# What Is a Causal Graph?
## 4 Interventional Causality and Extended Conditional Independence  
Our approach to this begins by introducing an explicit, non-graphical understanding of causality, expressed in terms of the probabilistic response of a system to an (actual or proposed) intervention. A causal effect of \(A\) on \(B\) exists if the distribution of \(B\), after an external intervention sets \(A\) to some value \(a\), varies with \(a\). Formally, we introduce a non-stochastic variable \(F_{A}\), having the same set of values as \(A\), such that \(F_{A}=a\) describes the regime in which \(A\) is set to value \(a\) by an external intervention (here, we shall only consider \"surgical interventions\", such that \(F_{A}=a\Rightarrow A=a\)). Then, \(A\) has _no_ causal effect on \(B\) just when the distribution of \(B\), given \(F_{A}=a\), does not depend on \(a\).  
If \(F_{A}\) were a stochastic variable, this would just be the usual property of independence of \(B\) from \(F_{A}\), notated as  
\[B\perp\!\!\!\perp F_{A}. \tag{3}\]  
Now, not only does this understanding of independence remain intuitively meaningful in the current case that \(F_{A}\) is a non-stochastic variable, but the formal mathematical properties of independence and conditional independence can be rigorously extended to such cases [14; 23; 28]; we term this _extended conditional independence (ECI)_. The standard axiomsof conditional independence apply, with essentially no changes, to ECI. (In ECI, we interpret \(A\perp\!\!\!\perp B\mid C\) as the property that the conditional distribution of \(A\), given \(B=b\) and \(C=c\), depends only on \(c\). Here, we can allow \(B\) and \(C\) to include non-stochastic variables; however, \(A\) must be fully stochastic.) Since much of the enterprise of statistical causality involves drawing causal conclusions from observational data, we further endow \(F_{A}\) with an additional state \(\emptyset\), read as \"idle\": \(F_{A}=\emptyset\) denotes the regime in which no intervention is applied to \(A\), but it is allowed to develop \"naturally\" in the observational setting. The distinction between \(A\) and \(F_{A}\) is that the state of \(A\) describes _what_ value was taken, while the state of \(F_{A}\) describes _how_ that value was taken.  
We regard the main thrust of causal statistical inference as aiming to deduce properties of a hypothetical interventional regime, such as the distribution of \(B\) given \(F_{A}=a\), from observational data, obtained under the idle regime \(F_{A}=\emptyset\). But since there is no logically necessary connexion between the distributions under different regimes, suitable assumptions will need to be made--and justified--to introduce such connexions.  
The simplest such assumption--which, however, will only very rarely be justifiable--is that when we consider the distribution of \(B\) given \(A\), we need to know _what_ was the value \(a\) that \(A\) took but not _how_ it came to take that value (i.e., not whether this was in the observational regime \(F_{A}=\emptyset\) or in the interventional regimes \(F_{A}=a\)), the conditional distribution of \(B\) being the same in both cases. That is,  
\[p(B=b\mid A=a,F_{A}=\emptyset)=p(B=b\mid A=a,F_{A}=a). \tag{4}\]  
This is the property of _ignorability_. When this strong property holds, we can directly take the observed distribution of \(B\) given \(A=a\) for the desired interventional distribution of \(B\) given \(F_{A}=a\): the distribution of \(B\) given \(A\) is a \"modular component\", transferable between different regimes.  
We note that (4) can be expressed succinctly using ECI notation as  
\[B\perp\!\!\!\perp F_{A}\mid A. \tag{5}\]  
As exemplified by (3) and (5), ECI supplies a simple and powerful language for expressing and manipulating causal concepts.