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Consider three random variables $X,Y,Z$, which are jointly normally distributed. I know that $Y$ is orthogonal to $X$ conditional on $Z$, in the sense that $\beta_{YX;Z}=0$ (i.e. the regression coefficient of Y on X conditional on Z is zero). I'd like to know if it is correct to state this orthogonality condition as: $$ X \bot Y | Z $$

In other words, I'm interested in understanding the precedence of the operator for orthogonality.

OO_SE
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    Side point: the notation is for *independence* which is not necessarily the same thing as *orthogonality*, although independence and uncorrelatedness / lack of covariance will coincide in the Gaussian case. (See also @whuber's comment on http://stats.stackexchange.com/questions/12128/what-does-orthogonal-mean-in-the-context-of-statistics ) – conjugateprior Jul 23 '13 at 11:39
  • @conjugateprior, yes, thanks for pointing it out. I did have in mind the Gaussian case when I formulated the question. I've edited it accordingly – OO_SE Jul 23 '13 at 13:51
  • The conditioning applies to both $X$ and $Y$ whether you like it or not. Put another way, there is no such thing as a random variable called $Y\mid Z$ that $X$ can be orthogonal to. Since $X$ and $Y$ continue to enjoy joint normality when conditioned on the value of $Z$, in which case orthogonality means independence as conjugateprior has pointed out to you already, what you are wanting to say is that $X$ and $Y$ are _conditionally independent_ given the value of $Z$. It does not necessarily follow that $X$ and $Y$ are also _unconditionally independent_ random variables. – Dilip Sarwate Jul 23 '13 at 14:24
  • @DilipSarwate: Thanks for the clarification regarding precedence -I understand now it was wrong to phrase the question that way. But regarding the notation, it is then correct to use the notation $X \bot Y|Z$ to say that $X$ and $Y$ are conditionally independent given Z... right? – OO_SE Jul 23 '13 at 14:53

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