In the context of Bayesian networks if two random variables (i.e. nodes) are d-separated, they are independent. However, is there any example of random variables being independent but not d-separated?
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$\begingroup$ Possible duplicate of Does statistical independence mean lack of causation? $\endgroup$ – usεr11852 Jan 26 '19 at 0:45
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$\begingroup$ I think this question is worth leaving open, but the link by @usεr11852 is well worth the read. $\endgroup$ – Alexis Jan 26 '19 at 1:06
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1$\begingroup$ @Alexis: I retracted my vote. $\endgroup$ – usεr11852 Jan 26 '19 at 1:09
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The No answer: Variables that are d-separated are always independent, and variables that are independent are d-separated. D-separation is a concept formalized by Pearl to understand association from the perspective of a causal DAG. The 'independence' you ask about is the language of statistics describing the same concept of association under different formalisms.
D-separation on a causal DAG, and statistical independence are both describing the same lack of association, so it is not possible to have statistical independence among d-connected variables.
The Yes answer: The comment by usεr11852 touches on the issue of faithfulness: there may be a zero measure of average causal effect (ACE), even if the sharp causal null hypothesis is false. For example, if (1) $A$ causes $Y$ (both dichotomous) because for some individuals $Y^{a=1}\ne Y^{a=0}$, but in some individuals $Y^{a=1} - Y^{a=0} = 1$ and in others $Y^{a=1} - Y^{a=0} = -1$ and (2) the number of individuals in each group is the same, then although there are individual causal effects, the ACE is zero, and the measure of association is unfaithful because $A$ & $Y$ are not d-separated.
See Chapter 6: Graphical Representations of Causal Effects (especially Fine Points 6.1 and 6.2) in Hernán, M. & Robins, J. (2019; forthcoming) Causal Inference. Boca Raton: Chapman & Hall/CRC
