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I have 3 random variables, A, B and C. Given A, variables B and C are independent. I have estimates of the variance of A, B and C and the covariance between A and B and A and C. Is it possible calculate an implied covariance between B and C?

My application is that I have a large number of random variables (N) and need to calculate an empirical covariance matrix from a limited number (M) of samples, where M < N. However, most of the random variables are independent of eachother given a few of the other variables, and it seems like I should be able to estimate the covariance matrix more efficiently by using the fact that many of the variables are conditionally independent.

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If we assume multivariate normality, "conditional independence" can be understood as zero partial correlation. This correlation between B and C given A is $r_{bc.a}=(r_{bc}-r_{ab}r_{ac})/\sqrt{(1-r_{ab}^2)(1-r_{ac}^2)}=0$.

And so, $r_{bc}=r_{ab}r_{ac}$. From usual formula of $r_{xy}=cov_{xy}/(sd_xsd_y)$ it then follows that $cov_{bc}=cov_{ab}cov_{ac}/sd_a^2$.

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Exactly what I needed. – John Salvatier Jul 16 '12 at 15:24
Is there a way to extend this to multiple variables like A (variables that given them, B and C are independent of each other)? The formula for partial correlation in the partial correlation article (en.wikipedia.org/wiki/Partial_correlation) doesn't seem to involve the term r_bc at all. – John Salvatier Jul 16 '12 at 16:41
Er, nevermind the second part of that, I was misreading the formula for sample partial correlation. – John Salvatier Jul 16 '12 at 16:45
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Partialling out more than one A-variable is done by applying the above formula of partial corr. recursively or by computing partial corr from linear regressional coefficients (which, in fact, are the "unnormalized" form of partial corr. coeffitients) – ttnphns Jul 16 '12 at 19:35

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