  This paper considers inference of causal structure in a class of graphical
models called "conditional DAGs". These are directed acyclic graph (DAG) models
with two kinds of variables, primary and secondary. The secondary variables are
used to aid in estimation of causal relationships between the primary
variables. We give causal semantics for this model class and prove that, under
certain assumptions, the direction of causal influence is identifiable from the
joint observational distribution of the primary and secondary variables. A
score-based approach is developed for estimation of causal structure using
these models and consistency results are established. Empirical results
demonstrate gains compared with formulations that treat all variables on an
equal footing, or that ignore secondary variables. The methodology is motivated
by applications in molecular biology and is illustrated here using simulated
data and in an analysis of proteomic data from the Cancer Genome Atlas.
