# Are the two do-calculus systems equivalent?

The version of the do-calculus system in The Book of Why by Judea Pearl is a little bit simpler than the original version. Are the two systems equivalent?

The simple version is as follows:

• Rule1 Ignoring observations: If $$W$$ blocks all paths from $$Z$$ to $$Y$$ after we have deleted all arrows leading into $$X$$, then $$P(y|\operatorname{do}(x),z,w)=P(y|\operatorname{do}(x),w)\quad\mbox{ if } (Y\perp Z\mid X,W)_{G_{\overline{X}}}$$

• Rule2 Action/observation exchange: If $$Z$$ blocks all back-door paths from $$X$$ to $$Y$$, then $$P(y|\operatorname{do}(x),z)=P(y|x,z)\quad\mbox{ if } (Y\perp X\mid Z)_{G_{\underline{X}}}$$

• Rule3 Ignoring actions: If there are no causal paths from $$X$$ to $$Y$$, then $$P(y|\operatorname{do}(x))=P(y)\quad\mbox{ if } (Y\perp X)_{G_{\overline{X}}}$$

The original version in the book Causality is as follows:

• Rule1 Ignoring observations $$P(y|\operatorname{do}(x),z,w)=P(y|\operatorname{do}(x),w)\quad\mbox{ if } (Y\perp Z\mid X,W)_{G_{\overline{X}}}$$

• Rule2 Action/observation exchange (backdoor) $$P(y|\operatorname{do}(x),\operatorname{do}(z),w)=P(y|\operatorname{do}(x),z,w)\quad\mbox{ if } (Y\perp Z\mid X,W)_{G_{\overline{X},\underline{Z}}}$$

• Rule3 Ignoring actions $$P(y|\operatorname{do}(x),\operatorname{do}(z),w)=P(y|\operatorname{do}(x),w)\quad\mbox{ if } (Y\perp Z\mid X,W)_{G_{\overline{X},\overline{Z(W)}}}$$ where $$Z(W)$$ is the set of $$Z$$-nodes that are not ancestors of any $$W$$-node.