Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by $e$-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy; e.g. at the extreme case, a variable with entropy $0$ can convey no information. We show how these constraints can be used to learn about the true causal model from an observed data distribution. In addition, we propose a measure of causal influence called the minimal mediary entropy, and demonstrate that it can augment traditional measures such as the average causal effect.