An example of prediction which complies with Demographic Parity and equalizes group-wise risks in the context of regression
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Let (X, S, Y) ∈ℝ^p ×{1, 2}×ℝ be a triplet following some joint distribution 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) = 𝔼[Y | X = x]. We provide a non-trivial example of a prediction x → f(x) which satisfies two common group-fairness notions: Demographic Parity (f(X) | S = 1) d= (f(X) | S = 2) and Equal Group-Wise Risks 𝔼[(f^*(X) - f(X))^2 | S = 1] = 𝔼[(f^*(X) - f(X))^2 | S = 2]. 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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