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On Universality of Classical Probability with Contextually Labeled Random Variables

One can often encounter claims that classical (Kolmogorovian) probability theory cannot handle, or even is contradicted by, certain empirical findings or substantive theories. This note joins several previous attempts to explain that these claims are unjustified, illustrating this on the issues of (non)existence of joint distributions, probabilities of ordered events, and additivity of probabilities. The specific focus of this note is on showing that the mistakes underlying these claims can be precluded by labeling all random variables involved contextually. Moreover, contextual labeling also enables a valuable additional way of analyzing probabilistic aspects of empirical situations: determining whether the random variables involved form a contextual system, in the sense generalized from quantum mechanics. Thus, to the extent the Wang-Busemeyer QQ equality for the question order effect holds, the system describing them is noncontextual. The double-slit experiment and its behavioral analogues also turn out to form a noncontextual system, having the same probabilistic format (cyclic system of rank 4) as the one describing spins of two entangled electrons.

In the literature on foundations of quantum physics (Accardi, 1982;Feynman, 1951;Feynman, Leighton, & Sands, 1975;Khrennikov, 2009b) and, more recently, psychology (Aerts, 2009(Aerts, , 2014Broekaert, Basieva, Blasiak, & Pothos, 2017;Busemeyer & Bruza, 2012;Moreira & Wichert, 2016;Pothos & Busemeyer, 2013), one can encounter statements that classical (Kolmogorovian) probability theory is inadequate because it does not have conceptual means to handle (sometimes even, is contradicted by) this or that empirical fact. Three of the most widespread assertions of this kind are as follows: Statement 1: Classical probability requires that certain (e.g., Bell-type) inequalities hold for certain systems of random variables, but we know from quantum mechanics and from behavioral experiments that they may be violated.
Statement 2: In classical probability, the joint occurrence of two events is commutative, but we know from quantum mechanics and from behavioral experiments that the order of two events generally matters for their joint probability.
Statement 3: Classical probability is additive, but we know from quantum mechanics and from behavioral experiments that this additivity can be violated.
This note has two objectives: (1) to show that the three statements above are mistakes brought in by ignoring contextual labeling of random variables; and (2) to demonstrate how the use of contextual labeling enables so-called contextuality analysis of systems of random variables, a relatively new form of probabilisitic analysis of considerable interest in empirical applications. Contextual labeling of random variables is the departing principle of Khrennikov's Växjö Model (Khrennikov, 2009a) and of the Contextuality-by-Default theory (Dzhafarov, 2017;Dzhafarov, Cervantes, & Kujala, 2017;Dzhafarov & Kujala, 2014a, 2016a, 2017a, 2017b. Let us preamble this discussion by stating our view of classical probability theory (CPT), one that we are not prepared to defend in complete generality, confining ourselves instead to merely illustrating it on the three statements above. This view is that CPT, on a par with classical logic and set theory, is a universal abstract mathematical theory. As an abstract mathematical theory, it does not make empirically testable predictions, because of which it cannot be contradicted by any empirical observation. As a universal theory, for any empirical situation, it has conceptual means to adequately describe anything that can be qualified as this situation's probabilistic features (in the frequentist sense). Moreover, as a conceptual tool, in the same way as classical logic and set theory, it is indispensable and irreplaceable in dealing with probabilistic problems: at the end, the results of any non-classical probabilistic analysis have to be formulated in terms of classical (frequentist) probabilities, distributions, and random variables. However, when applied to an empirical situation, CPT can (even must) be complemented by special-purpose computations identifying some of the random variables, distributions, and probabilities in this particular situation. To give a very simple example, CPT provides methods for deriving probabilities of events defined on the outcomes of rolling a die from a distribution of these outcomes, but it cannot predict this distribution. A special theory is needed to know, e.g., that if a die is manufactured in a particular way, then the distribution of its outcomes is uniform. We view quantum probability as such a special-purpose theory complementing classical probability. It can, of course, be formalized and presented as an abstract calculus, in the same way one can formalize a paraconsistent logic. However, just as one cannot replace classical logic with paraconsistent logic in analyzing anything, including the very paraconsistent logic itself, one cannot dispense with classical probability when discussing and analyzing quantum probability computations.
This view is not new. Ballentine (1986) defended a similar position in essentially the same way we are doing here. The difference is in that instead of using random variables, Ballentine confines himself to a more limited language of events, and he uses conditionalization in place of the more general contextualization (Dzhafarov & Kujala, 2014b; we discuss conditionalization in Section 3 below). Khrennikov (2009a), in describing his Växjö contextual model uses Ballentine's conditional-probability notation, but emphasizes that these are not conditional probabilities of CPT. Rather he calls them "contextual probabilities," and explains that "contextual probability [...] is not probability that an event, say B, occurs under the condition that another event, say C, has occurred. The contextual probability is probability to get the result a = α under the complex of physical conditions C" (Khrennikov, 2009a, p. 50). This seems to be precisely the same as the contextual labeling used in the Contextuality-by-Default theory. A very clear presentation of a position that is close to ours can be found in the arguments presented in an internet discussion by Tim Maudlin (2013).
Although Richard Feynman is often cited as arguing the opposite point of view, that classical probability is not compatible with quantum mechanics (Accardi, 1982;Costantini, 1993;Khrennikov 2009b), our own reading of Feynman (1951) makes us think that his point of view was less than unequivocal. While in one place he indeed speaks of "the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace," (p. 533) he says in the same paper that "the concept of probability is not altered in quantum mechanics. When I say the probability of a certain outcome of an experiment is p [...] no departure from the concept used in classical statistics is required. What is changed, and changed radically, is the method of calculating probabilities" (p. 533). Compare this with the special-purpose computations mentioned earlier. We will return to Feynman when discussing the double-slit experiment in Section 3.
We will not recapitulate all the arguments related to this issue, as they have been presented in many previous publications (Dzhafarov, Cervantes, & Kujala, 2017;Dzhafarov & Kujala, 2014a, 2014b, 2016a, 2017a, 2017b. We will use just one familiar example. Let R 1 , R 2 , R 3 , R 4 denote a set of binary (+1/ − 1) random variables with known distributions of (R 1 , R 2 ), (R 2 , R 3 ), (R 3 , R 4 ), and (R 4 , R 1 ). The necessary and sufficient condition for the existence of such a quadruple of random variables is given by the CHSH/Fine inequality (Bell, 1964;Clauser, Horne, Shimony, & Holt, 1969;Fine, 1982): where ⊕1 is cyclic shift 1 → 2 → 3 → 4 → 1, and · is expectation. One can easily construct examples of distributions of (R i , R i⊕1 ) for which this inequality is violated, indicating that such R 1 , R 2 , R 3 , R 4 do not exist (essentially by the same logic as in determining that there are no four numbers a, b, c, d with a = b, b = c, c = d, and d = a + 1).
The problem arises when we are being told that the existence of such R 1 , R 2 , R 3 , R 4 is predicted by quantum theory and corroborated by experiment. If we believe this, violations of (1) should indeed mean that CPT is inadequate, if not internally contradictory. We should not, however, believe this. R 1 , R 2 , R 3 , R 4 in (1) are random variables in the CPT sense; they are not within the language of quantum theory. To decide what classical random variables should describe outcomes of what quantum measurements, one needs to go outside this theory. The general rule is that a random variable is identified by what is being measured and how it is being measured. The latter includes all conditions under which the measurement is made, in particular, all other measurements performed together with the given one. In our example, the measurements are indicated by star symbols in the following matrix: The row labels c 1 , . . . , c 4 are called contexts, and here they are defined by which two quantities are being measured together: in c 1 it is q 1 and q 2 , in c 2 it is q 2 and q 3 , etc. In behavioral science the quantities q 1 , . . . , q 4 can be, e.g., four Yes-No questions posed to a large number of people divided into four groups: in the group c 1 each person is asked q 1 and q 2 , in the group c 2 each person is asked q 2 and q 3 , etc. In quantum mechanics the matrix above could describe the well-known EPR/Bell paradigm with two entangled spinhalf particles: q 1 and q 3 correspond to the two axes along which Alice measures spins in her particle, while q 2 and q 4 correspond to the two axes analogously used by Bob in his particle. Let us use the notation R j i for the outcome of a measurement of q i in context c j : Since the values of R i i and R i i⊕1 are empirically paired (two responses given by the same person, or the measurements by Bob and Alice made simultaneously), the random variables in each row of the matrix are jointly distributed. This is not true for any two measurements made in different contexts: their joint distribution is undefined, and we call them stochastically unrelated (not to be confused with being stochastically independent, which is a special case of being jointly distributed). In particular, R j i and R j ′ i measuring the same property q i in two different contexts are stochastically unrelated. In the Kolmogorovian language, It is clear now that CPT imposes no constraints whatever on the row-wise joint distributions. The CHSH/Fine inequality (1) cannot be derived for this matrix of contextually labelled random variables. However, it can be derived as a solution for the following problem: find necessary and sufficient conditions for the existence of a jointly distributed quadruple of random variables (S 1 , S 2 , S 3 , S 4 ) such that (2) for i = 1, . . . , 4. Such a vector (S 1 , S 2 , S 3 , S 4 ) is called a "reduced coupling" of the stochastically unrelated pairs R i i , R i i⊕1 (Dzhafarov & Kujala, 2016b). The reduced coupling (S 1 , S 2 , S 3 , S 4 ) is merely a shortcut for describing a special case of what we call a C-coupling (Dzhafarov, Cervantes, & Kujala, 2017;Dzhafarov & Kujala, 2016a, 2017a, 2017b. C is some property of a pair of random variables, and a C-coupling of the pairs R i i , R i i⊕1 in our example is a jointly distributed octuple of random variables S 1 1 , S 1 2 , S 2 2 , S 2 3 , S 3 3 , S 3 4 , S 4 4 , S 4 1 such that, for i = 1, . . . , 4, and, in addition, The reduced coupling is the one defined by C that means "are equal with probability 1" (applied to pairs of random variables). More generally, in the Contextuality-by-Default theory, C is chosen to mean "are equal with maximal possible probability." For this choice of C, the criterion for the existence of a C-coupling is a useful generalization of CHSH/Fine inequality (1) (Dzhafarov & Kujala, 2016a;. A system of random variables for which a C-coupling exists (does not exist) is called C-noncontextual (respectively, C-contextual). 1 That context is part of the identity of a random variable is the departure point for the Contextuality-by-Default theory, the term "identity" being understood in the Kolmogorovian sense, as the measurable function from a domain probability space to a codomain measurable space (for detailed explanations, see Dzhafarov & Kujala, 2016a, 2017a. One advantage provided by this contextual identification is that it allows for the possibility that random variables measuring the same property in different contexts, such as R 1 2 and R 2 2 in our example, are differently distributed. This can happen, e.g., if one of the two questions posed to a person influences her response to the other question, or if Alice can signal to Bob and thereby change his recordings. With noncontextual labeling, such as R 1 , R 2 , R 3 , R 4 in the opening formulation, to express the same fact one would have to say that R 2 is differently distributed depending on whether it is recorded together with R 1 or R 3 . This is at best an abuse of language, if not outright nonsensical, as the distribution of R 2 is part of its identity.

ON STATEMENT 2
"In classical probability the joint occurrence of two events is commutative, but we know from quantum mechanics and from behavioral experiments that the order of two events generally matters for their joint probability." This statement, like Statement 1, is common in both physics and psychology (Busemeyer & Bruza, 2012;Busemeyer, Pothos, Franco, & Trueblood, 2011;Jacobs, in press;Pothos & Busemeyer, 2013;Wang & Busemeyer, 2013). 2 In particular, this view is given as a justification for using quantum theory to formalize the order effects of successively posed questions (Wang & Busemeyer, 2013) and decision-making tasks in which the order of information presented may impact the decision . This is perhaps the most remarkable of the statements about the inadequacy of CPT, given that its falsity is so obvious. CPT would be a singularly helpless exercise had it not been able to depict the difference between ordered pairs of observations (a, b) and (b, a). The difference between the two is obvious on the basic set-theoretic level: an ordered pair (a, b) is an abbreviation for the set {{a, 1} , {b, 2}}, or {a, {a, b}}, because of which (a, b) and (b, a) are different sets, unless a = b. Moreover, since an ordered pair is merely a simple case of a process (indexed set), Statement 2 is a paradoxical denial of the existence, within the framework of CPT, of random processes that are not exchangeable.
Let us begin our explanation of this confusion by considering the probabilistic identity of responses R q to some question q. Its domain probability space can be thought of as a set X of potential responders to the question q, with some probability measure µ imposed on its power set (treated as sigma-algebra). Let the possible values of R q be Yes/No. Its distribution then is defined by x responds Yes to q}) . (6) By construction, q is part of the identity of R q , so if q is replaced with another question q ′ , the random variable R q will be replaced with another random variable R q ′ . Probability theory allows this new random variable to have another distribution, but, of course, being an abstract mathematical theory, does not predict what the distributions of R q and R q ′ can be: such a prediction is up to an empirical theory dealing with people's understanding of questions and their knowledge of answers.
Consider now two questions that have identical formulation but are asked in different tones of voice or with different noise or images in the background; or two questions that have the same content but differ in how they are formulated (e.g., "Is it 11 am now?" versus "Is 11 am the correct time at this moment?"). The usual experimental design, if one is interested in such differences, would be to partition X into two subsets X 1 and X 2 , asking the question q in one form of the members of X 1 and in another form of the members of X 2 . From the point of view of abstract probability theory, whatever the difference between the two questions substantively, formally the responses to them are two different random variables defined on two different domain probability spaces. They are, therefore, stochastically unrelated. One can choose (based on one's substantive, non-mathematical understanding of questions and answers) to consider the differences in formulations or in the tone of voice to be part of the questions themselves (in which case one will deal with random variables denoted R q and R q ′ ) or to formalize the differences as different contexts in which one and the same question is asked (in which case one will present the random variables as R c q and R c ′ q ). The two representations are interchangeable, but the latter one is preferable because, in accordance with the principles of the Contextuality-by-Default theory, it encodes the stochastic unrelatedness of R c q and R c ′ q in the very notation. (A combined notation, such as R c q and R c ′ q ′ , is possible too, as discussed by , but it is less interesting for subsequent contextuality analysis.) Using different orders of two questions has precisely the same logical status as differences in the tone of voice or background noise: it creates two random variables (or two pairs of jointly distributed random variables) that are stochastically unrelated to each other. The set X is partitioned into two subsets X AB and X BA , corresponding to the two orders, (q A , q B ) and (q B , q A ). And the random variables defined on these subsets and corresponding to a given question, say q A , can have different distributions. The latter is exceedingly obvious if one uses specially chosen questions. Consider, e.g., q A ="Is this the first question I am asking?" and q B ="Is this the second question I am asking?". If q A is asked first and q B second, the probability of the responses Yes and Yes should be close to 1. If the order of the questions is reversed, the probability of the responses Yes and Yes should be close to zero.
By analogy with two forms of the same question, one can now proceed in several different ways, but the one most informative for contextuality analysis is as follows. We define a jointly distributed pair R AB where X AB A = x ∈ X AB : x responds Yes to q A , X AB B = x ∈ X AB : x responds Yes to q B .
The joint distribution for R BA A , R BA B , stochastically unrelated to the previous pair, is defined similarly, and can be arbitrarily different from (7).
Using the Contextuality-by-Default representation, the system of the random variables just defined is We can now choose some property C for pairs of random variables, as explained in the previous section, and ask whether the system above has a C-coupling (or, in the terminology of Contextuality-by-Default, whether it is Cnoncontextual). With C chosen to mean "are equal with maximal possible probability," such a C-coupling exists if and only if (Dzhafarov & Kujala, 2016a;Dzhafarov, Zhang, & Kujala, 2015) R Here, Yes and No responses have been encoded as +1 and −1, respectively. The remarkable QQ law discovered by Wang and Busemeyer (2013) is equivalent to saying that the left-hand side expression in (9) is zero, from which it follows that according to this law this system of random variables is noncontextual. 3 See Dzhafarov, Kujala, Cervantes, Zhang, and Jones (2016) and Dzhafarov, Zhang, and Kujala (2015) for a detailed discussion.

ON STATEMENT 3
"Classical probability is additive, but we know from quantum mechanics and from behavioral experiments that this additivity can be violated." This statement, even though it has been repeatedly criticized, appears to be shared by many quantum theorists and is accepted explicitly in some psychological accounts that use quantum theory (Busemeyer & Bruza, 2012;Busemeyer, Wang, & Lambert-Mogiliansky, 2009;Khrennikov, 2009aKhrennikov, , 2010. Additivity, expressed in the language of random variables, is that if A and B are disjoint events in the codomain space of a random variable R, then This principle is sometimes analyzed in a more complex form, referred to as the "law of total probability" (Busemeyer, Wang, & Lambert-Mogiliansky, 2009;Khrennikov, 2006). The latter is a consequence of the additivity above and the set-theoretical distributivity, So the authors who claim it is violated have the option of attributing the violation to the failure of the distributivity in place of or in addition to failure of additivity (Busemeyer & Bruza, 2012;Khrennikov, 2010). We will discuss here the basic form (10) only. The claim of violations of this law in quantum mechanics comes from the double-slit experiment. We consider it in the following version: a source of particles emits them into a barrier with two slits (left and right, each of which can be closed or open), and a detector of the particles occupies a small area behind this barrier. One considers the probability with which an emitted particle reaches the detector, and discovers that this probability, when both slits are open, is not equal to (depending on the detector's location, can be greater or smaller than) the sum of these probabilities recorded with only the left slit open and with only the right slit open. Richard Feynman is often quoted as saying that this is "a phenomenon which is impossible, absolutely impossible, to explain in any classical way" (Feynman, Leighton, & Sands, 1975, Section 37-1). The words "in any classical way" in this quote are commonly interpreted as "by means of CPT." This interpretation may be correct, but it is also possible that Feynman meant that this phenomenon cannot be explained by means of classical mechanics, and that he viewed quantum probabilities as a special-purpose theory for computing probabilities in a specific physical situation. The second quote from Feynman (1951) given at the end of our introductory section seems to agree with this interpretation.
Whatever the case with Feynman, Ballentine (1986) presents a systematic analysis of the double-slit experiment in terms of CPT, and argues that the two are perfectly compatible if one treats the probabilities in (10) as conditional ones, conditioned on three different events. Translating this into the language of random variables, Ballentine's solution is to rewrite (10) as the hypothetical equality  (12) should generally be expected. Dzhafarov and Kujala (2014b) call this approach "conditionalization," in relation to Avis, Fischer, Hilbert, and Khrennikov (2009) where it was used systematically (see also Khrennikov, 2006). It is true that if R and Q are jointly distributed, then R conditioned on some value of Q and R conditioned on another value of Q are two random variables that possess no joint distributions, i.e., are stochastically unrelated. This means that conditionalization is a special case of contextual labeling, in fact an instructive case for introducing the notion of stochastic unrelatedness (Dzhafarov & Kujala, 2014b, 2016b. However, the choice among conditions c •• , c •× , c ×• need not be random. One can conduct an experiment with both slits open for a year, then for another year with the left slit closed, and so on. This should not change anything in the analysis of the double-slit experiment. As we mentioned in the introductory section, Khrennikov (2009a) pointed out the difference between contextual and conditional probabilities in presenting his general Växjö contextual model.
The analysis of the double-slit experiment within the framework of the Contextuality-by-Default theory begins with identifying the random variables in play, their contexts and the properties they measure. The contexts are the same as in Ballentine's analysis, c •• , c •× , c ×• , but for completeness we add c ×× (both slits closed). The measured properties are identified by which slit one considers (left or right) and by its condition (open or closed): q •· (left slit, open), q ·• (right slit, open), and similarly for the closed slits, q ×· , q ·× . This creates eight random variables that we can arrange as follows: where R c q can be interpreted as answering the question "Has the particle just emitted reached the detector through the slit q under condition c?" The possible values of R c q are Yes and No. Thus, R ×• ×· =Yes means that a particle reached the detector having passed through the closed slit on the left when the right slit is open. Physics (not probability theory) tells us that the probability of this happening is zero.
Our arrangement of the random variables shows that, surprisingly, the system they comprise is formally a cyclic system of rank 4 (Dzhafarov & Kujala, 2016a;. It is the same system as the one in the simplest EPR/B "Alice-Bob" paradigm, described in our analysis of Statement 1. If one uses one's knowledge or assumptions that (a) no particle can reach the detector through a closed slit, and that (b) a particle cannot reach the detector through both slits, then the joint distributions of all context-sharing pairs of random variables (the rows of the matrix above) are as follows: The physical interpretation of the joint distribution for c •• compared to that for, say, c •× is that, somehow, the way particles reach the detector having passed through the open left slit may be different depending on whether the right slit is open or closed. A physicist may tell us that this is because of the particle-wave duality and wave interference, but this is irrelevant for the probabilistic analysis. It is interesting to see whether the system just described is C-noncontextual (has a C-coupling) with C="are equal with maximal possible probability." The application of the general criterion (5) for noncontextuality of such a system yields where we have assumed that the detector is so small that the probabilities 1 − 2p, 1 − 2q, 1 − 2p ′ − 2q ′ are all positive. By simple algebra one can show that this inequality is always satisfied, that is, the double-slit system is Cnoncontextual. 4 If one allows for the possibility that a particle in c •• passes through both slits simultaneously, the matrix of joint probabilities for this context changes: One can check that, continuing to assume that the probabilities p, q, p ′ , q ′ , r ′ are very small, this does not change the conclusion that the system is noncontextual.

CONCLUDING REMARKS
In this concluding section we will briefly address three commonly raised concerns about the contextual notation and the principle that random variables in different contexts are different (and stochastically unrelated).
Question: In empirical situations where the contexts are known not to influence a measurement directly (like in the EPR/B Alice-Bob paradigm with spacelike separation of the measurements), what "causes" the random variable representing this measurement to change its identity?
Answer: The identity of a random variable is determined by its own distribution and also by the joint distribution of this random variable with all other random variables in the same context. Therefore, any change in these other variables "automatically" changes its identity. Here is a simple analogy. A person P is in a room with other people. P has some characteristics, such as "she is kind," or "she is tall." It is possible that she is the tallest person in the room, in which case she is also characterized by this fact. The statement "she is the tallest person in the room" therefore describes a property of P , part of her identity in addition to her being kind and tall. If one of the other people leaves, and someone enters who is taller than P , she "automatically" changes her identity, as she ceases to be the tallest person in the room. This change in P occurs even if she is not aware of the change in the room, or the room is so large that there are no physical means for her to notice this.
Question: If every condition recorded together with a random variable can be considered part of its context, does this not mean that any two realizations of the same random variable are in fact two different random variables, stochastically unrelated to each other?
Answer: If the realizations are separately indexed, e.g., by the ordinal position in a sequence of trials, each of them indeed must be viewed as a single realization of a unique random variable. There is, however, a choice of the point of view for subsequent analysis. One can view these unique random variables as ones with different measured properties (trial numbers) within a single context (sequence of trial numbers). Conversely, one can view them as random variables measuring the same thing (e.g., they all measure the response of a person to a flash) but in different contexts (trial number). We implicitly adopt the second point of view when we speak of the sequence as one of different realizations of "the same" random variable. See Dzhafarov andKujala (2015, 2016a) for detailed discussions. In the case considered, the choice of one of the two points of view makes no difference. If the realizations are treated as context-sharing, the random variables are jointly distributed, but this joint distribution is manifested in a single realization only. We need additional assumptions to reconstruct it, such as stochastic independence, ergodicity, martingale property, etc. If the realizations are treated as measuring the same property in different contexts, they are pairwise stochastically unrelated, and we need to couple them. The choice of a coupling here amounts to adopting the same additional assumptions.
Question: Is the contextual labeling with stochastic unrelatedness really classical, in the Kolmogorovian sense?
Answer: It is a matter of definition and understanding of history. In some publications one of us and Janne Kujala called our approach a "qualified" Kolmogorovian theory (Dzhafarov & Kujala, 2014c), and it can be presented in a way that sets it aside from a standard account of CPT (as in Dzhafarov & Kujala, 2016a). However, we prefer to speak of Contextuality-by-Default as part of the Kolmogorovian probability theory, with emphasis on multiple freely introducible domain probability spaces, stochastically unrelated random variables defined on these spaces, and their couplings understood as placing their copies on the same domain space. This preference is based on our disbelief that Kolmogorov himself and the brilliant probabilists working in his language could have overlooked the obvious fact that there cannot exist a joint distribution of all imaginable random variables.