0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.