In the context of cause-effect pairs, two random variables $X$ and $Y$ can be in one of four mutually exclusive pairwise relationships, denoted by the "ground truth" $G$:

1. $X$ causes $Y$ (X rightarrow Y)
2. $Y$ causes $X$ (X leftarrow Y)
3. $X$ and $Y$ are dependent, but not in a causal relationship (X leftrightarrow Y)
4. $X$ and $Y$ are independent ($X \perp Y$)

Formally, these relationships are defined using functional models with latent noise variables ($N_x, N_y$) and potential latent confounders ($Z$):

• G=[X rightarrow Y] Rightarrow exists f wedge exists N_y such that $Y := f(X, N_y)$
• G=[X leftarrow Y] Rightarrow exists f wedge exists N_x such that $X := f(Y, N_x)$
• G=[X leftrightarrow Y] Rightarrow exists f, g wedge exists N_x, N_y, Z such that $X := f(Z, N_x) \wedge Y := g(Z, N_y)$
• $G=[X \perp Y]$ means $X$ and $Y$ are independent, with no functional relationship.

This framework does not necessarily assume that $(X \perp N_y)$ nor that $(Y \perp N_x)$, allowing for additional latent confounders in the directed cases X rightarrow Y and X leftarrow Y.