  Testing whether a probability distribution is compatible with a given
Bayesian network is a fundamental task in the field of causal inference, where
Bayesian networks model causal relations. Here we consider the class of causal
structures where all correlations between observed quantities are solely due to
the influence from latent variables. We show that each model of this type
imposes a certain signature on the observable covariance matrix in terms of a
particular decomposition into positive semidefinite components. This signature,
and thus the underlying hypothetical latent structure, can be tested in a
computationally efficient manner via semidefinite programming. This stands in
stark contrast with the algebraic geometric tools required if the full
observable probability distribution is taken into account. The semidefinite
test is compared with tests based on entropic inequalities.
