Table 2. Overview of empirical fairness definitions.

Comparison of our generalized empirical fairness definitions (definition 2.7, 2.8, 2.14 and 2.15) and the corresponding existing definitions (e.g., Pessach & Shmueli, 2022).

	Derived generalized empirical definitions (multi cases)	Existing empirical definitions (binary cases)
Independence:	$\mathcal{Y}=\{0,1\}$, $\mathcal{S}$ arbitrary:	$\mathcal{Y}=\mathcal{S}=\{0,1\}$:
$DI$	$\underset{s_{k_1},s_{k_2}\in \mathcal{S}}{min}\frac{\mathbb{P}(\hat{Y}=1|S=s_{k_1})}{\mathbb{P}(\hat{Y}=1|S=s_{k_2})}$	$\frac{\mathbb{P}(\hat{Y}=1|S=0)}{\mathbb{P}(\hat{Y}=1|S=1)}$
Independence:	$\mathcal{Y},\mathcal{S}$ arbitrary:	$\mathcal{Y}=\mathcal{S}=\{0,1\}$:
$DP$	$\underset{y\in \mathcal{Y},s_{k_1},s_{k_2}\in \mathcal{S}}{max}$
	$\left|\mathbb{P}(\hat{Y}=y|S=s_{k_1})-\mathbb{P}(\hat{Y}=y|S=s_{k_2})\right|$	$\left|\mathbb{P}(\hat{Y}=1|S=0)-\mathbb{P}(\hat{Y}=1|S=1)\right|$
Separation:	$\mathcal{Y}=\{0,1\}$, $\mathcal{S}$ arbitrary:	$\mathcal{Y}=\mathcal{S}=\{0,1\}$:
$EO$	$\underset{s_{k_1},s_{k_2}\in \mathcal{S}}{max}$
	$\left|\mathbb{P}(\hat{Y}=1|S=s_{k_1},Y=1)-\mathbb{P}(\hat{Y}=1|S=s_{k_2},Y=1)\right|$	$\left|\mathbb{P}(\hat{Y}=1|S=0,Y=1)-\mathbb{P}(\hat{Y}=1|S=1,Y=1)\right|$
Separation:	$\mathcal{Y},\mathcal{S}$ arbitrary, $\mathrm{\forall }y\in \mathcal{Y}$:	$\mathcal{Y}=\mathcal{S}=\{0,1\}$:
$EOs$	$\underset{s_{k_1},s_{k_2}\in \mathcal{S}}{max}$
	$\left|\mathbb{P}(\hat{Y}=y|S=s_{k_1},Y=y)-\mathbb{P}(\hat{Y}=y|S=s_{k_2},Y=y)\right|$	$\left|\mathbb{P}(\hat{Y}=1|S=0,Y=1)-\mathbb{P}(\hat{Y}=1|S=1,Y=1)\right|$
		$\left|\mathbb{P}(\hat{Y}=1|S=0,Y=0)-\mathbb{P}(\hat{Y}=1|S=1,Y=0)\right|$