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I am reading Elements of Causal Inference by Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. In section 3.2 the book defines a structural causal model (SCM) $\mathfrak{C}$:

$$C:=N_C$$ $$E:= 4 \times C + N_E$$

where $N_C, N_E \stackrel{iid}{\sim} \mathcal{N}(0, 1)$ and graph $C \to E$ (which indicates the causal direction). The $:=$ sign signifies an asymmetric assignment operation. We can infer $P^{\mathfrak{C}}_{E} = \mathcal{N}(0, 17)$ and $P^{\mathfrak{C}}_{E \vert C = 2} = \mathcal{N}(8, 1)$rather intuitively, simply following the assignments. The book then asks the reader to show $P^{\mathfrak{C}}_{C \vert E = 2} = \mathcal{N}(\frac{8}{17}, \frac{1}{17})$. This does not follow the causal direction. Using the fact that the conditional bivariate normal probability distribution is normal with $$E(X_2 \vert X_1 = x_1) = \mu_2 + \rho \sigma_2 \left( \frac{x_1 - \mu_1}{\sigma_1} \right)$$ $$Var(X_2 \vert X_1 = x_1) = (1 - \rho^2) \sigma_2^2$$ only gets us as far as $$E(C \vert E = 2) =\frac{2\rho}{17}$$ Surely, I am missing something.

Zelazny
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1 Answers1

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Since $E$ conditioned on $C$ is normal, and both have normal marginals, we know that together they follow the bivariate normal distribution. Using $E(X_2|x_1=2)=\mu_2+\rho\sigma_2\left(\frac{2-\mu_1}{\sigma_1}\right)$ with $C=X_1,E=X_2$, we can solve for the parameter $\rho$ in $$\begin{split}8&=0+\rho\sqrt{17}\left(\frac{2-0}{1}\right)\\ 8&=2\rho\sqrt{17},\end{split}$$ getting $\rho=\frac{4}{\sqrt {17}}$. Now we have all the parameters of the bivariate distribution so we can find the conditional distribution of of $C$ given $E$, which will be normal with mean and variance as follows.

So then using the same equations with $X_1$ and $X_2$ switched around, we find that $$\begin{split}E(C|E=2)&=0+\frac{4}{\sqrt{17}}\cdot 1\cdot \frac{2-0}{\sqrt{17}}=\frac{8}{17}\\ Var(C|E=2)&=\left(1-\frac{16}{17}\right)\cdot 1=\frac 1{17}\end{split}$$

Vons
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