Multivariate Gaussian - Calculate Covariance-entry given other entries

Question

As the titel is suggesting, i want to calculate one covariance entry given some other entries. lets say i have a multivariate gaussian of the following form,

$p(X_1,X_2,X_3) = N(\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix},\begin{bmatrix} \Sigma_{1,1} & \Sigma_{1,2} & \Sigma_{1,3} \\ \Sigma_{2,1} & \Sigma_{2,2} & \Sigma_{2,3} \\ \Sigma_{3,1} & \Sigma_{3,2} & \Sigma_{3,3} \end{bmatrix})$.

how can i calculate $\Sigma_{1,3}$ given $\Sigma_{1,1}, \Sigma_{2,2}, \Sigma_{3,3}, \Sigma_{1,2}, \Sigma_{2,3}$, involving conditional independence of $X_1$ and $X_3$ given $X_2$.

Answer 1

Calculate the conditional covariance matrix of $X_1$ and $X_3$ given $X_2$ by applying the widely known Schur complement formula, as shown in What is the conditional probability of variables in a multivariate gaussian? . Then set the "1,3" element of the conditional covariance equal to 0, and solve for $\Sigma_{1,3}$

The result is $\Sigma_{1,3} = \Sigma_{1,2} \Sigma_{2,2}^{-1} \Sigma_{3,2}$