  A novel approach for analyzing "classical" alternatives to quantum mechanics
for explaining the statistical results of an EPRB-like experiment is proposed.
This perspective is top-down instead of bottom-up. Rather than beginning with
an inequality derivation, a hierarchy of model types is constructed, each
distinguished by appropriately parameterized conditional probabilities. This
hierarchy ranks the "classical" model types in terms of their ability to
reproduce QM statistics or not. The analysis goes beyond the usual
consideration of model types that "fall short" (i.e., satisfy all of the CHSH
inequalities) to ones that are "excessive" (i.e., not only violate CHSH but
even exceed a Tsirelson bound). This approach clearly shows that
noncontextuality is the most general property of an operational model that
blocks replication of at least some QM statistical predictions. Factorizability
is naturally revealed to be a special case of noncontextuality. The same is
true for the combination of remote context independence and outcome determinism
(RCI+OD). It is noncontextuality that determines the dividing line between
"classical" model instances that satisfy the CHSH inequalities and those that
don't. Outcome deterministic operational models are revealed to be the
"building blocks" of all the rest, including quantum mechanical, noncontextual,
and contextual ones. The set of noncontextual model instances is exactly the
convex hull of all 16 RCI+OD model instances, and furthermore, the set of all
model instances, including all QM ones, is equal to the convex hull of the 256
OD model instances. It is shown that, under a mild assumption, the construction
of convex hulls of finite ensembles of OD model instances is (mathematically)
equivalent to the traditional hidden variables approach. Plots and figures
provide visual affirmation of many of the results.
