What's the relationship between statement "Z causes both X and Y" and "X and Y are independent given Z"?

Question

Suppose I have two statements:

Statement 1: Random variable Z is the common cause for random variable X and Y (Z causes both X and Y)

Statement 2: Random variable X and Y are (conditionally) independent given Z.

What's the relationship between Statement 1 and Statement 2? Does Statement 1 imply statement 2, or does statement 2 imply statement 1, or they have other relationships?

Example: Z could be the grade level(or age) for a primary school student, X could be his height, Y could be his math ability. Thanks!

When you say in Statement 1 that Z is "the" comment cause for X and Y, do you mean that X does not directly cause Y? — Adrian Keister, Oct 30 '22 at 00:22
This question seems to be about the difference between causality and statistical associations. There are many related CV threads, incl. Is there an example of two causally dependent events being logically (probabilistically) independent?, Does statistical independence mean lack of causation?, Conditional probability and causality. — dipetkov, Oct 30 '22 at 09:41
@dipetkov Thanks! I've just read these threads, indeed many insightful comments. — ExcitedSnail, Nov 02 '22 at 21:45
@dipetkov You have closed this question in favour of https://stats.stackexchange.com/questions/454516/conditional-probability-and-causality, but the previous thread does not even mention conditional independence so I do not see how it can be taken to be a complete answer to the current question, which is specifically about conditional independence. Linking to previous questions about causality was very helpful but closing seems to me to be incorrect. — Gordon Smyth, Nov 14 '22 at 06:58
@GordonSmyth Thank you for the feedback. I don't quite agree: I pointed to not one but three previous threads that taken together I believe do answer the question substantially. Still, this is a complex topic, so it's good to have one more exposition on it. — dipetkov, Nov 14 '22 at 07:33
@dipetkov You closure message gives only one of the links. In any case, even we take all three threads pointed to in your comment, none of them mentions conditional independence. — Gordon Smyth, Nov 14 '22 at 09:20
@GordonSmyth I've voted to reopen, which hopefully puts this to rest. — dipetkov, Nov 14 '22 at 09:48

Gordon Smyth · Accepted Answer · 2022-10-30T09:10:22.873

2

Statement 1 is a scientific statement rather than a mathematical relationship. The idea of one random variable "causing" another doesn't have a strict mathematical definition, rather the role of the mathematician or statistician is to posit a mathematical model that captures the relationship in a way that is relevant to a particular scientific problem. For example, a mathematician might suppose that X and Y both have functional relationships with Z but not with each other, and that would capture the scientific idea of "causation".

Statement 2 is a mathematical statement, and it is one sensible and useful way to translate the first statement into mathematical terms. It is a very strong statement because it implies that Z accounts for all possible dependencies between X and Y. There cannot be any other common causes of X and Y that are not mediated by Z or correlated with Z. On the other hand, Statement 2 is more general than Statement 1 in the sense that X and Y could be conditionally independent given Z without being direct functions of Z.

edited Oct 30 '22 at 09:10

answered Oct 30 '22 at 01:39

Gordon Smyth

12,807

I don’t think your assertion that any causal model where Z is a common cause for X and Y is plausible. In particular, even within the standard model of causality, if Z’ were another common cause between X and Y, then X would not in general be conditionally independent of Y given Z – stats_model Oct 30 '22 at 02:06
@stats_model Yes, obviously. OP's Statement 1 says "the common cause" rather than "a common cause" so I assumed that no other common causes are present. Anyway, I have deleted the last paragraph of my answer so that it doesn't depend on such a strict interpretation of OP's words. – Gordon Smyth Oct 30 '22 at 03:17
@GordonSmyth Thank you very much! This is very helpful! – ExcitedSnail Nov 02 '22 at 21:44

What's the relationship between statement "Z causes both X and Y" and "X and Y are independent given Z"?

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