Multiple-context event spaces and distributions: A new framework for Bell's theorems

• Stochastic Bell's theorems are equivalent to deterministic versions for all cases.
• Generalization of parameter independence is ambiguous for three or more observers.
• Parameter independence is necessary for satisfaction of all Bell inequalities.

I describe a new framework for the articulation and analysis of Bell's theorems for arbitrarily complicated discrete physical scenarios. The framework allows for efficient proof of some new results, as well as generalizations of some older results already known for simpler cases. The generalized known results are: (1) satisfaction of all Bell inequalities is equivalent to the existence of a joint probability function for all possible measurement contexts and (2) stochastic versions of Bell's theorem are not stronger than deterministic versions. The new results are: (1) the straightforward generalization of parameter independence is inadequate when there are more than two observers, (2) parameter independence is a necessary condition for observable probabilities to satisfy the Bell inequalities, and (3) if the primary states can appear in isolation, and the observable probabilities satisfy all Bell inequalities, then parameter independence is a necessary condition for the primary states, too.