Citation: Dzhafarov EN, Kujala JV (2013) All-Possible-Couplings Approach to Measuring Probabilistic Context. PLoS ONE 8(5):
e61712.
https://doi.org/10.1371/journal.pone.0061712
Editor: Gerardo Adesso, University of Nottingham, United Kingdom
Received: October 22, 2012; Accepted: March 13, 2013; Published: May 6, 2013
Copyright: © 2013 Dzhafarov, Kujala. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work is supported by NSF grant SES-1155956 (www.nsf.gov). The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. No additional external funding has been received for this study.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Consider a system with two inputs, 
, and two random outputs, 
, about which it is assumed that 
is not influenced by 
, nor 
by 
. A necessary condition for this selectivity of influences is marginal selectivity [1]: changes in the values of 
do not influence the distribution of 
, and analogously for 
and 
. Let, for example, both inputs and outputs be binary: 
, 
, and 
attain values 
and 
each. Denoting by 
and 
the two outputs conditioned on 
(
), the distribution of 
is described by the joint probabilities 
(summing to 1) in the matrix
(1)Assuming all four combinations 
are possible, marginal selectivity in this example means
(2)for all 
.
The assumption of selective influences, however, is stronger. It requires that the joint distribution of the two outputs satisfies, for all 
,
(3)where 
stands for “has the same distribution as,” 
are some functions, and 
is a source of randomness that does not depend on 
[2]–[8]. In our example (1) this means
(4)In the quantum mechanical context (see below) 
is interpreted as “hidden variables.” Such a representation may or may not exist when marginal selectivity is satisfied. For instance, the latter is satisfied in the following four distributions,
(5)It can be shown, however, that no representation (3) here is possible as the joint probabilities violate the Bell/CHSH inequalities considered below (Section 1 of Theory and Text S1). At the same time, a representation in the form of (3) is possible for the similar distributions
(6)One can think of 
and 
in (5) and (6) as being involved in different kinds of probabilistic context for the “direct” dependence of, respectively, 
on 
and 
on 
.
We propose a principled way of quantifying and classifying conceivable contextual influences, whether within or outside the scope of (3). Our approach is neutral with respect to such issues as causality or what distinguishes direct influences from contextual. We merely accept as a given a diagram of direct input-output correspondences (e.g., 
) and study the joint distribution of the outputs at all possible values of the inputs. The interpretation of the diagram is irrelevant insofar as it is compatible with the observed pattern of marginal selectivity: as 
changes while 
remains fixed, the distribution of 
does not change, and as 
changes while 
remains fixed, the distribution of 
does not change. Note that the distribution of 
may but does not have to change in response to changes in 
, and analogously for 
and 
.
Our approach is maximally general in the sense of applying to arbitrary sets of inputs and outputs (see Section 5 of Theory). To demonstrate it by detailed computations, however, we focus primarily on binary 
influencing binary 
; and even more narrowly, on the “homogeneous” case with the two values of both 
and 
equiprobable at all values of the inputs 
(
),
(7)Marginal selectivity then is satisfied trivially (because all marginal distributions are fixed).
The example focal for this paper is Bohm's version of the Einstein-Podolsky-Rosen paradigm (EPR/B) [9]: a quantum mechanical system consisting (in the simplest case) of two entangled spin
particles separated by a space-like interval (see Fig. 1). The two inputs here are spin measurements on these particles: input 
has two values corresponding to spin axes 
chosen for one particle, and input 
has two values corresponding to spin axes 
for another particle. The two outputs are spin values recorded: having chosen axes 
and 
, 
, one records 
for the first particle and 
for the second, each being a random variable with values 
and 
. (Note that the spins of a given particle along two different axes are noncommuting (see Text S2), because of which if one spin value is determined precisely, +1 or −1, the other one has a nonzero uncertainty. This means that 
considered as measurements yielding precise values of spins are mutually exclusive, and this is the reason 
can be viewed as values of a single input 
; and analogously for 
[10], [11].) Marginal selectivity (2) in this context is known under a variety of other names, such as “parameter independence” and “physical locality” [12]. We confine ourselves to the case (7), with the two spin values +1 and −1 being equiprobable for both 
and 
.
Formally equivalent situations are abundant in behavioral and social sciences [8], [13]–[17], where the issue of selective influences was initially introduced in [18], [19], in the context of information processing architectures. An example of a system here (from our laboratory) can be a human observer who adjusts a visual stimulus until it matches in appearance another, “target” visual stimulus. Let the latter be characterized by two properties, 
and 
(e.g., amplitudes of two Fourier-components), each varying on two levels, 
and 
. Denoting by 
and 
the corresponding properties (amplitudes) of the adjusted stimulus in response to 
, we define a binary random output 
as having the value “high”
or “low”
according as the variable 
is above or below the median of its distribution; output 
is defined from 
analogously. Marginal selectivity in the form (7) is ensured here by construction.
In an example from a biological domain 
and 
could be activity levels of two neurons tuned to two stimulus properties, 
and 
, respectively. Making 
and 
vary on two levels each and defining 
with respect to the medians of 
by the same rule as above, we get precisely the same mathematical formulation.
The formal equivalence of these three examples should by no means be interpreted as a hint at their physical affinity. Unlike in the EPR/Bohm paradigm, no physical laws prohibit the activity level 
of a neuron tuned to stimulus property 
from being affected by stimulus property 
. Similarly, the amplitude 
of the first Fourier component of the adjusted stimulus in the second example may very well be affected by the amplitude 
of the second Fourier component of the target stimulus. Our only claim is that if these “secondary” influences do not change the marginal distributions of 
and 
(which in the two examples in question is ensured by the definition of 
and 
), they can be viewed within the framework of a formal treatment that also includes the (physically very different) case of entangled particles.
Theory
1 Forms of context (determinism)
In the following, symbols 
(possibly with primes) always take on values 
each, and each of the outputs 
takes on values 
with equal probabilities. Representation (3) is equivalent to the existence of a jointly distributed system
(8)such that every output pair 
is distributed as 
; in symbols,
(9)As this entails
all components of 
are random variables with equiprobable +1/−1, and (9) reduces to
(10)The existence of 
in (8) satisfying (9) is known as (a special case of) the Joint Distribution Criterion (JDC) [6], [7], [14], [20], [21]. It follows from (3) by
(11)Conversely, if (9) holds for some 
, then one can put 
and
(12)where 
stands for the “
th member” (in the list of arguments). The JDC is a deep criterion that provides a probabilistic foundation for our understanding of the classical (non)contextuality (or classical determinism in physics). In particular, it immediately follows from the JDC that if representation (3) for 
exists, the “hidden variables” 
can always be reduced to a single discrete random variable with 
possible values (corresponding to the possible values of 
).
Using the same notation as above,
(13)the JDC in our case (two binary inputs and two binary outputs with equiprobable values) is equivalent to four double-inequalities
(14)with 
, 
[6], [7]. (See Text S1 for a derivation.) They are often referred to as the Bell/CHSH inequalities (in the homogeneous form), CHSH acronymizing the authors of [4], although the first appearance of these inequalities dates to [5].
The theory of the EPR/B paradigm predicts and experimental data confirm violations of the Bell/CHSH inequalities [22], [23], but quantum mechanics imposes its own constraint on the same linear combinations of probabilities:
(15)This constraint is known as the Cirel'son inequalities [24], [25] (see Text S2 for a derivation). Since the class of vectors 
that satisfy these double-inequalities include those allowed by (14) as a proper subset, it is natural to expect that (15) represents some relaxation, or generalization of the JDC. No such generalization, however, has been previously proposed. Developing one is the main goal of this paper.
This generalization is not confined to quantum mechanical systems. In other (e.g., behavioral) applications, one cannot exclude a priori the possibility of the bounds 
and 
in
(16)being wider than in (15), or falling between the bounds in (14) and (15), or being more narrow than in (14). One can think of all kinds of other constraints imposed on the possible values of 
, from confining this vector to one specific value to allowing it to vary freely. The latter (“complete chaos”) is represented by the “no-constraint” constraint
(17)with 
attained if one of 
is 
and the rest are zero, and 
attained if three of 
are 
and the remaining one is zero. Recall that we only consider the outputs with equiprobable outcomes, so
(18)All these conceivable constraints on the possible values of 
represent different forms and degrees of contextual influences. It would be unsatisfactory if all these possibilities, whether or not empirically realizable, could not be treated within a unified probabilistic framework including JDC as a special case. We construct such a framework, based on the classical (Kolmogorov's) theory of probability and the probabilistic coupling theory [26].
2 Connections
It is easy to see that for any vector of probabilities 
one can find a jointly distributed system of +1/−1 variables
(19)such that
(20)for all 
. The JDC then amounts to additionally assuming that among all such vectors 
there is one with
(21)and this is the assumption that is rejected by quantum theory in the EPR/B paradigm. Once (21) is explicitly formulated, however, it becomes clear that it is not the only way of thinking of 
. Since 
and 
occur under mutually exclusive conditions, one cannot identify the distribution of 
with that of 
. The latter does not exist as a pair of jointly distributed random variables. There is therefore no privileged pairing scheme for realizations of 
and 
,and zero values for 
are as acceptable a priori as any other. Analogous considerations apply to 
and 
.
Our approach consists in replacing (21) with more general
(22)and characterizing the dependence of 
on 
by properties of the set of all 4-vectors 
that are compatible with or imply certain constraints imposed on the vectors 
. Having adopted a particular diagram of input-output correspondences (in our case, 
), we can also say that these sets of 
characterize the contextual role of 
for 
and 
, respectively.
We call 
a vector of connection probabilities. The connection probabilities are of a principally non-empirical nature: they are joint probabilities of events that can never co-occur. By contrast, due to (20) the components of 
are joint probabilities of events that do co-occur, and by observing these co-occurrences the probabilities in 
can be estimated. To emphasize this distinction we refer to 
as a vector of empirical probabilities.
To distinguish our approach from other forms and meanings of probabilistic contextualism, e.g., [27], [28], [29], we dub it the “all-possible-couplings” approach. The term “coupling” refers to imposing a joint distribution (say, that of 
) on random variables that otherwise are not jointly distributed (
and 
). For a rigorous and general discussion of couplings and connections see Section 5.
3 Extended Linear Feasibility Polytope (ELFP)
ELFP is the set of all possible 
for which there exists a vector 
in (19) with jointly distributed components 
such that (20) holds, and, in accordance with (22),
(23)for all 
. The existence of such an 
means the existence of a probability vector 
consisting of the 
joint probabilities
(24)
. Let 
denote the 
component vector consisting of 24 empirical probabilities
(25)and 24 connection probabilities
(26)
.
Define a 
Boolean matrix 
whose rows are enumerated in accordance with components of 
(i.e., by equalities 
, 
, or 
) and columns in accordance with components of 
(i.e., by equalities 
). An entry of 
contains 1 if and only if the corresponding random variables in the enumerations of its row and its column have the same values: e.g., if a row is enumerated by 
and a column by 
, then their intersection contains 1 if and only if 
.
It is easy to see that 
exists if and only if
(27)for some vector 
(componentwise) of probabilities. The vectors 
for which such a 
exists are exactly those within the polytope whose vertices are the columns of the matrix 
. The term ELFP is due to this construction extending that of the linear feasibility test in [10]. This test, among other applications, is the most general way of extending the Bell/CHSH criterion to an arbitrary number of particles, spin axes, and spin quantum numbers [10], [11], [30]–[32]. Its application to binary inputs/outputs (not necessarily with equiprobable outcomes) is shown in Text S1.
To describe ELFP by inequalities on 
, we introduce the 16-component sets
(28)
and 
denote the subsets of 
with, respectively, even (0,2, or 4) and odd (1 or 3) number of 
signs; 
and 
are defined analogously. ELFP is described by
(29)(see Text S3).
4 All, Fit, Force, and Equi sets
Let 
denote any constraint (e.g., inequalities) imposed on 
. Our approach consists in characterizing this constraint by solving the following four problems:
- Find the set
of all 
with 
subject to 
: i.e., 
if and only if
(30)
- Find the set
of connection vectors 
that fit (are compatible with) all empirical probability vectors 
satisfying 
: i.e., 
if and only if
(31)
- Find the set
of 
that force all compatible empirical probability vectors 
to satisfy 
: i.e., 
if and only if
(32)
- Find the set
of 
for which an empirical probability vector 
satisfies 
if and only if 
is in the ELFP set: i.e., 
if and only if
(33)Clearly, 
To illustrate, we focus on the following four benchmark constraints. The no-constraint, or “complete chaos” situation is given by
(34)equivalent to (17). The quantum mechanical constraint is given by
(35)equivalent to (15). The “classical” constraint is given by
(36)equivalent to the Bell/CHSH inequalities (14). Finally, we consider the constraint
(37)For all constraints except for 
the sets All, Fit, Force, and Equi are as shown in Table 1 (for derivations see Text S4).
Thus, 
is the set of all 
such that 
: if an 
is in this set, then any 
(with no constraints) is compatible with it. 
is characterized by 
: if an 
is in this set, then all compatible with it 
satisfy 
. 
is the set of all 
such that 
contains 
: for any such an 
, a 
is compatible with it if and only if it satisfies 
.
For each of these sets we compute 
, its volume normalized by that of 
, with 
being the dimensionality of the set (Fig. 2). Thus, the defining property of 
, 
, is satisfied if and only if either all 
are 
, or they all are 
, or two of them are 
and two 
. Hence 
. For nonzero volumes, the derivation is described in Text S4. Each panel of Fig. 2 can be viewed as a “profile” of the corresponding constraint. Each of the first three volumes in a panel can be viewed as characterizing the “strictness” of a constraint, in three different meanings. The intuition of a stricter constraint is that it corresponds to a smaller 
, larger 
, and smaller 
. Characterizing constraints imposed on empirical probabilities by multidimensional volumes is not a new idea [33], but our computations are different: they are aimed at sets of nonempirical connection probabilities in relation to constraints imposed on empirical probabilities.
The constraint 
has to be handled separately. Clearly, 
. 
is described by
(38)and 
is a polynomial function of 
and 
, these two quantities forming the triangle 
. The polynomial and its values are shown in Fig. 3 (see Text S5, for computational details). 
is clearly empty, hence so is 
.
5 All-possible-couplings approach on the general level
We show here how the approach presented so far generalizes to arbitrary sets of inputs and random outputs. We use the term sequence to refer to any indexed family (a function from an index set into a set), with index sets not necessarily countable. We present sequences in the form 
, 
, or 
. A random variable is understood most broadly, as a measurable mapping between any two probability spaces. In particular, any sequence of jointly distributed random variables is a random variable. For brevity, we omit an explicit presentation of probability spaces and distributions. In all other respects the notation and terminology closely follow [15], [11].
An input is a set of elements called input values. Let 
be a sequence of inputs. A treatment is a sequence 
that belongs to a nonempty set 
(so that 
for all 
). If 
, 
, and 
, then 
and 
is the restriction of 
to 
, i.e., the sequence 
.
An output is a random variable. Let 
be a sequence of outputs such that
is a random variable for every 
, i.e., the random variables 
across all possible 
possess a joint distribution;- if
, 
, and 
, then 
.
Property 2 is (complete) marginal selectivity [8]. 
is called an empirical random variable, and 
is the sequence of empirical random variables.
Remark 1. The interpretation is that for every 
, each 
may “directly” influence 
but no other output in 
. The fact that inputs in 
and outputs in an empirical random variable 
are in a bijective correspondence is not restrictive: this can always be achieved by an appropriate grouping of inputs and (re)definition of treatments 
[10].
Remark 2. The special case considered in the previous sections corresponds to 
,
(39)
(40)and (abbreviating 
as 
and 
as 
)
(41)where each 
is a binary random variable with 
.
Given a sequence of empirical random variables 
, a sequence of random variables
(42)(not necessarily jointly distributed) is called a connecting set for 
if each 
is a coupling for
(43)where 
. This means that 
is a random variable of the form
(44)with
(45)for all 
such that 
. 
is called an 
connection. The indexation in 
is to ensure that if 
, then 
and 
are stochastically unrelated. An identity 
connection 
is one with 
for any 
.
Remark 3. It is generally convenient not to distinguish identically distributed connections. By abuse of language, the distribution of 
(or some characterization thereof) can also be called 
connection. We used this language in the previous sections when we represented 
connections (without introducing them explicitly) by probabilities 
and called 
a connection vector. See Remark 4.
A jointly distributed sequence
(46)is called an Extended Joint Distribution Sequence (EJDS) for 
if for any 
and any 
,
(47)where 
, and
(48)for any 
.
Remark 4. For the special case considered in the previous sections, a connecting set for 
is (conveniently replacing 
, 
, and 
with 
, 
, and 
, respectively)
(49)such that
(50)for 
. An EJDS for 
is a random variable (using analogous abbreviations)
(51)such that
(52)and
(53)for 
. In the previous sections each 
was represented by 
and each 
by 
.
An EJDS for 
reduces to the Joint Distribution Criterion set (JDC set) of the theory of selective influences [11], [14] if all connections in 
are identity ones. Note that no connection has an empirical meaning: for distinct 
, the variables 
and 
corresponding to 
and 
do not have an empirically observable (or theoretically privileged) pairing scheme.
Let 
be any set whose elements are sequences of empirical random variables 
. 
can be viewed as the set of all possible empirical random variables satisfying certain constraints. We define the sets AllX, FitX, ForceX, and EquiX as follows:
- AllX is the set of all pairs
such that
(54)
- FitX is the set of all
such that
(55)
- ForceX is the set of all
such that
(56)
that is, 
if and only if
(57)
The all-possible-couplings approach in the general case consists in characterizing any 
(interpreted as a type of contextuality or determinism) by AllX, FitX, ForceX, and EquiX. A straightforward generalization of this approach that might be useful in some applications is to replace 
in all definitions with a subset of 
, or several subsets of 
tried in turn. Thus one might consider connections involving only particular 
(e.g., only singletons), or one might require that some of the connections are identity ones.
Conclusion
The essence of the proposed mathematical framework is as follows. We consider all possible couplings for empirically observed vectors of random outputs. In the case of two binary inputs/outputs these vectors are pairs
(58)the couplings 
for them have the form (19), with the coupling relation (20). We assume that the joint distributions (in our case described by pairwise joint probabilities) of the empirically observed 
are subject to a certain constraint, given to us by substantive considerations outside the scope of our approach: for instance, if a system consists of entangled particles, a constraint, say (15), is derived from the quantum theory. Due to (20), the constraint is imposed on
(59)We investigate then the unobservable “connections”, the subvectors of the components of 
that correspond to outputs obtained at mutually exclusive values of the inputs (i.e., never co-occurring). In our case these are the pairs
(60)corresponding to, respectively,
(61)We then characterize the constraint imposed on the empirical pairs (59) by describing the “fitting” or “forcing” (or both “fitting and forcing”) distributions of the unobservable connections (60). By fitting distributions of (60) we mean those that are compatible with any (59) subject to the constraint in question, the compatibility meaning that all these eight pairs can be embedded into a single 
(with jointly distributed components). By forcing distributions of (60) we mean those that are compatible with (59) only if the latter are subject to the given constraint.
The value of this approach is in providing a unified language for speaking of probabilistic contextuality. At the cost of greater computational complexity but with no conceptual complications the computations involved in our demonstration of the all-possible-couplings approach can be extended to more general cases: arbitrary marginal probabilities (satisfying marginal selectivity), nonlinear constraints, and greater numbers of inputs, outputs, and their possible values. The language for a completely general theory, involving unrestricted (not necessarily finite) sets of inputs, outputs, and their values, is presented in Section 5 of Theory.
