Contextuality and hidden variable models

Contextuality in systems of random variables has been originally formulated in quantum physics in terms of hidden variable models (HVMs). These formulations are contingent on the systems of random variables being consistently connected, which means that any two variables answering the same question in different contexts have the same distribution. Outside quantum physics, and specifically in systems of random variables describing behavioral phenomena, consistent connectedness is virtually never observed. This has necessitated, in the last decade, an extension of the notion of contextuality to arbitrary systems of random variables, resulting in the theory called Contextuality-by-Default. However, for inconsistently connected systems the possibility of interpreting contextuality in terms of HVMs is lost, and this is considered by some a major problem. It can be shown, however, that any inconsistently connected system can be recast as a consistently connected one, so that the two systems describe precisely the same empirical or theoretical situations, and they are contextual or noncontextual together. The consistently connected rendering of a system is amenable to formulation in terms of HVMs. The similarities of this formulation with and differences from the HVM representations of the traditionally considered quantum-mechanical systems elucidate the subtle interplay of the mathematical and the empirical in describing phenomena by systems of random variables. [1] Kujala, J.V., Dzhafarov, E.N., & Larsson, J.-A. (2015). Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems. Physical Review Letters 115, 150401. [2] 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. [3] Dzhafarov, E.N. (2022). Contents, contexts, and basics of contextuality. In Shyam Wuppuluri and Ian Stewart (Eds). From Electrons to Elephants and Elections, The Frontiers Collection. pp. 259-286. Cham, Switzerland: Springer.