Matrix representations and independencies in directed acyclic graphs

TL;DR

This paper introduces a novel matrix-based approach to determine conditional independencies in directed acyclic graphs, providing an alternative to traditional path-based separation criteria.

Contribution

It presents a new matrix representation method that offers an equivalent path criterion for separation, expanding the tools for analyzing DAGs.

Findings

Matrix condition equivalent to existing path criteria

New path criterion for separation based on matrices

Provides a different perspective on DAG independencies

Abstract

For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types of path in graphs. They are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense. We introduce and discuss an alternative approach using binary matrix representations of graphs in which zeros indicate independence statements. A matrix condition is shown to give a new path criterion for separation and to be equivalent to each of the previous two path criteria.