  This paper considers fair probabilistic binary classification where the
outputs of primary interest are predicted probabilities, commonly referred to
as scores. We formulate the problem of transforming scores to satisfy fairness
constraints that are linear in conditional means of scores while minimizing a
cross-entropy objective. The formulation can be applied directly to
post-process classifier outputs and we also explore a pre-processing extension,
thus allowing maximum freedom in selecting a classification algorithm. We
derive a closed-form expression for the optimal transformed scores and a convex
optimization problem for the transformation parameters. In the population
limit, the transformed score function is the fairness-constrained minimizer of
cross-entropy with respect to the true conditional probability of the outcome.
In the finite sample setting, we propose a method called FairScoreTransformer
to approach this solution using a combination of standard probabilistic
classifiers and ADMM. We provide several consistency and finite-sample
guarantees for FairScoreTransformer, relating to the transformation parameters
and transformed score function that it obtains. Comprehensive experiments
comparing to 10 existing methods show that FairScoreTransformer has advantages
for score-based metrics such as Brier score and AUC while remaining competitive
for binary label-based metrics such as accuracy.
