# Calculating induced covariances

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 $cov_{bc}=cov_{ab}cov_{ac}/sd_a^2$.