The idea is to ensure that a classifier is fair by ensuring that membership in the class is independent from membership in a protected group (such as race, sex, etc.).

More formally, let the class attribute be $y = \{+, -\}$, and the protected attribute be $S = \{s, \overline{s}\}$, where $s$ denotes a sample inside the protected class, and $\overline{s}$ denotes a sample outside the protected class.¹⁾

There are a few ways to concretize the notion of fairness, the simplest of which is maybe statistical parity²⁾, which says that if a class prediction is fair, it should satisfy: $$ P(\hat{y}|S=s) - P(\hat{y}|S=\overline{s}) \leq \epsilon $$ Which can be intuitively understood to mean that for any outcome $\hat{y}$, the probability of the outcome conditioned on it being in the protected class ought to be about the same as (upto some tolerance $\epsilon$) the probability if it's not in the protected class.