The authors consider that pairs of variables X and Y can be in one of four types of relationships:

• X causes Y ( X → Y )
• Y causes X ( X ← Y )
• X and Y are dependent, but not in a causal relationship ( X :left_right_arrow: Y )
• X and Y are independent ( X ⊥ Y ). Any pair of variables ( X , Y ) is associated with one and only one such relationships, called “ground truth” G . By definition, we have: $G=[X\rightarrow Y] \Rightarrow \exists f \wedge \exists N_y s.t. Y:=f(X,N_y)$ $G=[X\leftarrow Y] \Rightarrow \exists f \wedge \exists N_x s.t. X:=f(Y,N_x)$ $G=[X\leftrightarrow Y] \Rightarrow \exists f,g \wedge \exists N_x, N_y, Z s.t. X:= f(Z,N_x)\wedge y:= f(Z,N_y)$ $G= X ⊥ Y$ X and Y are independent, no functional relationship

The authors do not necessarily assume that ( X ⊥ N y ) nor that ( Y ⊥ N x ), thus there can be additional latent confounders in the first two cases X → Y and X ← Y.