# What Is a Causal Graph?
## 5 Augmented DAG  
We can now introduce formal graphical causal semantics, so moving onto the second rung of the ladder of causation. This is based on ECI represented by an _augmented DAG_, which is just like a regular DAG except that some of its nodes may represent non-stochastic variables, such as regime indicators. We may indicate a non-stochastic variable by a square node, and a stochastic variable by a round node; however, this distinction does not affect how we use the DAG. Just as certain collections of purely probabilistic conditional independence properties can usefully and precisely be represented (by moralisation) by means of a regular DAG, so we may be able to construct an augmented DAG to represent (by exactly the same moralisation criterion) causal properties of interest expressed in terms of ECI. Consider, for example, the simple augmented DAG  
\[F_{A}\to A\to B.\]  
This represents (by moralisation, as always) the ECI property of (5), and so is a graphical representation of the ignorability assumption. Note that it is the whole structure that, with ECI, imparts causal meaning to the arrow from \(A\) to \(B\): the direction of that arrow would not otherwise be meaningful in itself. **Example 2**. : _Consider the following augmented DAG modifications of the first three DAGs of Example 1 to allow for an intervention on \(A\):_1. \(F_{A}\to A\to B\to C\);
2. \(F_{A}\to A\leftarrow B\gets C\);
3. \(F_{A}\to A\gets B\to C\). We saw in Example 1 that, without the node \(F_{A}\), these DAGs are all Markov equivalent. With \(F_{A}\) included, we see that DAGs 2 and 3 are still Markov equivalent since they have the same skeleton and the same unique immorality \(F_{A}\to A\gets B\), but they are no longer Markov equivalent to DAG 1, which does not contain that immorality. All three DAGs represent the \(\text{ECI}\ A\perp\!\!\!\perp C\mid(F_{A},B)\), which says that, in any regime, \(A\) is independent of \(C\) given \(B\) (in fact, this only has bite for the idle regime since, in an interventional regime \(F_{A}=a\), \(A\) has a degenerate distribution at \(a\), so that the conditional independence is trivially satisfied); however, while DAGs 2 and 3 both represent the \(\text{ECI}\)\((B,C)\perp\!\!\!\perp F_{A}\), which implies that \(A\) does not cause either \(B\) or \(C\), DAG 1 instead represents the \(\text{ECI}\)\((B,C)\perp\!\!\!\perp F_{A}\mid A\), expressing the ignorability of the distribution of \((B,C)\) given \(A\). Note, moreover, that the Markov equivalence of DAGs 2 and 3 means that we can not interpret the direction of the arrow between \(B\) and \(C\) as meaningful in itself. In particular, in DAG 3, the arrow \(B\to C\) does not signify a causal effect of \(B\) on \(C\); in this approach, causality is only described by \(\text{ECI}\) properties and their moralisation-based graphical representations. **Example 3**. : _To endow the instrumental variable problem of Figure 4 with causal content--specifically, relating to the causal effect of \(X\) on \(Y\)--we might replace it with the augmented DAG of Figure 6, where the node \(F_{X}\) now allows us to consider an intervention on \(X\)._  
_This DAG still represents the probabilistic conditional independencies (1) and (2) in any regime. But now it additionally represents genuine causal properties:_  
\[(U,Z) \perp\!\!\!\perp F_{X} \tag{6}\] \[Y \perp\!\!\!\perp F_{X}\mid(Z,U,X) \tag{7}\]  
_Property (6) says that \(X\) has no causal effect on \(U\) and \(Z\), these having the same joint distribution in all regimes (including the idle regime). Property (7) entails the modular conditional ignorability property that the distribution of \(Y\) given \((z,u,x)\) (which, in fact, depends only on \((u,x)\), from (1)) is the same, both in the interventional regime, where \(X\) is set to \(x\), and in the observational regime, where \(X\) is not controlled.