An example of prediction which complies with Demographic Parity and equalizes group-wise risks in the context of regression

• URL: http://arxiv.org/abs/2011.07158v1
• Date: Fri, 13 Nov 2020 22:46:05 GMT
• Title: An example of prediction which complies with Demographic Parity and equalizes group-wise risks in the context of regression
• Authors: Evgenii Chzhen and Nicolas Schreuder
• Abstract summary: Bayes optimal prediction $f*$ which does not produce Disparate Treatment is defined as $f*(x) = mathbbE[Y | X = x]$. We discuss several implications of this result on better understanding of mathematical notions of algorithmic fairness.
• Score: 3.9596068699962323
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $(X, S, Y) \in \mathbb{R}^p \times \{1, 2\} \times \mathbb{R}$ be a triplet following some joint distribution $\mathbb{P}$ with feature vector $X$, sensitive attribute $S$ , and target variable $Y$. The Bayes optimal prediction $f^*$ which does not produce Disparate Treatment is defined as $f^*(x) = \mathbb{E}[Y | X = x]$. We provide a non-trivial example of a prediction $x \to f(x)$ which satisfies two common group-fairness notions: Demographic Parity \begin{align} (f(X) | S = 1) &\stackrel{d}{=} (f(X) | S = 2) \end{align} and Equal Group-Wise Risks \begin{align} \mathbb{E}[(f^*(X) - f(X))^2 | S = 1] = \mathbb{E}[(f^*(X) - f(X))^2 | S = 2]. \end{align} To the best of our knowledge this is the first explicit construction of a non-constant predictor satisfying the above. We discuss several implications of this result on better understanding of mathematical notions of algorithmic fairness.

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