Decomposable context-specific models

TL;DR

It is proved that the moralization operation applied to the graphical representation of a context-specific model does not affect the implied independence relations, thus affirming that these models are algebraically described by a finite collection of decomposable graphical models.

Abstract

We introduce a family of discrete context-specific models, which we call decomposable. We construct this family from the subclass of staged tree models known as CStree models. We give an algebraic and combinatorial characterization of all context-specific independence relations that hold in a decomposable context-specific model, which yields a Markov basis. We prove that the moralization operation applied to the graphical representation of a context-specific model does not affect the implied independence relations, thus affirming that these models are algebraically described by a finite collection of decomposable graphical models. More generally, we establish that several algebraic, combinatorial, and geometric properties of decomposable context-specific models generalize those of decomposable graphical models to the context-specific setting.

Figures (7)

• Figure 1: A CStree for $p=3$ and its minimal context DAGs.
• Figure 2: Staged tree that is not a CStree.
• Figure 3: All CStrees with $p=3$ and variable ordering $123$ which do not represent a DAG.
• Figure 4: A balanced CStree with a non-perfect minimal context.
• Figure 5: The context subtree of the tree in Figure \ref{['fig:motivatingEx']} for the context $X_3=0$, and its minimal context DAGs.

Theorems & Definitions (64)

• Theorem 1.1: Context-specific Hammersley-Clifford
• Theorem 1.2
• Theorem 1.3
• Example 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Example 2.5
• Remark 2.6
• Lemma 2.7