Question on solving OLS

Question

I'm reading this article on Structural Causal Models (SCM) and the author is giving this example:

where $m=1$ is the single source environment in this case, ~ denotes the target domain and all noise variables $\epsilon$ follow independent Gaussian distributions with mean zero, and we want to find the coefficients that satisfy the following optimization problem:

The output stated in the paper is the following:

I'm having some hard time trying to arrive at this result. My current logic is that, given the constraint, $E[\beta_1 X_1^{(1)}+\beta_2 X_2^{(1)} + \beta_3 X_3^{(1)}] = E[\beta_1 \tilde{X_1}+\beta_2 \tilde{X_2} + \beta_3 \tilde{X_3}] \Rightarrow $ (substitute the equations of every X's in and knowing that $E[\epsilon] = 0$ for all the noises) $E[\beta_1 + \beta_2 + \beta_3] = E[-\beta_1 -\beta_2 + \beta_3]$. This equation won't be satisfied given the output in the picture. Or am I doing the calculation wrong? Any help is appreciated!

Answer 1

Edit: The example appears to be wrong

I am using the setup described in 3.4 and 3.5.1 of Domain adaptation under structural causal models which matches your description.

In the Source environment we have:

$E_{X\sim\mathcal{P}}[\beta_1 + \beta_2 + \beta_3] = \frac{1}{3} + \frac{1}{3} - \frac{2}{3} = 0$

In the target environment we have:

$E_{X\sim\widetilde{\mathcal{P}}}[-\beta_1 - \beta_2 + \beta_3] = -\frac{1}{3} - \frac{1}{3} - \frac{2}{3} = -\frac{4}{3}$.

The constraint $E_{X\sim\mathcal{P}}[X^\top \beta] = E_{X\sim\widetilde{\mathcal{P}}}[X^\top \beta]$ for DIP$^{(m)}$-mean in 3.4 would therefore not be fulfilled.

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Perhaps the interventions were intended to be different?

Looking at figures 3 and 4, it seems the authors usually define the target interventions such that $\widetilde{a} = -a^{(1)}$. The scenario in figure 2 seems to be the exception and the numbers they provide would be correct for $\widetilde{a} = -a^{(1)}$. Perhaps that's what they had in mind?