Fair Bayes-Optimal Classifiers Under Predictive Parity

1 Introduction

• Fairness matters because biased training data can produce systematically biased predictions against groups underrepresented due to historical bias.The paper situates this concern in high-stakes applications including healthcare and criminal prediction.Source: §1 · p. 1
• Fair Bayes-optimal classifiers provide a benchmark for maximizing accuracy subject to fairness, while GWTRs select people with the highest conditional probabilities in each group.Prior work established GWTR optimality for many other fairness metrics.Source: §1 · pp. 1–2
• Predictive parity is less studied than independence- and separation-based fairness, and the paper addresses gaps in understanding its fair Bayes-optimal classifiers and how to learn them.It equalizes the probability of success among positive predictions across protected groups.Source: §1 · p. 2
• When group performances vary moderately, all fair Bayes-optimal classifiers under predictive parity are GWTRs; without the condition, they may not be GWTRs and may cause within-group unfairness for the minority group.The latter case occurs when the minority group is more qualified than the majority group.Source: §1 · p. 2
• FairBayes-DPP is a two-stage plug-in algorithm that checks the sufficient condition and, if it holds, estimates optimal fair thresholds for each group.The paper reports computational efficiency and solid experimental performance.Source: §1 · p. 2

2.1 Fairness Measures

• Fairness measures include independence, separation, and sufficiency; sufficiency is less studied and harder to enforce because conditioning on predictions creates a complex constraint.Separation can also be difficult to validate when labels are unavailable for some groups, such as declined loan applicants.Source: §2.1 · pp. 2–3

2.2 Algorithms Aimed at Fairness

• Fairness algorithms include preprocessing and in-processing approaches; post-processing adjusts model outputs, commonly with group-specific thresholds, and FairBayes-DPP uses this approach for predictive parity.In-processing examples include penalized optimization and adversarial training.Source: §2.2 · p. 3

3 Problem Formulation and Notations

• A randomized classifier takes ordinary features X and protected attribute A as input and returns the probability of predicting the positive label.The ground-truth label is Y.Source: §3 · p. 3
• A GWTR uses group-specific thresholds on conditional label probabilities, selecting individuals with the highest such probabilities within each group.The paper presents this selection property as a minimal requirement for within-group fairness.Source: §3 · p. 3
• Predictive parity requires the probability of a positive outcome among positive predictions to be equal across protected groups.The definition equates each group's positive predictive value with the overall value.Source: §3 · pp. 3–4
• Difference in Predictive Parities measures the difference between a group's positive predictive value and the overall positive predictive value.The paper denotes the measure DPP.Source: §3 · p. 4

4.1 GWT Fair Bayes-Optimal Classifiers under Predictive Parity

• Condition 4.1 requires each group’s positive predictive value above the cost threshold to be at least the overall positive-label rate of every group.It holds, for example, when every group’s overall positive-label rate is no greater than cost parameter c.Source: §4.1 · p. 4
• When Condition 4.1 holds, every fair Bayes-optimal classifier under predictive parity is a group-wise thresholding rule.The proof constructs a lower-error fair thresholding rule from any fair classifier that is not one.Source: §4.1 · p. 4

4.2 Fair Bayes-optimal Classifiers under Predictive Parity do not Need to be Thresholding Rules

• When condition (3) holds and the stated probability-mass condition is met, no fair Bayes-optimal classifier is a group-wise thresholding rule for sufficiently large minority-group prevalence.Theorem 4.3 specifies the prevalence bound using δ1 and δ2.Source: §4.2 · p. 5
• In the non-thresholding case, predictive parity may label the most qualified minority-group individuals unqualified to improve overall accuracy.The paper explains that choosing a lower standard for a highly qualified minority group can allow more qualified individuals in other groups to be selected.Source: §4.2 · p. 5

• The two-stage plug-in method first estimates group-conditional label probabilities using standard learning algorithms, then checks the empirical version of Condition 4.1.Source: §5 · p. 5
• If the empirical condition fails, the algorithm recommends considering other fairness measures because predictive parity may not be appropriate in that case.Source: §5 · pp. 5–6
• If the condition holds, FairBayes-DPP matches groups’ estimated positive predictive values by adjusting their thresholds, then searches a grid for the threshold minimizing training cost-sensitive risk.The estimated positive predictive values are only approximately matched, and their empirical monotonicity is approximate.Source: §5 · p. 6
• Experiments report that FairBayes-DPP removes disparities and preserves accuracy; Table 1 reports accuracy and DPP for the true fair optimum and its logistic-regression estimator on a synthetic example.Source: §5 · p. 7, Table 1 · p. 7

6.1 Synthetic Data

• The synthetic study compares the learned fair classifier with the analytically derived fair Bayes-optimal classifier using logistic regression and a 0.001-spaced threshold grid.It uses 50,000 training and 5,000 test samples; the model is evaluated over 100 simulations.Source: §6.1 · p. 7
• On synthetic data, FairBayes-DPP closely tracks the fair Bayes-optimal classifier and controls test accuracy and DPP effectively.When group label prevalences differ substantially, it mitigates unconstrained-classifier disparity while preserving model accuracy.Source: §6.1 · p. 7

6.2 Empirical Data Analysis

• Adult and COMPAS use a 70/50/30 train-validation-test split, while CelebA experiments use a pretrained ResNet50 and validation-selected thresholds.The reported experiments repeat 100 times for Adult and COMPAS, and 10 times for CelebA.Source: §6.2 · p. 8
• For Adult and COMPAS, FairBayes-DPP has smaller disparity than unconstrained learning, with one-sided-test p-values of 3.90 × 10−4 and 3.09 × 10−8, respectively.The method achieves almost the same accuracy as the unconstrained classifier on these datasets.Source: §6.2 · p. 8
• On CelebA, FairBayes-DPP mitigates gender bias with almost no loss of accuracy across the evaluated attributes.The analysis includes 27 attributes meeting subgroup-sample prevalence bounds and identifies “Young” as violating Condition 4.1.Source: §6.2 · p. 9

7 Summary and Discussion

• When group performances vary moderately, all fair Bayes-optimal classifiers under predictive parity are group-wise thresholding rules, and a post-processing method estimates the optimal rule.The proposed method removes disparity in unconstrained classifiers while preserving similar test accuracy.Source: §7 · p. 9
• When the sufficient condition fails, a fair Bayes-optimal classifier under predictive parity may lead to within-group unfairness for the minority group.The authors caution that fairness measures require careful analysis before use because improper use may have unintended consequences.Source: §7 · p. 9

A Proof of Theorem 4.2

• For each group, the proof chooses a threshold and tie-breaking probability so the conditional success probability among positive predictions can match a target value.The function g_a(t, τ) denotes the probability of a positive label conditional on a randomized threshold prediction being positive in group a.Source: A Proof of Theorem 4.2 · p. 13
• The risk comparison establishes that a classifier not matching the constructed group-wise threshold rule has strictly greater cost-sensitive risk.The proof uses the sign of the threshold-based integrand and the equality condition for the risk bound.Source: A Proof of Theorem 4.2 · pp. 15–16
• The proof strategy replaces any fair classifier that is not a group-wise thresholding rule with a fair thresholding rule of smaller risk.The argument aims to show that at least one fair Bayes-optimal classifier is a group-wise thresholding rule.Source: A Proof of Theorem 4.2 · p. 16
• The construction handles three ranges of the shared positive predictive value relative to the cost parameter and groups’ base rates.For the intermediate range, the proof divides into cases according to whether the cost parameter or the maximum group base rate is larger.Source: A Proof of Theorem 4.2 · p. 17

B Proof of Theorem 4.3

• A constructed non-threshold classifier satisfies predictive parity and has lower cost-sensitive risk than any fair group-wise thresholding rule.This establishes that no fair Bayes-optimal classifier under predictive parity is a group-wise thresholding rule in the constructed setting.Source: B Proof of Theorem 4.3 · p. 19

C Fair and Unconstrained Bayes-optimal Classifiers of the Synthetic Model

• The synthetic model specifies group and label prevalences and generates features from group- and label-conditional bivariate Gaussian distributions.Its conditional label probabilities η_a(x) are given by Bayes’ rule using the corresponding Gaussian densities.Source: C Fair and Unconstrained Bayes-optimal… · p. 19
• In the synthetic model, the fair Bayes-optimal classifier uses a numerically computed threshold t⋆ and the corresponding threshold function T0(t⋆).Both T0(t) and the minimizing threshold t⋆ can be found numerically.Source: C Fair and Unconstrained Bayes-optimal… · p. 20

D Experimental Settings and More Simulation Results

• Adult and COMPAS use three-layer networks with 32-neuron hidden layers, while CelebA uses a pretrained ResNet-50.The experiments also use dropout with p = 0.5 and Adam with default parameters.Source: D Experimental Settings and More… · p. 20
• The experiments use dataset-specific training details summarized for three datasets in Table 3.Source: D Experimental Settings and More… · p. 20, Table 3 · p. 21

D.1 Synthetic Data

• Synthetic experiments vary sample size, minority-group proportion, cost parameters, and protected-attribute cardinality, with 100 repetitions per scenario.Source: §D.1 · p. 21

D.1.1 Sample Size

• As training size rises from 5,000 to 25,000, FairBayes-DPP has smaller disparity than unconstrained learning and improves as estimation error falls.Source: Figure 2 · p. 21, §D.1.1 · p. 21

D.1.2 Proportion of Minority Group

• As P(A = 0) increases, accuracy rises for both methods; at very small P(A = 0), η0 variability lowers unconstrained accuracy and destabilizes FairBayes-DPP.Source: §D.1.2 · p. 21, Figure 3 · p. 22

D.1.3 Cost Parameter

• Across c = 0.4–0.8, FairBayes-DPP mitigates disparity across a wide range of cost parameters.Source: Figure 4 · p. 22, §D.1.3 · p. 22

D.1.4 Multi-class Protected Attribute

• For |A| = 3 and |A| = 5, FairBayes-DPP preserves accuracy while mitigating bias in the multi-class synthetic experiments.Source: Figure 5 · p. 23, §D.1.4 · p. 23

D.2 CelebA Dataset

• Across the remaining 20 CelebA attributes, FairBayes-DPP mitigates gender bias effectively in most cases while preserving model accuracy.Source: §D.2 · p. 23, Table 5 · p. 24