Compatible priors for model selection of high-dimensional Gaussian DAGs

Graphical models represent a powerful framework to incorporate conditional independence structure for the statistical analysis of high-dimensional data. In this paper we focus on Directed Acyclic Graphs (DAGs). In the Gaussian setting, a prior recently introduced for the parameters associated to the (modified) Cholesky decomposition of the precision matrix is the DAG-Wishart. The flexibility introduced through a rich choice of shape hyperparameters coupled with conjugacy are two desirable assets of this prior which are especially welcome for estimation and prediction. In this paper we look at the DAG-Wishart prior from the perspective of model selection, with special reference to its consistency properties in high dimensional settings. We show that Bayes factor consistency only holds when comparing two DAGs which do not belong to the same Markov equivalence class, equivalently they encode distinct conditional independencies; a similar result holds for posterior ratio consistency. We also prove that DAG-Wishart distributions with arbitrarily chosen hyperparameters will lead to incompatible priors for model selection, because they assign different marginal likelihoods to Markov equivalent graphs. To overcome this difficulty, we propose a constructive method to specify DAG-Wishart priors whose suitably constrained shape hyperparameters ensure compatibility for DAG model selection.