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!