In his book "Introductory Econometrics", Jeffrey Woolridge states "The most we can know about how X affects Y is contained in the conditional distribution of Y given X".
Is this statement true?
Would Judea Pearl agree with this statement?
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asked Feb 2, 2019 at 16:52
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3$\begingroup$ Wow... big conflation of causation with association there. $\endgroup$– AlexisCommented Feb 2, 2019 at 17:19
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1 Answer
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What does "X affects Y" mean? If it means "X is the cause of Y", then Pearl would not agree with it, because, in general, $P(Y=y \mid do(X=x)) \neq P(Y=y \mid X=x)$. However, I think that by "X affects Y" the author meant how $Y$ is dependent on $X$. In other words, the conditional probability of $Y$ given $X$ tells you if these random variables are dependent or not (provided that you know the marginals $P(X)$ and $P(Y)$), but it does not tell you if they are causally related.
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1$\begingroup$ Thank you for the response. I agree with you. What about the part that says that this is most we can know about how X affects Y. Setting causality aside for a moment, isn't it the case that there is no more information about how X affects Y in the conditional distribution of Y given X than there is in the joint distribution of X and Y? In fact, knowing the joint distribution of X and Y, we can always build the conditional distribution of Y given X, but not vice versa. I would contend that the most we can know about how X affects Y is contained in the joint distribution of X and Y. $\endgroup$ Commented Feb 2, 2019 at 17:06
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$\begingroup$ Good point. But isn't knowledge of how Y affects X potentially informative (however marginally) about how X affects Y? If so, from the joint distribution we can obtain both, whereas the conditional distribution of Y given X only gives us how X affects Y (albeit more directly). $\endgroup$ Commented Feb 2, 2019 at 17:14
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$\begingroup$ Would your answer change if we were to put back our "Causal" hat? $\endgroup$ Commented Feb 2, 2019 at 17:22
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$\begingroup$ Thank you for the follow up, nbro. I take my last question back. So, if I were to try to summarize your answers, you're saying if we're looking at dependence (i.e. how X "affects" Y) then the conditional distribution of Y given X contains exactly the same information as the joint distribution of X and Y. In this case, his statement is true. If we're looking at causality (i.e. how X "causes" Y) then the conditional distribution of Y given X is the wrong distribution to be looking at because P(Y=y|X=x) is different than P(Y=y|do(X=x)). In this case, his statement is false. A fair synthesis? $\endgroup$ Commented Feb 2, 2019 at 18:10