  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.
