SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Note that  $\alpha >\beta >\gamma >\delta$ , which already hints at a possible ordering of the variables by  $R^2$ . Computing the  $R^2$  yields  $R^2 (X,1)\approx 0.59$ ,  $R^2 (X,2)\approx 0.59$ ,  $R^2 (X,3)\approx 0.53$ , and  $R^2 (X,4) = 0.5$ . This means that  $X_{1}$  (the root node) and  $X_{2}$  have the same  $R^2$  while, conversely and perhaps surprisingly, the variables  $X_{2},X_{3}$ , and  $X_{4}$ , which all have the same fraction of cause-explained variance, have different and descending  $R^2$ . Even  $X_{2}$  and  $X_{3}$  have different  $R^2$  despite having the same fraction of cause-explained variance, and both being neither a root cause nor a leaf node. The  $R^2$ -sortability of this example is  $^{1 / 22}$ . In finite sample settings where the equality of the  $R^2$  of  $X_{1}$  and  $X_{2}$  is not exact, sorting by increasing  $R^2$  is thus likely to give a causal ordering that is either equal or very close to the inverse of the true causal order. The inverse causal order is markedly different from a random ordering  $(\mathbf{v}_{R^2} = 0.5)$ , and shows that the  $R^2$  carry information about the causal ordering. As this example shows, controlling the CEV does not guarantee a  $R^2$ -sortability close to 0.5, and we recommend evaluating and reporting  $R^2$ -sortability even when node variances or the cause-explained variance fractions are controlled.