Minimal Trails in Restricted DAGs

TL;DR

This work addresses the problem of characterizing minimal trails activated by a subset $Z$ in directed acyclic graphs that exclude active cycles, with applications to copula-based Bayesian networks. The authors define TRAILS$(X,Y|Z)$, the converging-connection set ConvCon$(T)$, closest descendants, subtrails, and a partial order $<_{TRAIL}$ to compare trails, proving structural properties of minimal trails under the no-active-cycle constraint. A key contribution is showing how converging nodes interact with the activation set $Z$, especially when $Y 1d Z$ has local relationships, leading to precise subgraph configurations and d-separation implications. These results yield a framework for efficient conditional independence reasoning in constrained DAGs and PCBNs, potentially aiding computations in copula-based BN models.

Abstract

In this paper, the properties of minimal trails in a directed acyclic graph that is restricted not to contain an active cycle are studied. We are motivated by an application of the results in the copula-based Bayesian Network model developed recently. We propose a partial order on the set of trails activated by a certain subset of nodes, and show that every minimal trail, according to such an order, has a simple structure.

Figures (17)

• Figure 1: Directed acyclic graph with seven nodes.
• Figure 2: Active cycle, where $\rightleftharpoons$ represents arcs that form only diverging or serial connections.
• Figure 3: A graph containing an active cycle.
• Figure 4: Subgraph in ${G}$ with the active cycle colored in red.
• Figure 5: Subgraph of $G$ with common descendant.

Theorems & Definitions (39)

• Definition 2.1: Active cycle
• Definition 2.2: d-separation
• Definition 3.1
• Definition 3.2
• Definition 3.3: Closest descendant
• Definition 3.4: Subtrails
• Definition 3.5: Smaller trail
• Remark 3.1
• Lemma 4.1
• proof