The back-door criterion states that
A set of variables $X$ satisfies the backdoor criterion relative to $(A, Y)$ in a DAG if
My question is why do we need the first condition that $X$ does not contain a descendant of $A$?
A variable that is a descendant of $A$ is a potential mediator; therefore conditioning on it risks blocking part of the causal effect of interest. Descendants of $A$ may also be colliders (e.g. if the descendant (let's call it $D$) of $A$ shares an uncontrolled common cause with $Y,$ then conditioning on $D$ opens the non-causal path $A\to D \leftarrow U\to Y.$
Hope that helps.
The first condition of the backdoor criterion is necessary to make sure that the relationship between $A$ and $Y$ is genuine and thus is not confounded by other variables.
In causal inference, the goal is to estimate the causal effect of a treatment or exposure variable $A$ on an outcome variable $Y$ by accounting for other variables that may confound the relationship. Confounding occurs when there are common causes of both the treatment and the outcome, leading to an apparent association between them that is not truly causal.
When $X$ contains a descendant of $A$, it means that there is a path from $A$ to $X$ that passes through other variables. In this case, the relation between $A$ and $Y$ may be confounded by these intermediate variables. Including such variables in $X$ would introduce bias in estimating the causal effect of $A$ on $Y$.
By excluding descendants of $A$ from $X$, we ensure that there are no intermediate variables that confound the causal relationship between $A$ and $Y$. This condition helps isolate the effect of $A$ on $Y$ and allows for a more accurate estimation of the causal effect using the backdoor adjustment or control for confounding variables.
Thus, in summary, the first condition of the backdoor criterion is necessary to prevent confounding and ensure that $X$ captures only variables that are not influenced by $A$, allowing for unbiased estimation of the causal effect of $A$ on $Y$.
