A Generalization of the Tetrad Representation Theorem
Glenn R. Shafer, Alex Kogan · 2002
The tetrad representation theorem, due to Spirtes, Glymour, and Scheines (1993), gives a graphical condition necessary and sufficient for the vanishing of tetrad differences in a linear correlation structure. This note simplifies their proof and generalizes the theorem. In order to make the ideas as accessible as possible to mathematicians who might develop them further, we begin with a thorough exposition of their purely graphtheoretical aspects. Part I. Treks and Choke Points in a Directed Acyclic Graph We assume that the reader is familiar with the most basic definitions of graph theory. Recall that a graph is an object consisting of nodes and edges between them. It is directed if its edges are directed (marked with arrows). We assume that we are working with a finite directed graph, in which edges are always between distinct nodes and there is at most one edge between any pair of distinct nodes. We use the usual definitions of parent, child, descendant, and ancestor; if there is an edge between X and Y with its arrowing pointing from X to Y, we say that X is a parent of Y and Y is a child of X. We call a node exogenous if it has no parents, endogeneous if it does have parents, and barren