Telling cause from effect from purely observational data (focusing on pairs (X,Y) of variables rather than multi-variate case) has been a challenge at the NIPS 2008 workshop on Causality. The scenario can be described as follows:

• Given observations $( x_1 , y_1 ), …, ( x_k , y_k )$ iid (independent and identically distributed) drawn from some distribution $P_{X , Y}$ , infer whether X causes Y (a) or Y causes X (b) , given the promise that exactly one of these alternatives is true.

The NIPS 2013 challenge extended this setting to three classes: $X\rightarrow Y (a)$ , $Y\rightarrow X (b)$, or the null class $\{(c),(d)\}$, which consisted of the unconnected case $X Y$ or the case of pure confounding $X \leftrightarrow Y$ (this notation is used as shorthand for X ← H → Y , where H is unobserved, but where neither X or Y cause each other).