On Parameter Priors for Discrete DAG Models

We investigate parameter priors for discrete DAG models. It was shown in previous works that a Dirichlet prior on the parameters of a discrete DAG model is inevitable assuming global and local parameter independence for all possible complete DAG structures. A similar result for Gaussian DAG models hinted that the assumption of local independence may be redundant. Herein, we prove that the local independence assumption is necessary in order to dictate a Dirichlet prior on the parameters of a discrete DAG model. We explicate the minimal set of assumptions needed to dictate a Dirichlet prior, and we derive the functional form of prior distributions that arise under the global independence assumption alone.