  We use disparate impact, i.e., the extent that the probability of observing
an output depends on protected attributes such as race and gender, to measure
fairness. We prove that disparate impact is upper bounded by the total
variation distance between the distribution of the inputs given the protected
attributes. We then use pre-processing, also known as data repair, to enforce
fairness. We show that utility degradation, i.e., the extent that the success
of a forecasting model changes by pre-processing the data, is upper bounded by
the total variation distance between the distribution of the data before and
after pre-processing. Hence, the problem of finding the optimal pre-processing
regiment for enforcing fairness can be cast as minimizing total variations
distance between the distribution of the data before and after pre-processing
subject to a constraint on the total variation distance between the
distribution of the inputs given protected attributes. This problem is a linear
program that can be efficiently solved. We show that this problem is intimately
related to finding the barycenter (i.e., center of mass) of two distributions
when distances in the probability space are measured by total variation
distance. We also investigate the effect of differential privacy on fairness
using the proposed the total variation distances. We demonstrate the results
using numerical experimentation with a practice dataset.
