Do-Calculus (p213)

Authors: Aymen Merrouche and Pierre-Henri Wuillemin.

This notebook follows the example from “The Book Of Why” (Pearl, 2018) chapter 7 page 213

import pyagrum as gum
import pyagrum.lib.notebook as gnb

the causal diagram

The corresponding causal diagram is the following:

We’re facing the following situation and we want to measure the causal effect of $X$ on $Y$:

fd = gum.fastBN("w->z->x->y;w->x;w->y")
fd

We suspect the presence of some unmeasured confounders, that could explain the correlation between $W$ and $X$ and between $W$ and $Y$:

fdModele = gum.CausalModel(fd, [("u1", ["w", "x"]), ("u2", ["w", "y"])], False)
# (<latent variable name>, <list of affected variables’ ids>).
gnb.show(fdModele)

Even with two umeasured confounders :

• We can measure the causal effect of $Z$ on $Y$ using the back-door adjustment:

print(" + Back-door doing Z on Y :" + str(fdModele.backDoor("z", "y")))

 + Back-door doing Z on Y :{0}

• We can measure the causal effect of $W$ on $X$ using the front-door formula:

print(" + Front-door doing W on X :" + str(fdModele.frontDoor("w", "x")))

 + Front-door doing W on X :{1}

• In order to measure the causal effect of $X$ on $Y$, we can use neither the back-door adjustment nor the front-door formula:

print(" + Backdoor doing X on Y :" + str(fdModele.backDoor("x", "y")))
print(" + Frontdoor doing X on Y :" + str(fdModele.frontDoor("x", "y")))

 + Backdoor doing X on Y :None
 + Frontdoor doing X on Y :None

• In this case, the only way to measure the causal effect of $X$ on $Y$ is to use the do-calculus:

gnb.showCausalImpact(fdModele, on="y", doing="x")

Causal Model

$$\begin{equation*}P\left(y \mid \text{do}(x)\right) = \frac {\sum_{w}{P\left(y\mid w,x\right) \cdot P\left(x\mid w,z\right) \cdot P\left(w\right)}}{\sum_{w,y'}{P\left(y'\mid w,x\right) \cdot P\left(x\mid w,z\right) \cdot P\left(w\right)}}\end{equation*}$$

Explanation : Identified via do-calculus (ID/IDC).

		x
y	z	0	1
0	0	0.3236	0.5246
	1	0.2020	0.5021
1	0	0.6764	0.4754
	1	0.7980	0.4979

Impact