Consider a pair of random variables $( X , Y )$ with ground truth $G ∈\{ X → Y , X ← Y , X \leftrightarrow Y , X ⊥ Y \}$, drawn from an unknown mother distribution. We call an estimator $\hat{G}$ of $G$ based on $( X , Y )$.

The authors define a causation coefficient as follows: A causation coefficient $C ( X , Y )$ is a real scalar value, such that the larger $C ( X , Y )$, the more confident we are that $G = X → Y$ . A causation coefficient must have the following desired properties:

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