A new characterization of discrete decomposable models

Decomposable graphical models, also known as perfect DAG models, play a fundamental role in standard approaches to probabilistic inference via graph representations in modern machine learning and statistics. However, such models are limited by the assumption that the data-generating distribution does not entail strictly context-specific conditional independence relations. The family of staged tree models generalizes DAG models so as to accommodate context-specific knowledge. We provide a new characterization of perfect discrete DAG models in terms of their staged tree representations. This characterization identifies the family of balanced staged trees as the natural generalization of discrete decomposable models to the context-specific setting.