I have real-world quantities $(X, Y, Z)$ whose joint distribution I would like to model as a multivariate Gaussian $G = (\mu, \Sigma)$.
By some method, I come up with an estimate $G_{x, y} = (\mu_{x, y}, \Sigma_{x, y})$ for the joint distribution of $(X, Y)$.
Independently (by some other method), I come up with an estimate $G_{x, z} = (\mu_{x, z}, \Sigma_{x, z})$ for the joint distribution of $(X, Z)$.
Is there a sensible way to combine the 2D estimates $G_{x, y}$ and $G_{x, z}$ into the 3D distribution $G$, and under what assumptions? For instance, it might be helpful to assume $\text{cov}(Y, Z) = 0$.
Thanks in advance.