# What Is a Causal Graph?
## 7 Causes of Effects  
So far we have only considered modelling and analysing problems relating to the \"effects of causes (EoC)\", where we consider the downstream consequences of an intervention. An entirely different class of causal questions relates to \"causes of effects (CoE)\" [17], where, in an individual case, both a putative causal variable \(X\) and a possible consequence \(Y\) of it have been observed, and we want to investigate whether \(X\) was indeed the cause of \(Y\). Such problems arise in the legal context, for example, when an individual sues a pharmaceutical company claiming that it was because she took the company's drug that she developed a harmful condition. At the simplest level, we may only have information about the joint distribution of \((X,Y)\) and their values, \(x\) and \(y\), for the case at hand. But this is not enough to address the CoE question, which refers, not to unknown facts or variables, but to an unknown relationship: was it causal? Trying to understand this question takes us into new territory: the third rung of the ladder of causation. Although by no means totally satisfactory, this question is most commonly understood as relating to a counterfactual situation. Suppose both \(X\) and \(Y\) are binary, and we have observed \(X=Y=1\). We might express the \"event of causation\" as follows:  
_The outcome variable would have been different (i.e., \(Y=0\)) if the causal variable had been different (i.e., \(X=0\))._  
But the hypothesis here, \(X=0\), contradicts the known fact that \(X=1\); it is _counterfactual_. Since in no case can we observe both \(X=1\) and \(X=0\)--what has been called \"the fundamental problem of causal inference\" [29]--it would seem impossible to address this question, at any rate on the basis of purely factual knowledge of the joint distribution of \((X,Y)\). So, a more complicated framework is required, necessitating more complex assumptions and analyses [30]. One approach builds on the idea of \"potential responses\", popularised by [PERSON] [31; 32], which were initially proposed for addressing EoC questions--which, as we have seen, can progress perfectly well without them. They do, however, seem essential for formulating counterfactual questions. For the simple example above, we duplicate the response \(Y\), replacing it by the pair \((Y_{0},Y_{1})\), with \(Y_{x}\) conceived of as a potential response that would be realised if in fact \(X=x\). Then, the _probability of causation_ (PC) can be defined as the conditional probability, given the data, of the event of causation:  
\[\text{PC}=\text{P}(Y_{0}=0\ |\ X=1,Y_{1}=1). \tag{8}\]  
There is, however, a difficulty: on account of the fundamental problem of causal inference, no data of any kind could ever identify the joint distribution of \((Y_{0},Y_{1})\), so PC is not estimable. It turns out that data supplying the distribution of the observable variables \((X,Y)\) can be used to set interval bounds on PC, and these bounds can sometimes be refined if we can collect data on additional variables [13; 33], but only in very special cases can we obtain a point value for PC. _Graphical Representation_  
Because PC cannot be defined in terms of observable domain variables, we cannot represent a CoE problem by means of a regular or augmented DAG on these variables. It is here that the expanded SCM version appears to come into its own. Thus, consider the simple SCM of Figure 11.  
In the observational regime, we have \(X=f_{X}(E_{X})\), \(Y=f_{Y}(X,E_{Y})\), and \(E_{Y}\perp\!\!\!\perp X\). The deterministic structure of a SCM supports the introduction of potential responses (something that can not be performed with a purely probabilistic DAG): \(Y_{x}=f_{Y}(x,E_{Y})\). Indeed, we could replace \(E_{Y}\) with the pair \(Y:=(Y_{0},Y_{1})\), and the function \(F_{Y}\) with the \"look-up\" function \(L_{Y}\), where \(L_{Y}(x,Y)=Y_{x}\). Then, \(X,Y_{0}\), and \(Y_{1}\) are all functions of \((E_{X},E_{Y})\), and thus have a joint distribution entirely determined by that of \((E_{X},E_{Y})\).