SlowGuess/NeurIPS_2023_OCR huggingface dataset:
1. Observe in Figure 2a that the only path between  $X_{A}^{\perp}$  and  $A$  is blocked by  $Y$ , so counterfactual fairness implies demographic parity. Because the only block in that path is  $Y$ , counterfactual fairness does not imply equalized odds. And the only path between  $Y$  and  $A$  (a parent-child relationship) is unblocked and does not contain  $X_{A}^{\perp}$ , so counterfactual fairness does not imply calibration.
2. Observe in Figure 2b that the only path between  $X_{A}^{\perp}$  and  $A$  is unblocked (because  $S$  is necessarily included in the predictive model), so counterfactual fairness does not imply demographic parity. Because that path contains  $Y$ , counterfactual fairness implies equalized odds. And the only path between  $Y$  and  $A$  is unblocked (because  $S$  is necessarily included in the predictive model) and does not contain  $X_{A}^{\perp}$ , so counterfactual fairness does not imply calibration.
3. Observe in Figure 2c that the only path between  $X_{A}^{\perp}$  and  $A$  is unblocked (because  $S$  is necessarily included in the predictive model), so counterfactual fairness does not imply demographic parity. Because that path does not contain  $Y$ , counterfactual fairness does not imply equalized odds. And the only path between  $Y$  and  $A$  is unblocked (because  $S$  is necessarily included in the predictive model) and contains  $X_{A}^{\perp}$ , so counterfactual fairness implies calibration.
$$
f_{naive}:= \operatorname *{argmin}_{f}\mathbb{E}[ \ell (f(X,A),Y)]
$$
Proof. Notice that  $f_{CF}$  does not depend on  $A$  directly because the realization of  $A$  is not in the definition. To show that  $f_{CF}$  also does not depend on  $A$  indirectly (i.e., through  $X$ ), consider that a purely spurious association means that  $Y \perp X \mid X_A^\perp, A$ . Therefore, the naive predictor:
$$
\begin{array}{l} f _ {n a i v e} (X, A) = \mathbb {P} (Y = 1 | X, A) \\ = \mathbb {P} (Y = 1 | X _ {A} ^ {\perp}, A) \\ \end{array}
$$
Because  $X_A^\perp$  is the component of  $X$  that is not causally affected by  $A$ , there is no term in  $f_{CF}$  that depends on  $A$ , which means  $f_{CF}$  is counterfactually fair.