On universality of classical probability with contextually labeled random variables

Abstract

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.

Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables

“…Nevertheless, as was shown by Khrennikov and coauthors [ 30 , 31 ] and by Dzhafarov and coauthors [ 32 , 33 , 34 , 35 , 36 , 37 , 38 ], the Bohm–Bell type experiments can be modeled with the aid of the CP-representation of quantum observables. However, such CP-models are not so straightforward as In this paper, we present a general CP-model based on the conditional probability scheme which was explored in [ 31 ] (in the very concrete situation).…”

True contextuality in a psychophysical experiment

“…However, this is not considered a "direct influence," because in the given system of random variables these unnoticeable changes do not carry information. If one can actually observe the changes R 3 1 → −R 3 1 , the system of random variables one deals with changes dramatically, and the CbD analysis then changes accordingly (Dzhafarov, Cervantes, & Kujala, 2017;Dzhafarov & Kon, 2018;Dzhafarov & Kujala, 2018). in quantum physics by Bell (1964Bell ( , 1966 and Kochen and Specker (1967). They demonstrated that one could meaningfully address, using only observable measurements, the question famously discussed in Bohr's (1935) critique of Einstein, Podolsky, and Rosen (1935).…”

Sciences of Observation

“…If Alice sees smoke coming out of a system to which S is connected, it will affect her report on the pointer state of S. To return to the language of physics, it will affect the probability distribution over her possible reports, by assigning high probabilities to reports that had low or zero probability before. Dzhafarov has termed this ubiquitous effect of context on response probabilities "contextuality by default" [100,101]. Within this framework, the ideal of a fixed probability distribution on outcomes, impervious to the modulatory effects of other observations, becomes a minor special case.…”