  A review of the main fairness definitions and fair learning methodologies
proposed in the literature over the last years is presented from a mathematical
point of view. Following our independence-based approach, we consider how to
build fair algorithms and the consequences on the degradation of their
performance compared to the possibly unfair case. This corresponds to the price
for fairness given by the criteria $\textit{statistical parity}$ or
$\textit{equality of odds}$. Novel results giving the expressions of the
optimal fair classifier and the optimal fair predictor (under a linear
regression gaussian model) in the sense of $\textit{equality of odds}$ are
presented.
