 Let \(V_i\) and \(V_j\) be two distinct variables in \(\bV\), with corresponding exogenous error variables \(U_i\) and \(U_j\), representing causes of \(V_i\) and \(V_j\) that are excluded from the \(\bV\). Suppose \(V_i\) and \(V_j\) share at least one common cause that is excluded from \(\bV\). In this case, we would not expect \(U_i\) and \(U_j\) to be probabilistically independent, and the theorem of Pearl and Verma (1991) would not apply. In this case, the causal relationship among the variables in \(\bV\) would not be appropriately represented by a DAG, but would require an acyclic directed mixed graph (ADMG) with a double-headed arrow connecting \(V_i\) and \(V_j\). We will discuss this kind of case in more detail in  Section 4.6  below.