Recently I've learnt about something called partial correlation coefficient (denoted as $\rho_{i,j|1...i,j...n}$ or in short, say $\rho_{i,j}$), which is like Pearson correlation between variables $X_i$ and $X_j$, but with assumption that all other variables in $X$ are fixed, i.e. we can say it is "conditional correlation" conditioned on all variables except $X_i, X_j$. One can compute it as $$\rho_{i,j} = \frac{-C_{i,j}}{\sqrt{C_{ii}C_{jj}}},$$ where $C_{i,j}$ is an algebraic complement of element $i,j$ in matrix $C = corr(X)$.
The question is where and when is it used? I understand intuitively how to interpret this coefficient, but never seen such thing before in any data analysis case (like, in Medium posts or textbooks) and just wondering why.
