Bayesian networks (BNs) are probabilistic graphical models which concisely represent the dependence structure between discrete variables through a directed acyclic graph (DAG) [29, 41]. Any conditional independence between variables embedded in the model can be directly read from the underlying DAG through the so-called D-separation criterion [30]. However, in practical applications, it has been found that often the symmetric assumption of conditional independence is too restrictive and models graphically depicting asymmetric independence are needed. Various notions of asymmetric conditional independence have been since defined, including context-specific [4], partial [32] and local [7], and formal studies of their properties appeared [11, 12, 36, 43]. Although extensions of BNs embedding and representing asymmetric conditional independence have been defined [16, 22, 31, 33, 42], they often lose the intuitiveness associated to DAGs and no software is available for their use in practice.

Bayesian networks are a widely-used class of probabilistic graphical models capable of representing symmetric conditional independence between variables of interest using the topology of the underlying graph. They can be seen as a special case of the much more general class of models called staged trees, which can represent any type of non-symmetric conditional independence. Here we formalize the relationship between these two models and introduce a minimal Bayesian network representation of the staged tree, which can be used to read conditional independences in an intuitive way. Furthermore, we define a new labeled graph, termed asymmetry-labeled directed acyclic graph, whose edges are labeled to denote the type of dependence existing between any two random variables. Various datasets are used to illustrate the methodology, highlighting the need to construct models which more flexibly encode and represent non-symmetric structures.