A Proofs of Main Results

(conclusion 1). (conclusion 2). Z contains and only contains exogenous noises w.r.t. " means source and " Based on Theorem 6, we can readily give proof to Theorem 2. Note that in our setting where " is equivalent to " Theorem 7 (Trek-separation for directed graphical models, Theorem 2.8 in [ We now show that Theorem 2 can also be proved by trek-separation theorem: Proof of Theorem 2 (another version). 's noise components that is not shared in Therefore, the direction between X and Y is unidentifiable. GIN( Z, Y) must hold, with solution ω .