Consider a pair of random variables (X, Y) with ground truth G in {X rightarrow Y, X leftarrow Y, X leftrightarrow Y, X perp Y} drawn from an unknown mother distribution, and an estimator $\hat{G}$ of $G$ based on (X, Y).

A causation coefficient C(X, Y) is defined as a real scalar value where a larger C(X, Y) indicates higher confidence that G = [X rightarrow Y]. It must have the following properties:

1. [Anti-symmetry] $C(X, Y) = -C(Y, X)$
2. [Discriminant] $C(X, Y) > \theta \ge 0 \Rightarrow \hat{G} = [X \rightarrow Y]$ (and $\hat{G} \neq [X \rightarrow Y]$ otherwise)
3. [Arbitrary units] $C(aX+b, cY+d) = C(X, Y)$ for all a, b, c, d in mathbb{R} where $a \neq 0$ and $c \neq 0$.