causal effect of context on the values of observables. Whereas inconsistent connectedness is an empirical (statistical) property of the measurements, direct influence as defined here is a theoretical (model-dependent) property of the physical system.
This article proposes a quantitative definition of direct influence within any probabilistic causal model, as the probability of all latent (hidden) states of the model in which a change of context changes the value of an observable. The degree of inconsistent connectedness of a measurement system imposes a minimal amount of direct influence needed to model each observable in any pair of contexts in which it is measured. We define a measurement system as M-contextual if modeling the full system requires direct influences stronger than these minimum values. This formalizes a proposal by Cavalcanti [7] that “A causal model should not allow causal connections stronger than needed to explain the observed deviations from the no-disturbance condition” (p. 6). For consistently connected systems, the minimal direct influences are zero, and the definition of M-contextuality coincides with that of standard contextuality.
Concerning the no-conspiracy principle, Ehtibar Dzhafarov gives the following philosophical-level statement of the principle (personal communication, June 2018): “Direct influences of a reasonable substantive theory (in physics or psychology) are not revealed in the distributional differences only under special, precariously set circumstances. As a rule, there are no “hidden” direct influences.” A conceptually similar and logically weaker principle is that of no-fine-tuning introduced by Wood and Spekkens [32] and elaborated by Cavalcanti [7], which holds that empirical conditional independence between measurement outcomes arises only when there is no causal connection: causal parameters cannot be fine-tuned such that their effects exactly balance out. Building on the present definition of direct influence, we formalize the idea of hidden direct influences as direct influences that work in opposite directions for different latent states, thus leaving the marginal distributions of observables unaffected. We then interpret the no-conspiracy principle as a prohibition against models with hidden influences. The primary results of this article are proofs that M-contextuality and CbD-contextuality, as properties of a measurement system, are both equivalent to the non-existence of a model of that system without hidden influences (Theorems 2, 5, & 6).
The definition proposed here for hidden direct influence agrees with that of non-communicating signaling given in Atmanspacher and Filk’s recent criticism of CbD [4]. Likewise, their observation that the criterion of CbD-contextuality accounts for communicating but not non-communicating signaling anticipates the result of the present article that a CbD-contextual system is one that cannot be modeled without hidden direct influences. We discuss in the concluding section how the formalism offered here reconciles the position of Atmanspacher and Filk with that of Dzhafarov, Kujala, and colleagues, at least at a mathematical level. More generally, the value of the present results is that they show a formal equivalence between three conceptually different approaches to contextuality: (1) the assumption underlying M-contextuality that direct influence in causal models is limited to that implied by inconsistent connectedness, (2) the no-conspiracy and no-fine-tuning principles, and (3) the probabilistic couplings approach of CbD. This correspondence will hopefully facilitate understanding and further development of both CbD and model-based approaches to contextuality.
The remainder of this article is organized as follows. Section 2 gives notation and definitions for standard contextuality. Section 3 describes causal probabilistic models and their relationship to standard contextuality. Section 4 defines a quantitative measure of direct influence in causal models, defines hidden direct influences, and offers a formalization of the no-conspiracy principle. Section 5 defines M-contextuality, proves that it agrees with standard contextuality for consistently connected systems (Theorem 1), and proves that regardless of consistent connectedness M-contextuality is equivalent to the non-existence of a model without hidden influences (Theorem 2). Section 6 recasts the preceding results for systems defined by a set of separate observers, as in Bell scenarios, relating direct influence to signaling among observers. Section 7 gives examples. Section 8 derives a translation between the model-based approach and CbD and proves the final main result (Theorem 5), that M-contextuality and CbD-contextuality are equivalent. As a corollary (Theorem 6), we also show that CbD-contextuality can be given a causal interpretation, in that a CbD-contextual system is one that is incompatible with a particular class of probabilistic causal models, namely those without hidden influences.
2 Standard Contextuality
Definition 1** (Measurement system).**
A measurement system consists of a set of observables , a set of possible values for each observable, a set of contexts , a relation with indicating that observable is measured in context , and a set of random variables . The subset is jointly distributed with distribution for each , and and are stochastically unrelated (i.e., are not measured together) whenever . Note the specification of determines , , , , and , and therefore we can refer to the entire measurement system as . The only technical requirements for the present results to hold are that and are both countable (i.e., no larger than the infinite set of natural numbers) and that each is Hausdorff and second-countable (this includes finite outcome spaces, -dimensional Cartesian space , and separable Hilbert space).
Although most literature on contextuality treats the as known distributions, in empirical practice one has access only to samples from those distributions. Therefore one might argue we should refer not to random variables but to individual observations, , where () indexes the instances in which the experiment was performed in condition . One advantage of the model-based approach is that it explicitly distinguishes the physical measurements from theoretical random variables (denoted below) used to model those measurements. This distinction makes the model-based approach naturally suited to handling sampling error, by standard model-evaluation methods of null-hypothesis significance testing or Bayesian model comparison. For ease of exposition, we set aside sampling error for the majority of the article, treating the as exactly known and referring to random variables rather than specific measurements , and comment on model fitting and evaluation in the concluding section.
Definition 2** (Consistent connectedness).**
A measurement system is consistently connected if each observable has the same marginal distribution within every context in which it is measured. That is, whenever , where indicates agreement in distribution.
Definition 3** (Standard contextuality).**
A measurement system is contextual in the standard sense if it is consistently connected but the distributions are not compatible with a joint distribution over all the observables. More precisely, each is a probability measure on the Cartesian product . A joint distribution over all the observables is a probability measure on , and for each it implies a marginal distribution on the observables measured in that context, given by the push-forward measure where is the natural projection . If there exists a such that for all , then the system is noncontextual; otherwise it is contextual.
The intuitive idea behind contextuality is that the distribution of each observable is unaffected by the context (consistent connectedness), but nevertheless context exerts some sort of effect on the observables that prevents them from being pieced together into a single jointly distributed system. Importantly, the existence of a global distribution in the sense of Definition 3 immediately implies the system is consistently connected. Therefore consistent connectedness is a necessary property of any traditionally noncontextual system. For a system that is inconsistently connected, the standard notion of contextuality does not apply.
3 Causal-model Characterization
of Contextuality
Following Cavalcanti [7], we analyze contextuality of a measurement system in terms of how it can be explained by probabilistic causal models [27].
Definition 4** (Causal probabilistic model).**
A causal probabilistic model is a set of jointly distributed random variables , with a dependency structure whereby each variable has a set of parents denoted (possibly ). The relation between and given by defines a directed acyclic graph, and the model’s joint distribution factors as .
For present purposes, we are interested in models of the physical system that generates some set of measurements . For any such model, we can classify its variables into three types: variables the experimenter sets (context), variables that are measured (observables), and unobserved latent variables representing theoretical constructs of the model. This classification enables a causal model of a physical system to be put into a simple canonical form that underlies most of the analysis in this article, as follows. Context variables can be collected into a single variable ranging over the set of contexts , with because context is an independent variable set by the experimenter. Observable variables are denoted for each observable . Those latent variables that do not depend on anything else in the model, , can be collected into a single random variable , which we variously refer to as the source state, latent state, or hidden state of the system prior to measurement. Any other latent variables, meaning intermediate ones that depend on and mediating ones that depend on or , may play an explanatory role in interpreting the theory but are unnecessary for predictions, . That is, we can marginalize over these other variables, so that encompasses all the internal mechanisms one might theorize for the physical system, on which the observables might depend. The result is a model described fully by , , and .
This canonical structure can be further simplified in two ways. First, because the determination of context is assumed to be under control of the experimenter and only distributions conditioned on are of interest, we can dispense with as a random variable and treat it more simply as an index variable, meaning without any probabilities associated to its values. Second, we assume each is a deterministic function of and . This assumption incurs no loss of expressive power because one can always incorporate all stochasticity in the model into the definition of , by replacing it with the underlying sample space. More precisely, because all variables in a probabilistic model are jointly distributed, they can be described as functions on a probability space . Then is a function on and each is a function on . By replacing with the identity function on , we can write each as a function of and . We call models of the resulting structure canonical causal models and use them as our primary focus here.
Definition 5** (Canonical causal model).**
A canonical causal model (or simply canonical model) comprises a random variable representing the hidden state of the system being modeled, an index variable representing the contexts in which measurements can be made, and a set of functions taking values in and representing the measurement outcome for each observable. The dependency structure is thus , , as shown in Figure 1a.
It should be apparent that canonical causal models extend the class of hidden-variables models used in classic work on contextuality (e.g., [5]), by allowing context to directly influence the observables (i.e., ). Prohibiting such dependencies yields the class of (noncontextual) hidden-variables models, which we refer to here as context-free models.
Definition 6** (Context-free causal model).**
A context-free causal model (or simply context-free model) is a canonical model in which each is independent of . That is, , and for all . See Figure 1b.
We can now formalize the relationship between canonical models and measurement systems:
Definition 7** (Model for a system).**
A canonical causal model is a model for a measurement system if it matches all of the individual contexts’ data distributions. That is, for each context , , where the equality here is an equality of distributions, as probability measures on .
The definition of a model for a system is closely related to the concept of a coupling as used in CbD [25]. A coupling for is a set of jointly distributed random variables such that the subset is distributed according to for each (see Definition 15 in Section 8). We prove in Section 8 (Proposition 5) that there exists a natural translation between models and couplings for any measurement system that preserves their essential properties regarding contextuality. However, we suggest the model-based approach offers two conceptual advantages. First, the model-based approach emphasizes the ontological distinction between the measurements as physical events, and the random variables as theoretical constructs meant to explain those physical events (when taken together with the other components of ) [7]. Second, a canonical causal model contains explicit causal structure via the latent variable , which enables formal definition of direct influences of on (see Section 4). This in turn enables us to distinguish direct influence, as a theoretical property of the physical system and its dynamics (i.e., of the process generating the data), from inconsistent connectedness, as a purely statistical property of the data distributions.
The following three results summarize the relationship between causal models and standard contextuality. Proposition 1 states that the framework of canonical causal models constitutes a universal language capable of describing any measurement system (contextual or not). Proposition 2 recapitulates Fine’s theorem [20] that noncontextuality is equivalent to compatibility with a hidden-variable theory (i.e., a context-free model). Proposition 3 states that consistent connectedness is equivalent to the analogous property of the individual observables. The proofs of these and all subsequent propositions and theorems are provided in the Supplementary Material.
Proposition 1**.**
For any measurement system , there exists a canonical causal model such that is a model for .
Proposition 2** (Fine [20]).**
A measurement system is noncontextual iff there exists a context-free model of that system.
Proposition 3**.**
A measurement system is consistently connected iff there exists a context-free model for the single-observable subsystem for all .
4 Direct Influence in Causal Models
Proposition 2 shows how standard contextuality can be understood in the framework of causal models. If a system of measurements can be explained by a causal model in which the value of each observable depends only on the state of the physical system prior to measurement, and not on the context (including which other observables are measured), then the system is noncontextual. If a measurement system is consistently connected but cannot be modeled without assuming the observables depend on context, then it is contextual. Furthermore, Proposition 3 shows that consistent connectedness has a similar relationship to models of individual observables: A system is consistently connected iff each separate observable can be modeled in a way that it is not dependent on context. Therefore a contextual system is one in which some dependence of the observables on the context is required to model the full system, even though no such dependence is needed to model any observable on its own.
These observations suggest an extension of the concept of contextuality to inconsistently connected systems. Specifically, we define a formal measure of direct influence of on each within any canonical causal model. We then propose to define any measurement system as contextual if modeling the complete system requires one or more direct influences to be greater than is necessary when they are considered separately. Zero direct influence will correspond to a context-free model. Therefore our extended definition of contextuality coincides with the standard one for the case of consistently connected systems: In that case, each separate observable can be modeled with zero direct influence, and the system is contextual iff the full system can also be modeled with zero direct influence.
In a context-free model, a change between two contexts never changes the outcome, regardless of the state of the physical system: for all , , and . We build on this property to define a quantitative measure of direct influence in any canonical causal model.
Definition 8** (Direct influence).**
Given a canonical model , the direct influence on each for any pair of contexts and is defined as . Thus represents the probability that a change of context between and would change the value of observable , with respect to the distribution over hidden states.
It is important to note that direct influence as defined here is a characteristic of a model, not of the set of measurements being modeled. Thus the direct influence attributed to a system is a theoretically relative construct (i.e., it depends on one’s theory of the underlying physical system), as opposed to consistent connectedness which is a purely empirical property of the measurements alone. Nevertheless, we can use direct influence to define contextuality, a property of the measurement system alone, by quantifying over models (see Section 5).
An important property related to direct influence is whether a model contains direct influences in opposing directions, which we refer to as hidden influences. We provide here the definition as it applies to observables with countably many possible values. The general definition, provided in the Supplementary Material, encompasses the definition here and is conceptually similar but more technical.
Definition 9** (Aligned model vs. hidden influences).**
A canonical model is aligned if, for any observable , value , and pair of contexts , either or . If, alternatively, both of these sets have positive probability (for some , , , ), we say the model contains hidden influences. That is, the switch from context to changes the value of observable to for some states and away from for other states.
Given this definition, Cervantes and Dzhafarov’s no-conspiracy principle [8] can be formalized as a prohibition against models with hidden direct influences, or equivalently a restriction to aligned models.
5 Model-based Contextuality
Given the formal measure defined above of direct influence of context upon an observable within a canonical causal model, we can now propose a definition of model-based contextuality that applies to inconsistently connected systems as well as to consistently connected ones. If a system is inconsistently connected, one can consider how great each direct influence must be, to model the difference in distribution for any observable between any pair of contexts. One can then ask whether these minimal direct influences are mutually compatible: That is, does modeling the full system require direct influences to be greater than is necessary individually? This approach formalizes the generalized no-fine-tuning principle proposed by Cavalcanti [7], that direct influences should be no greater than required by violations of consistent connectedness.
Definition 10** (M-contextuality).**
A measurement system is M-noncontextual if there exists a canonical model for that simultaneously minimizes all direct influences. That is, for each and , achieves the minimum value of over all models for . If such a model does not exist, is M-contextual.
For consistently connected systems, this definition reduces to that of standard contextuality. Indeed, the minimal direct influences for a consistently connected system are all zero, and consequently M-contextuality becomes equivalent to the nonexistence of a context-free model, which is equivalent to standard contextuality by Proposition 2.
Theorem 1**.**
For consistently connected systems, M-contextuality is equivalent to standard contextuality.
Our main result regarding M-contextuality is a characterization in terms of the existence of aligned models. The proof is based on explicit determination of the lower bound on direct influence for each observable and context pair and on showing that the models meeting that bound are precisely those without hidden direct influences.
Theorem 2**.**
A measurement system is M-contextual iff there does not exist an aligned canonical model for .
According to Theorem 2, an M-contextual system is one for which all models must contain hidden direct influences. Thus M-contextuality embodies the no-conspiracy principle: If one excludes hidden influences a priori, then the criterion of M-noncontextuality is simply that a system can be modeled by a classical probabilistic model. Put differently, if there is no aligned model for a system, then, following the logic of the no-conspiracy principle [8], we conclude the system cannot be explained by direct influences alone and that therefore there are contextual influences in addition to the direct ones.
6 Partitionable systems
In many measurement systems of interest, the observables can be partitioned such that exactly one observable from each subset is measured in any context. This situation arises in quantum physics when there are multiple observers and each observer can measure one out of a set of pairwise incompatible observables, as in Bell scenarios.
Definition 11** (Partitionable measurement system).**
A partitionable measurement system is one in which the set of observables can be partitioned as , the set of contexts is , and iff for each . That is, every context corresponds to a choice of exactly one observable (denoted ) from each subset . We say indexes observers, and is the set of (pairwise incompatible) observables available to observer .
When a measurement system is partitionable, it admits an alternative form of causal model, with one outcome variable per observer that depends on which measurement that observer chooses as well as (potentially) on the choices of all other observers. We refer to this as a partitioned model.
Definition 12** (Partitioned model).**
Given a partitionable measurement system and a canonical model for , the corresponding partitioned model is defined as follows. The hidden state of the system is modeled by the same random variable, , as in . The context is decomposed into a set of variables , with ranging over and indicating which observable is measured by observer . The measurement outcome for observer is represented by a variable defined by .
The general dependency structure of is and , as illustrated in Figure 1c. Note that the range of is the union of outcome spaces , although the image of is contained in for any . Therefore can be taken as a distribution on , which under Definition 12 matches the distribution from . Therefore is a model for whenever is.
A partitioned model expresses the same causal theory of a system as the corresponding canonical model , and therefore notions of direct influence and aligned models can be straightforwardly translated to partitioned models. The direct influence of interest here is the influence on one observer’s outcome due to the other observers’ choices of observables, which we refer to as signaling.
Definition 13** (Signaling).**
Given a partitioned model , an observer , and contexts and with , signaling to observer is defined as . That is, is the probability of a latent state in which switching the measurement choices of other observers (i.e., changing for one or more ) changes the outcome for observer .
It is easy to see that signaling in a partitioned model is equal to direct influence in the corresponding canonical model . That is, whenever . We can also define hidden signals versus aligned partitioned models, paralleling the definition of hidden direct influences and aligned canonical models. As with Definition 9, the definition given here applies to observables with countably many possible values; the general definition is provided in the Supplementary Material.
Definition 14** (Aligned vs. hidden signals).**
A partitioned model is aligned if, for any observer , value , and pair of contexts and with , either or . If, alternatively, both sets have positive probability (for some , , , ), we say contains hidden signals. That is, a specific change of choices of observables for other observers can change observer ’s measurement outcome to for some states and away from for other states.
The distinction between aligned and hidden signals is the same as that between communicating and non-communicating signals [4], because the aligned signals affect the marginal distribution of the receiving observer’s (’s) measurements, whereas purely hidden signals do not. It is easy to see that a canonical model is aligned iff the corresponding partitioned model is aligned. Likewise, is context-free iff has no signaling. The latter condition means that whenever , so that each observer’s outcome can be written as , and conforms to the simplified dependency structure for every (see Figure 1d). These observations imply a partitioned analogue of Proposition 2, whereby standard contextuality for consistently connected partitionable systems is equivalent to the nonexistence of a partitioned model without signaling, as in the original analyses of Bell scenarios [9]:
Proposition 4**.**
A consistently connected partitionable measurement system is noncontextual iff there exists a partitioned model for that has no signaling.
The definition of M-contextuality can also be expressed in terms of partitioned models:
Theorem 3**.**
A partitionable measurement system is M-noncontextual iff there exists a partitioned model that simultaneously minimizes all signaling. That is, for each , , and with , achieves the minimum value of over all partitioned models for .
Finally, the analogue of Theorem 2 for partitionable systems states that M-contextuality is equivalent to the nonexistence of a model without hidden signaling:
Theorem 4**.**
A partitionable measurement system is M-contextual iff there does not exist an aligned partitioned model for .
7 Examples
PR box
The Popescu-Rohrlich (PR) box [28] is a measurement system with four binary observables and four contexts, , with marginals for all (Table 1), and correlations , , , (Table 2). This system is consistently connected and contextual (hence also M-contextual). Consequently, the measurements for each observable, , can be modeled without direct influence, but the system as a whole cannot.
Define a model for the PR box by taking to range over with uniform distribution, and defining the by
[TABLE]
Then for , but (where and are the two applicable contexts for each ). Furthermore, the direct influence on is hidden: For , a switch from context to context changes from to , whereas for the same switch of context changes from to . These opposing influences cancel out in the margin. This situation illustrates the general fact that causal models for contextual systems always exist, but they must include hidden direct influences.
The PR box is a partitionable system, and can be written as and , . That is, observer measures observable or , and observer measures observable or . The contexts are now denoted . The model corresponds to a partitioned model with outcome variables defined by
[TABLE]
Signaling in this model is zero everywhere except for . That is, observer ’s measurement outcome is unaffected by observer ’s choice of observable, regardless of the state of the physical system, and a similar statement holds for observer 2 when she measures observable 2. However, when observer measures observable , the outcome depends on which variable observer measures. Moreover, this signaling goes in opposite directions depending on the latent state . That is, contains signaling, and that signaling is (perfectly) hidden. In the language of Atmanspacher and Filk [4], this is noncommunicating signaling that cannot transmit information between observers.
An M-noncontextual inconsistently connected system
Consider a system with the same observables, contexts, and correlation structure as the PR box, but with unbalanced marginals as shown in Table 3. The inconsistent connectedness of this system places a lower bound on the direct influences in any model thereof, which can be shown to be , , , and . The question is whether these lower bounds can be achieved simultaneously, and the answer is affirmative. Let be the model for this system defined by ranging over with uniform distribution, and the as given in Table 4. This model is immediately seen to achieve the lower bound on all . Note also that the direct influences are all aligned. For example, switching from context to context can change from to (when ), but it cannot change from to . These properties of show the system is M-noncontextual.
The partitioned model corresponding to (not shown) has similar properties. Signaling is present for both observers under both measurement settings, and in all cases the signaling is the minimum possible. Furthermore this signaling is aligned: Any change in one observer’s choice of observable changes the other observer’s measurement outcome only in one direction (or not at all) across all latent states of the system. Consequently, the signaling shows up fully in a change in the latter observer’s marginal probability. In the language of Atmanspacher and Filk [4], this is purely communicating signaling that transmits information between observers.
8 Relationship to CbD
The contextuality-by-default approach to contextuality is originally motivated by the observation that the traditional approach of treating as one and the same random variable for all contexts is mathematically incoherent, even for consistently connected systems [16]. CbD offers a rigorous alternative to this traditional approach through the construct of a probabilistic coupling [30].
Definition 15** (Probabilistic coupling).**
A set of jointly distributed random variables is a coupling for the measurement system if for all contexts , meaning as distributions on .
This definition enables the definition of contextuality to be shifted from the mathematically unsound question of a joint distribution for (where the superscript “” indicates the context is simply ignored) to well-defined questions regarding the joint distribution of . For a consistently connected system , the relevant coupling (if it exists) satisfies for all . This formalizes the idea of treating and as the same random variable. The theory also enables the concept of contextuality to be extended to inconsistently connected systems, by considering couplings in which and are not always equal. We follow the CbD 2.0 version of the theory, which is based on multimaximal couplings [17].
Definition 16** (Multimaximal coupling).**
For a set of stochastically unrelated random variables , a coupling is multimaximal if, for all and , is maximal among all couplings of .
Definition 17** (CbD-contextuality).**
A measurement system is CbD-noncontextual if there exists a coupling for such that, for each observable , is a multimaximal coupling for . Otherwise, is CbD-contextual.
It is easy to see that CbD-contextuality agrees with standard contextuality for consistently connected systems. If is consistently connected, a multimaximal coupling for is one that satisfies for all . Such a coupling is equivalent to a joint distribution over all of the observables (i.e., ) that has the distributions as marginals.
The main results of this section rest on a correspondence between couplings and canonical causal models:
Proposition 5**.**
Given a coupling for a measurement system , there exists a canonical model for such that for all and . Likewise, given a canonical model for a measurement system , there exists a coupling for such that the same relationship holds.
This correspondence implies the equivalence between M-contextuality and CbD-contextuality:
Theorem 5**.**
A measurement system is M-contextual iff it is CbD-contextual.
Finally, the equivalence between M-contextuality and CbD-contextuality, together with Theorem 2, provides an interpretation of CbD-contextuality in terms of aligned models:
Theorem 6**.**
A measurement system is CbD-contextual iff there does not exist an aligned model for .
Thus CbD-contextuality has a simple interpretation in terms of direct influence, specifically that a measurement system is CbD-contextual whenever modeling that system would require hidden direct influences or, equivalently, hidden signaling for partitionable systems. In other words, a CbD-contextual system is one that be modeled only by violating the no-conspiracy principle.
9 Conclusions
The present results show an equivalence between three approaches to extending contextuality analysis to inconsistently connected systems: Cavalcanti’s generalized no-fine-tuning principle [7] for causal probabilistic models, formalized here in terms of minimal direct influence (M-contextuality); Cervantes and Dzhafarov’s no-conspiracy principle [8], formalized here as a restriction to aligned causal models; and the purely mathematical framework of CbD [17]. Thus, the requirement of multimaximal couplings in CbD is equivalent to the assumption that direct influences (in the causal sense used here) are no greater than needed to explain the marginal distributions of individual observables, as well as to the assumption that direct influences never cancel out, even partially. Although there is a sense in which any mathematical definition is as good as any other, the fact that three different principles all converge on the same classification of systems into contextual and non-contextual might be taken as a stronger indication of the likely scientific utility of this classification.
Another main contribution of this article, beyond the correspondence between the particular definitions analyzed, is the translation between the CbD and model-based approaches. Under this translation, alternative conditions on couplings correspond to alternative criteria for direct influence. For example, the maximal couplings used in earlier versions of CbD [25] correspond to canonical causal models minimizing for each observable . Likewise, other measures of causal influence in probabilistic models [23] might translate to interesting criteria on couplings. This strategy of translation might enable insights from either framework to inform development in the other, to better understand the properties of contextual inconsistently connected systems, or to devise more refined definitions.
Although the translation between canonical models and couplings is fairly trivial, with the latent state of the model corresponding to the sample space of the coupling (Proposition 5), the model-based approach offers a number of conceptual advantages. First, it maintains the standard interpretation of contextuality as the impossibility of explaining a system classically, meaning with a hidden-variables theory in which measurements that were not made are nevertheless well-defined [19]. Second, it distinguishes direct influence, a property of the theoretical data-generating process, from inconsistent connectedness, a property of the data distribution [7]. Consequently, third, it provides a formalism for expressing how an observable might directly depend on context, thus enabling potential extensions wherein theoretical assumptions regarding the physical system impose additional constraints on this dependence.
Expanding on the third point, the definition of M-contextuality is based on causal models that may embody rich theories of the physical system underlying a set of measurements. Because the definition quantifies over all such models, it depends only on the measurements themselves (i.e., or samples therefrom). It is thus a meta-theory, providing conditions regarding what types of models are and are not mathematically possible for a given system. However, the model-based approach can also accommodate theoretical considerations directly. That is, one could incorporate domain-specific assumptions regarding direct influences, based on theoretical principles applicable to the physical system under study, and define a system as contextual whenever any model of that system would require stronger direct influences than allowed by the assumed theory. This is the standard logic with Bell scenarios, where the underlying theory is special relativity and the constraint is that signaling between spacelike-separated observers must be absent. The present approach can also accommodate more nuanced constraints, including limits on the magnitude of direct influence or signaling even when it is allowed to be nonzero. A further generalization of the analysis presented here would be to consider model architectures other than the canonical and partitioned ones. For example, rather than allowing direct influence of context on each observable, , or signaling between observers, , one could consider models with direct influence between observables, , to see what additional insight they might provide into the mathematical nature of contextuality.
As a fourth advantage, the model-based approach is naturally suited to the fact that contextuality analysis of an empirical measurement system is analysis of a finite dataset. As with any situation of hypothesis testing based on sample data, it requires statistical inference, in this case inference of whether the data could have been generated by a model from some class. As noted in Section 2, the data form a set , with each a sample from an unknown distribution . The set of possibilities for is a product of -simplices, where is the number of observables measured in context . The subset of possibilities consistent with a context-free model (or equivalently a partitioned model with no signaling) is defined by a set of linear inequalities and hence is a polytope within that space. Likewise, the subset of possibilities consistent with an aligned model (canonical or partitioned) is defined by a different polytope, containing the first. Given a dataset , the questions of contextuality, M-contextuality, and CbD-contextuality are questions of whether is a sample from a set of true distributions lying in the appropriate polytope, which can be answered with existing inferential methods [10, 21, 26]. Thus the model-based approach can accommodate sampling error in a principled way, without conflating it with direct influence.
The present results help to clarify and qualify Atmanspacher and Filk’s argument that CbD-contextuality is an inadequate definition because it accounts only for communicating and not non-communicating (hidden) signaling, whereas “signaling whatsoever, “hidden” or not, cannot create true quantum contextuality” [4]. Expositions of CbD make clear that this distinction is not meaningful within that framework, since it defines direct influence solely in distributional terms [8, 16]. Thus the two groups are essentially speaking different languages. The present approach bridges this gap, by distinguishing the probabilistic concept of inconsistent connectedness from the causal concept of direct influence, and by showing how one can translate between probabilistic couplings and causal models (Proposition 5). Using this translation, Theorem 6 shows Atmanspacher and Filk’s statement is correct: CbD-contextuality implies that a system cannot be explained entirely by communicating (aligned) direct influence or signaling, but it does not imply the system cannot be explained by hidden influences or signaling. On the other hand, the latter observation is not specific to CbD but applies to all possible theories of contextuality: If unconstrained, direct influence can always explain any pattern of data (Proposition 1), as Bohmian mechanics demonstrates for EPR-Bell scenarios [6]. Therefore, contextuality is of interest only under some restriction on direct influence. The no-conspiracy principle is one such restriction. In that regard, Theorem 6 unifies the positions of Atmanspacher and Filk and of Dzhafarov, Kujala, and colleagues: CbD-contextuality is equivalent to the proposition that a system cannot be explained by direct influence alone, if one excludes hidden influences a priori. The conceptual justification and scientific utility of this exclusion will likely be a matter of further debate.
Relating Causal and Probabilistic Approaches to Contextuality
Supplementary Material
Matt Jones ([email protected])
University of Colorado Boulder
Proof of Proposition 1
For each , extend to a probability measure on , for example by choosing an arbitrary distribution for each observable and then taking the product measure. Identify the set of possible values for (i.e., the space of hidden states) with the Cartesian product , and let be the product measure on this space obtained from the measures . Finally, define . For any , the conditional distribution is equal to the distribution obtained from by projecting (taking copy from the outside product and marginalizing over all in the inside product), which by construction is .
Proof of Proposition 2
Sufficiency: Noncontextuality of implies there exists a distribution over all the observables such that its projection to the observables within each context equals the distribution . Define a context-free model with ranging over with probability measure , and with for every and . Then the joint distribution equals , and thus for any context , the distribution equals .
Necessity: If is a context-free model for , then is the same distribution as , which in turn is the same as because the do not depend on . Therefore equals the projection of to for all , implying is noncontextual.
Proof of Proposition 3
In any context-free model for , is independent of , implying (as distributions on ) for all . Therefore has the same distribution for all , implying consistent connectedness. Conversely, if is consistently connected then for each we can define the probability measure on that is the distribution shared by all . A model of is then trivially constructed by letting range over with distribution and taking for all .
General Definition of Aligned Canonical Models and Hidden Influences
Given an observable with arbitrary outcome space and two contexts , a canonical causal model is said to have hidden direct influences with respect to when there exists a measurable set such that , , and for every measurable subset , either and , or else . A model is aligned if it has no hidden direct influences for any . This definition is equivalent to Definition 9 in the main text when is discrete, as can be seen by identifying with .
Proof of Theorem 1
Let be a consistently connected measurement system. By Proposition 3, for each there exists a context-free model for . The model satisfies for all , and it can be arbitrarily extended to a model for the full system. Therefore, is M-noncontextual iff there exists a model for with all direct influences equal to zero.
If there exists a context-free model for , all direct influences in this model are zero and therefore is M-noncontextual. Conversely, assume is M-noncontextual a let be a model for with all direct influences equal to zero. For each and contexts , define . By assumption, . Because and are assumed to be countable, , where . Now define a new model by restricting the range of and the domain of every (in the first argument) to . By construction, is a context-free model for .
Proof of Theorem 2
Fix and and , and let and respectively be the distributions of and , as probability measures on . By the Hahn-Jordan decomposition theorem applied to the signed measure , there exist a partition of the outcome space and positive measures and such that and . Moreover, and are unique. Define , which is necessarily a positive measure, and define . We prove the following three statements:
The minimal direct influence across all models for is given by . 2. 2.
If a model for satisfies , then it contains no hidden influences with respect to . 3. 3.
Conversely, if a model for contains no hidden influences with respect to , then it satisfies .
Together, these three statements imply that any model for is aligned iff it minimizes all direct influences, which in turn implies the theorem.
Proof of Statement (i). Let be any canonical model for . The direct influence in is constrained by
[TABLE]
Therefore is a lower bound for . To construct a model meeting this bound, let range over and define and for all . Let be the diagonal embedding , and define the push-forward measure , so that for all measurable . Define a second measure on , generated by
[TABLE]
for all measurable . Now define the distribution on by . For any measurable ,
[TABLE]
A similar calculation shows . Therefore is a model for the subsystem , which can be arbitrarily extended to a model for the full system . The direct influence is given by
[TABLE]
Proof of Statement (ii). Assume has hidden influences with respect to , and let be as given above in the General Definition of Hidden Influences. Define and . Because and , it cannot be that and . Without loss of generality, assume the former equality, , is false. Then the definition of hidden influences implies . Because , the sets and are disjoint, so we can bound the direct influence as . The proof of Statement 1 shows the former of these probabilities is at least , and therefore we have . Thus we have shown any model with hidden influences cannot satisfy .
Proof of Statement (iii). We first prove that for any measurable . To see this, assume the contrary, that with for some . Using alignment of , the probability can be squeezed into successively smaller subsets of so as to produce a contradiction. Specifically, define a property with being the statement that is a measurable subset of with . Note that is preserved under countable intersection and that implies . If we define , then the countable intersection property just stated implies there exists a set meeting this bound: and . If , then alignment of implies there are no hidden influences within ; that is, there exists such that or , and also or . Because , , which implies that the former relation in each of the two disjunctions just given holds: and . This in turn implies and , contradicting the definition of . On the other hand, if then , contradicting the fact that . Therefore the supposed set with cannot exist.
Next, let be a countable basis for , using the assumption that is second-countable. Take any and with . Because is Hausdorff, there exists be an open neighborhood of not containing . Because is a basis for , there exists some with and hence also . This shows that is a subset of . Therefore
[TABLE]
which in turn implies
[TABLE]
A parallel argument shows . Therefore the total direct influence in for is given by
[TABLE]
General Definition of Aligned Partitioned Models and Hidden Signals
Given an observer , an observable with arbitrary outcome space , and contexts and with , a partitioned model is said to have hidden signals with respect to when there exists a measurable set such that , , and for every measurable subset , either and , or else . A partitioned model is aligned if it has no hidden signals for any . This definition is equivalent to Definition 14 in the main text when is discrete for all , as can be seen by identifying with .
Proof of Proposition 4
If is noncontextual, then Proposition 2 implies there exists a context-free canonical model for . The corresponding partitioned model is easily seen to be a model for with no signaling. Conversely, if there is a partitioned model for that has no signaling, the corresponding canonical model is context-free, and Proposition 2 then implies is noncontextual.
Proof of Theorem 3
Let be any partitioned model for , with the corresponding canonical model. As observed in the main text, direct influence in and signaling in exactly correspond, in that whenever . Therefore minimizes all signaling iff minimizes all direct influences. The theorem then follows from the definition of M-noncontextuality, as the existence of such an .
Proof of Theorem 4
If is an aligned partitioned model for , then the corresponding canonical model is also aligned, implying is M-noncontextual by Theorem 2. Conversely, if is M-noncontextual, there exists an aligned canonical model for by Theorem 2, and the corresponding partitioned model is also aligned.
Proof of Proposition 5
First part: Let be the sample space for the jointly distributed random variables composing , such that each is a function . Define by letting range over with distribution and defining each by for and choosing arbitrary values for for (for all ). Then for any context and measurable subsets ,
[TABLE]
Therefore is a model for . For any and , the claimed equality holds:
[TABLE]
Second part: Given , let be the range of with the associated probability measure on and the sigma-algebra of measurable sets of values for . Then defines a sample space. For each and , define a random variable on this sample space by . Then derivations similar to those above show that is a coupling for and that for all and .
Proof of Theorem 5
If is M-noncontextual, then there exists a canonical causal model for that simultaneously minimizes all direct influences. The corresponding coupling provided by Proposition 5 minimizes for all and . Therefore is multimaximal for all , implying is CbD-noncontextual. Conversely, if is CbD-noncontextual then there exists a coupling for such that is multimaximal for all , implying is minimal for all . The corresponding canonical model provided by Proposition 5 minimizes for all and , implying is M-noncontextual.
Proof of Theorem 6
The theorem follows directly from Theorems 2 and 5: CbD-contextuality is equivalent to M-contextuality, which is equivalent to the non-existence of an aligned model.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Aerts, D., Argu lles, J.A., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S., & Veloz, T. (2018). Spin and wind directions I: Identifying entanglement in nature and cognition. Foundations of Science, 23 , 323-335.
- 2[2] Aerts, D., Gabora, L., Sozzo, S. (2013). Concepts and their dynamics: A quantum-theoretic modeling of human thought. Topics in Cognitive Science, 5 , 737– 772.
- 3[3] Aspect, A., Grangier, G., & Roger, G. (1981). Experimental tests of realistic local theories via Bell’s theorem. Physical Review Letters, 47 , 460–463.
- 4[4] Atmanspacher, H., & Filk, T. (in press). Contextuality revisited – Signaling may differ from communicating. In A. de Barros and C. Montemayor (eds.), Quanta and mind: Essays on the connection between quantum mechanics and the consciousness . Synthese Library.
- 5[5] Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox. Physics, 1 , 195–200.
- 6[6] Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden variables” I & II. Physical Review, 85, 166–193.
- 7[7] Cavalcanti, E. G. (2018). Classical causal models for Bell and Kochen-Specker inequality violations require fine-tuning. Physical Review X, 8 , 021018.
- 8[8] Cervantes, V.H., & Dzhafarov, E.N. (2018). Snow Queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices. Decision, 5 , 193-204.