We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability theory. In the contextualization approach each random variable is "automatically" labeled by all conditions under which it is recorded, and the random variables across a set of mutually exclusive conditions are probabilistically coupled (imposed a joint distribution upon). Analysis of all possible probabilistic couplings for a given set of random variables allows one to characterize various relations between their separate distributions (such as Bell-type inequalities or quantum-mechanical constraints). In the conditionalization approach one considers the conditions under which the random variables are recorded as if they were values of another random variable, so that the observed distributions are interpreted as conditional ones. This approach is uninformative with respect to relations between the distributions observed under different conditions because any set of such distributions is compatible with any distribution assigned to the conditions.
We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability theory. In the contextualization approach each random variable is "automatically" labeled by all conditions under which it is recorded, and the random variables across a set of mutually exclusive conditions are probabilistically coupled (imposed a joint distribution upon). Analysis of all possible probabilistic couplings for a given set of random variables allows one to characterize various relations between their separate distributions (such as Bell-type inequalities or quantum-mechanical constraints). In the conditionalization approach one considers the conditions under which the random variables are recorded as if they were values of another random variable, so that the observed distributions are interpreted as conditional ones. This approach is uninformative with respect to relations between the distributions observed under different conditions because any set of such distributions is compatible with any distribution assigned to the conditions.
From behavioral sciences to biology to quantum mechanics, one encounters situations where (i) a system outputs several random variables in response to several inputs, (ii) for each of these responses only some of the inputs may "directly" influence them, but (iii) other inputs provide a "context" for this response by influencing its probabilistic relations to other responses. These contextual influences are very different, say, in classical kinetic theory and in the entanglement paradigm of quantum mechanics, which are traditionally interpreted as representing different forms of physical determinism. One can mathematically construct systems with other types of contextuality, whether or not empirically realizable: those that form special cases of the classical type, those that fall between the classical and quantum ones, and those that violate the quantum type. We show how one can quantify and classify all logically possible contextual influences by studying various sets of probabilistic couplings, i.e., sets of joint distributions imposed on random outputs recorded at different (mutually incompatible) values of inputs.
Contextual situations are those in which seemingly ‘the same’ random variable changes its identity depending on the conditions under which it is recorded. Such a change of identity is observed whenever the assumption that the variable is one and the same under different conditions leads to contradictions when one considers its joint distribution with other random variables (this is the essence of all Bell-type theorems). In our Contextuality-by-Default approach, instead of asking why or how the conditions force ‘one and the same’ random variable to change ‘its’ identity, any two random variables recorded under different conditions are considered different ‘automatically.’ They are never the same, nor are they jointly distributed, but one can always impose on them a joint distribution (probabilistic coupling). The special situations when there is a coupling in which these random variables are equal with probability 1 are considered noncontextual. Contextuality means that such couplings do not exist. We argue that the determination of the identity of random variables by conditions under which they are recorded is not a causal relationship and cannot violate laws of physics.
Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach ("contextuality-by-default") is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide.
We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.
We describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central notion is that of all possible couplings for stochastically unrelated outputs indexed by mutually incompatible values of inputs. A system is characterized by a pattern of which outputs can be "directly influenced" by which inputs (a primitive relation, hypothetical or normative), and by certain constraints imposed on the outputs (such as Bell-type inequalities or their quantum analogues). The set of couplings compatible with these constraints represents a form of contextuality in the dependence of outputs on inputs with respect to the declared pattern of direct influences.
Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.