  Specification of the strongest possible Bell inequalities for arbitrarily
complicated physical scenarios -- any number of observers choosing between any
number of observables with any number of possible outcomes -- is currently an
open problem. Here I provide a new set of tools, which I refer to as
"multideviations", for finding and analyzing these inequalities for the fully
general case. In Part I, I introduce the multideviation framework and then use
it to prove an important theorem: the Bell distributions can be generated from
the set of joint distributions over all observables by deeming specific degrees
of freedom unobservable. In Part II, I show how the theorem provides a new
method for finding tight Bell inequalities. I then specify a set of new tight
Bell inequalities for arbitrary event spaces -- the "even/odd" inequalities --
which have a straightforward interpretation when expressed in terms of
multideviations. The even/odd inequalities concern degrees of freedom that are
independent of those involved in parameter independence, raising the
possibility of a new Bell's theorem with stronger philosophical implications.
Also, contrary to expectations, the violation of the inequalities by quantum
mechanics increases in size with the number of systems.
