  Statistical algorithms are usually helping in making decisions in many
aspects of our lives. But, how do we know if these algorithms are biased and
commit unfair discrimination of a particular group of people, typically a
minority? \textit{Fairness} is generally studied in a probabilistic framework
where it is assumed that there exists a protected variable, whose use as an
input of the algorithm may imply discrimination. There are different
definitions of Fairness in the literature. In this paper we focus on two of
them which are called Disparate Impact (DI) and Balanced Error Rate (BER). Both
are based on the outcome of the algorithm across the different groups
determined by the protected variable. The relationship between these two
notions is also studied. The goals of this paper are to detect when a binary
classification rule lacks fairness and to try to fight against the potential
discrimination attributable to it. This can be done by modifying either the
classifiers or the data itself. Our work falls into the second category and
modifies the input data using optimal transport theory.
