Existence of hidden variables having only upper probabilities

Patrick Suppes, Mario Zanotti

Part I. Foundations of Probability: 1. Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualititative probability ordering 2. Necessary and sufficient qualitative axioms for conditional probability 3. On using random relations to generate upper and lower probabilities 4. Conditions on upper and lower probabilities to imply probabilities 5. Qualitative axioms for random-variable representation of extensive quantifiers Part II. Causality and Quantum Mechanics: 6. Stochastic incompleteness of quantum mechanics 7. On the determinism of hidden variable theories with strict correlation and conditional statistical independence of observables 8. A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distribution 9. When are probabilistic explanations possible? 10. Causality and symmetry 11. New Bell-type inequalities for N>4 necessary for existence of a hidden variable 12. Existence of hidden variables having only upper probabilities Part III. Applications in Education: 13. Mastery learning of elementary mathematics: theory and data.

Miklós Rédei

Eugene Paul Wigner

Patrick Suppes, Mario Zanotti

Great attention has been given to the exact formulation of locality conditions in the existence of hidden variables for quantum mechanical configurations that do not satisfy Bell's inequalities. In contrast, almost no attention has been given to examining what happens if the requirements for a probability measure are relaxed. The purpose of this paper is to show that local hidden variables do exist if the requirement of a probability measure is weakened to that of having only an upper probability measure. In the past several decades, upper and lower measures have been brought into the theory of statistical inference and the measurement of beliefs rather extensively, but in almost all cases considered in statistical theory or the belief framework, it is assumed that a probability measure exists, or in many cases, that an entire family of probability measures is given such that the sup and inf of the family defines the upper and lower measures. Sometimes still more is required, namely, that the upper–lower measure be a capacity (in the sense of Choquet) of infinite order. This is, for example, true of Dempster's theory of statistical inference. For an extensive recent analysis of the various theories of imprecise statistical models or belief functions using upper and lower probabilities, Walley (1991) is an excellent reference. In contrast and as is well known, in standard quantum mechanical cases the Bell inequalities cannot be satisfied and this implies that there exists no probability measure.

Eugene Paul Wigner

J. Acácio de Barros, Patrick Suppes

In this paper we analyze the existence of joint probabilities for the Bell-type and GHZ entangled states. We then propose the usage of nonmonotonic upper probabilities as a tool to derive consistent joint upper probabilities for the contextual hidden variables. Finally, we show that for the extreme example of no error, the GHZ state allows for the definition of a joint upper probability that is consistent with the strong correlations.