We introduce a novel class of labeled directed acyclic graph (LDAG) modelsfor finite sets of discrete variables. LDAGs generalize earlier proposals forallowing local structures in the conditional probability distribution of anode, such that unrestricted label sets determine which edges can be deletedfrom the underlying directed acyclic graph (DAG) for a given context. Severalproperties of these models are derived, including a generalization of theconcept of Markov equivalence classes. Efficient Bayesian learning of LDAGs isenabled by introducing an LDAG-based factorization of the Dirichlet prior forthe model parameters, such that the marginal likelihood can be calculatedanalytically. In addition, we develop a novel prior distribution for the modelstructures that can appropriately penalize a model for its labeling complexity.A non-reversible Markov chain Monte Carlo algorithm combined with a greedy hillclimbing approach is used for illustrating the useful properties of LDAG modelsfor both real and synthetic data sets.
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