MHB Counterfactual Expectation Calculation

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The discussion focuses on calculating the expected salary of workers at a specific skill level given a certain number of years of college education, using a causal model involving Gaussian variables. The key steps involve applying Theorem 4.3.2 to derive the expected salary based on the relationship between education, skill, and salary. The process includes updating probabilities based on evidence, modifying the model to reflect educational attainment, and predicting outcomes using the adjusted model. Critical insights include the relationships between variables and the need for regression coefficients to express expectations accurately. The participant seeks clarification on the hint provided and the next steps in the calculation process.
Ackbach
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$\newcommand{\doop}{\operatorname{do}}$
Problem: (This is from Study question 4.3.1 from Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell.) Consider the causal model in the following figure and assume that $U_1$ and $U_2$ are two independent Gaussian variables, each with zero mean and unit variance.

Find the expected salary of workers at skill level $Z=z$ had they received $x$ years of college education. [Hint: Use Theorem 4.3.2, with $e:Z=z,$ and the fact that for any two Gaussian variables, say $X$ and $Z,$ we have $E[X|Z=z]=E[Z]+R_{XZ}(z-E[Z]).$ Use the material in Sections 3.8.2 and 3.8.3 to express all regression coefficients in terms of structural parameters, and show that $$E[Y_x|Z=z]=abx+\frac{bz}{1+a^2}.]$$

View attachment 9643

Here, $X$ is education, $Z$ is skill, and $Y$ is salary. The accompanying SEM is
\begin{align*}
X&=U_1\\
Z&=aX+U_2\\
Y&=bZ.
\end{align*}

My Work So Far:
We are called on to compute $E[Y_x|Z=z].$
Now Theorem 4.3.2 states: Let $\tau$ be the slope of the total effect of $X$ on $Y,$
$$\tau=E[Y|\doop(x+1)]-E[Y|\doop(x)] $$
then, for any evidence $Z=e,$ we have
$$E[Y_{X=x}|Z=e]=E[Y|Z=e]+\tau(x-E[X|Z=e]).$$
For our problem, with $e:Z=z,$ we have
$$E[Y_{X=x}|Z=z]=E[Y|Z=z]+\tau(x-E[X|Z=z]).$$
Not sure where to go from there.

Now I know that this is a non-deterministic counterfactual problem, which means the process should be:

1. Abduction: Update $P(U)$ by the evidence to obtain $P(U|E=e).$
2. Action: Modify the model, $M,$ by removing the structural equations for the variables in $X$ and replacing them with the appropriate functions $X=x,$ to obtain the modified model, $M_x.$
3. Prediction: Use the modified model, $M_x,$ and the updated probabilities over the $U$ variables, $P(U|E=e),$ to compute the expectation of $Y,$ the consequence of the counterfactual.

So, for abduction, am I right in thinking that the only evidence we're using right now is $Z?$ In that case, we want to determine the $U_1$ and $U_2$ that correspond to $Z=z.$ We have the two equations
\begin{align*}
X&=U_1\\
z&=aX+U_2,
\end{align*}
or
\begin{align*}
X&=U_1\\
z-aX&=U_2.
\end{align*}
Without knowing the pre-condition value of $X,$ it's not clear how to continue. How do I continue? I'm also really not understanding the hint. Any thoughts about the hint?

Thanks for your time!

Note: I have cross-posted this at Cross-Validated:

https://stats.stackexchange.com/questions/457740/counterfactual-expectation-calculation
 

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I have obtained access to the full solutions manual, after contacting Wiley about it. I will not type up the solution in full, but simply note a few critical pieces of information I was missing in order to answer this question:

1. $E[x|z]=\beta_{xz}\,z,$ because of the model, and the relationship between $x$ and $z.$ Here $\beta_{xz}$ is the regression coefficient, as in $X=\beta_{xz}Z.$
2. Reversing regression coefficients requires knowing the variances: $\beta_{xz}\sigma_z^2=\beta_{zx}\sigma_x^2.$
3. The slope of the total effect, $\tau,$ you can read off the diagram as $\tau=ab.$
4. Variances add like this: if $Z=aX+U_2,$ then $\sigma_z^2=a^2\sigma_x^2+\sigma_{U_2}^2.$

This is sufficient information to obtain the desired result.
 
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