Trek separation for Gaussian graphical models

Understand

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices.

• Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables.
• We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases.
• Our new trek separation criterion generalizes the familiar $d$-separation criterion.