# Question on solving OLS

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!

• I take it you are reading Domain adaptation under structural causal models, is that correct? Mar 20 at 9:27
• Asside from verification whether Scriddie's link is the article that you are reading, could you also add more details from the article. The meaning of $m$, $^{(1)}$ and $\tilde{}$ are not clear. Mar 20 at 12:46
• @SextusEmpiricus thanks I just made the edits! Mar 20 at 15:52

## 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.

## 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?

• Thanks for the response! But for the target domain, it’s still $+\beta_3$ so wouldn’t it be -1/3 - 1/3-2/3=-4/7? Mar 20 at 15:49
• I think you meant it would be $-\frac{4}{3}$, is that correct? Looking at it again, I think you're right and the example may just be wrong. I have updated the answer to reflect this. Mar 20 at 16:36
• @Scriddle thank you! I’m also having some difficulty deriving this least square solution as well, would you mind showing how the author arrive at this result? Mar 20 at 20:42
• Are you referring to $\beta_{\text{OLSSrc}}$ and $\beta_{\text{OLSTar}}$? I think it's just a standard OLS . Because the data generating process is given and not very complicated, one can also see that the weights are equal to the intervention values for the variables that are causal parents of $Y$. Does that answer your question? Otherwise maybe it's best to raise a new question. Mar 21 at 8:05
• I think a new question that specifies the problem of how to derive the DIP coefficients (analytically or numerically, depending on what you are interested in) would be a good idea. Mar 21 at 20:16