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We use disparate impact, i.e., the extent that the probability of observingan output depends on protected attributes such as race and gender, to measurefairness. We prove that disparate impact is upper bounded by the totalvariation distance between the distribution of the inputs given the protectedattributes. We then use pre-processing, also known as data repair, to enforcefairness. We show that utility degradation, i.e., the extent that the successof a forecasting model changes by pre-processing the data, is upper bounded bythe total variation distance between the distribution of the data before andafter pre-processing. Hence, the problem of finding the optimal pre-processingregiment for enforcing fairness can be cast as minimizing total variationsdistance between the distribution of the data before and after pre-processingsubject to a constraint on the total variation distance between thedistribution of the inputs given protected attributes. This problem is a linearprogram that can be efficiently solved. We show that this problem is intimatelyrelated to finding the barycenter (i.e., center of mass) of two distributionswhen distances in the probability space are measured by total variationdistance. We also investigate the effect of differential privacy on fairnessusing the proposed the total variation distances. We demonstrate the resultsusing numerical experimentation with a practice dataset.

Optimal Pre-Processing to Achieve Fairness and Its Relationship with Total Variation Barycenter