  Many widely studied graphical models with latent variables lead to nontrivial
constraints on the distribution of the observed variables. Inspired by the Bell
inequalities in quantum mechanics, we refer to any linear inequality whose
violation rules out some latent variable model as a "hidden variable test" for
that model. Our main contribution is to introduce a sequence of relaxations
which provides progressively tighter hidden variable tests. We demonstrate
applicability to mixtures of sequences of i.i.d. variables, Bell inequalities,
and homophily models in social networks. For the last, we demonstrate that our
method provides a test that is able to rule out latent homophily as the sole
explanation for correlations on a real social network that are known to be due
to influence.
