I recently read Introduction to Causal Inference from a Machine Learning Perspective by Brady Neal for fun. I’m familar with applied causal inference techniques such as IV, DID, and RDD, but I’m not familiar with causal graphs. So, I decided to start from Neal’s lecture notes when I saw the helpful flowchart he made:
In chapter 6, Neal introduces Pearl’s do-calculus, which allows us to identify any causal estimand that is identifiable as long as we have the correct causal graph. The three rules of do-calculus are:
Without rigorous proofs, it can be difficult to understand why these three rules are complete. At first, they seemed quite arbitrary to me, even with Neal’s explanation. I couldn’t see the connections among them. But after thinking about it more, I was able to gain some intuition that really helped me understand and remember them.
The goal of do-calculus is identification - turning causal estimand into statistical estimand. And the way to do this is to rely on conditional independence. We know that association can be divided into causal association and non-causal association. Therefore, we consider three types of independence:
- no association
- no non-causal association
- no causal association.
The three rules correspond to these three types of independence, respectively.
Rule 1 says if there is no association between \(Y\) and \(Z\) conditioning on some intervened variables \(T\) and observed variables \(W\), then we can remove \(Z\).
Rule 2 says if there is no non-causal association between \(Y\) and \(Z\) conditioning on some intervened variables \(T\) and observed variables \(W\), then we can transform \(do(Z)\) into \(Z\) (because causal association is the only association left). Note that causal association must flow though the edges going out of \(Z\), and by removing those edges, we focus on the non-causal independence.
Rule 3 says if there is no causal association between \(Y\) and \(Z\) conditioning on some intervened variables \(T\) and observed variables \(W\), then we can remove \(do(Z)\). Because we want to get rid of non-causal association in the graph, we remove all edges flowing into \(Z\). But note that when removing \(do(Z)\) from the conditioning set, we are recovering the inflow edges of \(Z\). And if \(Z\) contains colliders that are ancestors of \(W\), then conditioning on \(W\) (i.e., conditioning on the descendants of a collider) would introduce new association that can affect \(Y\). We don’t have this problem in Rule 2’s situation because if some variable wants to associate with \(Y\) through a collider in \(Z\), then \(Z\) and \(Y\) must have non-causal association, which violates our assumption (the if clause in Rule 2).
Therefore, we need to divide \(Z\) into 2 categories: \(Z(W)\), nodes of \(Z\) that aren’t ancestors of \(W\), and the rest. For \(Z(W)\), we assume \(Z\) and \(Y\) have no causal association; for the rest, we assume they have no association, which is a stronger assumption.
I hope this helps give you a better understanding of the three rules of do-calculus.
Citation
@online{zhang2023,
author = {Jia Zhang},
title = {Intuition for {Do-Calculus}},
date = {2023-01-01},
url = {https://jiazhang42.github.io/mysite/blog/do-calculus/},
langid = {en}
}