Let $X,Y,\epsilon:\Omega\to \mathbb R$ be random variables. Let's say that $X=\text{sign} (Y) +\epsilon$. Then $X$ is not independent of $Y$. However, we have all the information about $Y$ that we need for $X$ if we just know whether $Y\in \mathbb R_{< 0}$ or $Y\in \mathbb R_{\geq 0}$. That is, once we know that $Y$ is in one of those subsets, then $X$ is independent of further information about $Y$.

Is there a generally accepted name for this type of "partial" independence?

  • $\begingroup$ How about conditional independence? $\endgroup$ – whuber Oct 11 '18 at 14:39
  • $\begingroup$ @whuber, I may be misinterpreting you, but is "How about" supposed to mean "you should have known that it is conditional independence"? In any case, conditional independence is more general, since $X$ and $Y$ can be conditionally independent conditional on an unrelated third variable $Z$, rather than conditional on $Y$ being in a subset of its domain. $\endgroup$ – user600667 Oct 11 '18 at 15:47
  • $\begingroup$ Define the third variable as the indicator of the conditioning event. I am trying to understand whether, and if so how, your question differs from inquiring about conditional independence. From your comment it sounds like there is no difference. $\endgroup$ – whuber Oct 11 '18 at 16:15
  • $\begingroup$ @whuber, like I said, conditional independence is more general. I'm wondering if there is a name for this special case......... $\endgroup$ – user600667 Oct 11 '18 at 16:51

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