How to infer correlations from correlations

I have a question regarding correlation inference. Consider, I have two sets of variables X and Y. For an x element of X I know the correlation to an unknown variable z. I also have the covariance matrix X and Y. Moreover, I know a set of (y1,y2,y3) element of Y, which model best my variable x. So how to infer the correlation of (y1,y2,y3) and z?

I need a function: f(corr(x,z), corr(x,(y1, y2, y3)), cov(X,Y)) --> corr((y1,y2,y3),z)

This might be a very simple question, but I have no idea how to do it or for what I can search in the literature.

Given corr(A,B) and corr(A,C) you can obtain bounds on corr(B,C) (and similar such calculations involving more variables), but the bounds are in general quite wide. Indeed, typically such calculations aren't very informative at all.

Specifically, by looking at the relationship between the ordinary pairwise correlation and the partial correlation:

$$\rho_{BC\cdot A } = \frac{\rho_{BC} - \rho_{AB}\rho_{AC}} {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}$$

you can rearrange the formula to back out bounds for $\rho_{BC}$:

$$\rho_{BC}=\rho_{AB}\rho_{AC}+\rho_{BC\cdot A } {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}$$

and noting that the partial correlation must lie between -1 and 1, this implies that $\rho_{BC}$ is bounded to lie in

$$\rho_{AB}\rho_{AC}\pm {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}\,.$$

e.g. Let's say $\rho_{AB}=0.8$ and $\rho_{AC}=0.6$.

Then $\rho_{BC}= 0.6 \times 0.8 \pm \sqrt{(1-.64)(1-.36)}=0.48\pm 0.48 = (0,0.96)$

With more variables the situation becomes more complex; in some situations it's easier to work with Cholesky decompositions.

If you impose additional structure on the problem then in some situations those bounds might reduce.