This article develops an axiomatic basis for the relationship between conditional independence and graphical models in statistical analysis. In particular, the following relationships are established: (1) every axiom for conditional independence is an axiom for graph separation, (2) every graph represents a consistent set of independence and dependence constraints, (3) all binary factorizations of strictly positive probability models can be encoded and determined in polynomial time using their correspondence to graph separation, (4) binary factorizations of non-strictly positive probability models can also be derived in polynomial time albeit less efficiently and (5) unconditional independence relative to normal models can be axiomatized with a finite set of axioms.
"Logical and Algorithmic Properties of Conditional Independence and Graphical Models." Ann. Statist. 21 (4) 2001 - 2021, December, 1993. https://doi.org/10.1214/aos/1176349407