# The Book of Why, Table 8.1, Counterfactuals example

I'm reading "The Book of Why" by Judea Pearl and I'm getting an answer for a problem that doesn't match what the book says. This is my first foray into Structural Causal Models and their use, so I want to be absolutely certain that there isn't a gap in my understanding when I try to solve the problem; hence this question.

Here's the problem that I'm stuck on:

The Data

Bert: $EX = 9$, $ED = 1$, $S_1 = 92500$.

Caroline: $EX= 9$, $ED = 2$, $S_2 = 97000$

The equations

According to more data in the table, the following equations hold:

$$S_{ED} = 65000 + 2500 EX + 5000 ED + U_S\\ EX = 10 - 4 ED + U_{EX}$$

where $U_S$ and $U_{EX}$ are background variables that are different for Bert and Caroline.

The Question

The question requires that we calculate the value of $S_1(Bert) - S_1(Caroline)$ (basically subtract from 92500, the value of $S_{ED}$ for Caroline if her $ED = 1$). The problem is posed between pages 273 and 282.

My confusion

The book says that the final value should be 5000. My numbers are either 500, if I assume that the value of EX remains 9 like before (which would be wrong because the value of EX should be updated according to the second equation?), or -9500, if I simply plug in the new value of ED and solve the equations.

Am I missing something here? It's very possible as this whole exercise is couched in the realm of operating with causal model diagrams, and I might have missed an update step to the equations...

You are not missing anything, the correct answer is $-9500$!

First find the idiosyncratic characteristics of Caroline, by updating her background variables $U$ given the evidence $EX = 9$, $ED = 2$, $S = 97000$. This is the Abduction step,

$$97000 = 65000 + 2500(9) + 5000(2) +U_{s} \implies U_s = -500\\ 9 = 10-4(2) + U_{EX} \implies U_{EX} = 7$$

Now we perform the action, which is setting Caroline's new education to 1, that is, $do(ED = 1)$, and properly mutilating the graph and structural equations. Finally we perform prediction under the new system (using the new updated information about the exogenous variable and the interventional graph),

$$EX_1(Caroline) = 10-4*(1) + 7 = 13 \\$$ And, \begin{align} S_{1}(Caroline) &= 65000 + 2500\times EX_{1}(Caroline) + 5000(1) -500 \\ &=65000 + 2500(13) + 5000 - 500\\ &= 102000 \end{align}

Thus, $S_1(Bert) - S_1(Caroline) = 92500 - 102000 = -9500$.

PS: Pearl is aware of the typo and this will probably be in the errata and corrected soon.

• This kind of statements should be backed up with some kind of source for the claim. – Tim Aug 13 '18 at 9:01
• @Tim Personal communication and now twitter.com/yudapearl/status/1028828169028005888 – Carlos Cinelli Aug 13 '18 at 15:03
• Thanks, maybe you could edit and put the link into the body of the answer? – Tim Aug 13 '18 at 15:07
• @Tim this is just algebra, anyone can verify the correct answer by themselves. – Carlos Cinelli Aug 13 '18 at 15:08
• I was referring to PS that was preciously the only part of the answer. – Tim Aug 13 '18 at 15:09