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In a causal graph (DAG), $A\to B$ means $A$ causes $B.$ Even correlation can be defined with causal relationships (for example, maybe $A$ is correlated with $B$ because $C$ causes both $A$ and $B$).

However, how are eligibility criteria defined?

For example, $A$ can take a loan only if he/she is more than $23$ years old. How should I connect Age with Loan in a causal graph? $\operatorname{Age}\to\operatorname{Loan}$ seems wrong, as age does not cause taking a loan, it just allows it.

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It would still be $\operatorname{Age}\to\operatorname{Loan}.$ The exact functional relationship between Age and Loan is not specified in the DAG but in the Structural Causal Model (SCM) - the equations that show how the variables are related to each other. The DAG merely indicates that there is a causal relationship, and the direction of that relationship. While you can normally infer the DAG from the SCM, the reverse is certainly NOT the case.

You could use logical operators in the SCM to show the eligibility criterion you have, perhaps something implicit like the following: $$\operatorname{Loan}\implies (\operatorname{Age}>23).$$ Here the $\implies$ symbol means logical "implies." If you don't like the fact that Age is on the RHS, you could put the two variables on a somewhat more equal footing by writing this instead: $$(\operatorname{Age}\le 23)\lor\operatorname{Loan}=\operatorname{True}.$$ Here the symbol $\lor$ means "inclusive OR."

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  • $\begingroup$ Thanks for the answer. Where can I learn more about Structural Causal Model? Is there a specific chapter of one of Pearl's books? $\endgroup$
    – pentavol
    May 30, 2021 at 20:17
  • $\begingroup$ Absolutely! I would recommend Causal Inference in Statistics: A Primer, as a great intro.If you're exceptionally brave (braver than me, in fact), you could also try Causality: Reasoning, Models, and Inference, but that is an exceedingly difficult graduate-level text. $\endgroup$ May 30, 2021 at 21:05

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