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I am new to statistics, reading the elements of causal inference:

On section 4.2.1 Additive Noise Models, appears:

Left regress Y on X. Right regress X on Y

enter image description here

The fitted functions are shown in the top row, the corresponding residuals are shown in the bottom row. Only the direction X->Y yields independent residuals... Therefore the correct direction should be X->Y.

Overall I understand the example, but I am not able to relate to it. For example if I think about smoking->cancer, or yearsExperience->salary.

The way I see it, if I do cancer=f(smoking)+noise it should be pretty similar to doing smoking=f(cancer)+noise.

My question is: what is the intuition behind getting non-independent residuals for the wrong direction?

My intuition tells me that I can predict cancer seeing smoking the same as predict smoking seeing cancer (and similarly for yearsExperience and salary).

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    $\begingroup$ I highly doubt that one can infer the direction of causality between two variables based on these checks in general. Are the authors of Elements of causal inference making any assumptions about the joint distribution $P(Y,X)$ and the class of causal models generating it prior to describing these checks? $\endgroup$
    – CloseToC
    Commented Feb 7, 2020 at 13:07
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    $\begingroup$ I found the book in question on Google Books, books.google.co.uk/books?id=XPpFDwAAQBAJ , and see it was published over 2 years ago, but there are 0 reviews. I tried Google Scholar scholar.google.com/… and again found no reviews. That tells me something... $\endgroup$
    – onestop
    Commented Feb 12, 2020 at 20:03
  • $\begingroup$ You can predict one variable from another (though not necessarily very well) if they are correlated. But correlation does not imply causation. $\endgroup$
    – onestop
    Commented Feb 12, 2020 at 20:08
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    $\begingroup$ @onestop tells you what? The book is pretty niche, and it also can be downloaded for free, so that might explained lack of reviews. It's written by Peters, Janzing and Scholkopf and it won ASA Causality in Statistics Education Prize with Judea Pearl as a committee member, so I wouldn't be concerned about the accuracy of the content $\endgroup$
    – rep_ho
    Commented Feb 13, 2020 at 11:39
  • $\begingroup$ Can you elaborate your question, what is it that goes beyond your intuition? Yes, correlation smoking to cancer is the same as cancer to smoking, but here we are not talking about correlations or predictions but about causality $\endgroup$
    – rep_ho
    Commented Feb 13, 2020 at 11:55

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Additive noise models are an assumption

Additive noise models (ANM) express an assumption about the functional form of causal relationships. In your question, you say that "cancer <- f(smoking) + noise should be pretty similar to smoking <- f(cancer) + noise". In a way that's exactly what the ANM assumption is saying – they are pretty similar, except that in the causal direction the noise is independent. ANMs postulate that in the true causal model, noise is added to the effect, which leads to an asymmetry that can be exploited to learn the causal structure.

An intuition about additive noise models

Now you might say "why would the ANM assumption apply?" One motivation is to introduce stochasticity. Without noise of some form, the causal relationships would be completely deterministic, which would be hard to justify. Seeing that we need some stochasticity and having decided to use a noise term, why would it be additive? The intuition here is that the noise is some independent variation of the effect variable that does not interact with the values of the cause. Such variation could be caused by other variables that we have decided to view as outside our causal system but that still independently have some (usually weak) effect on our system variables.

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