Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bayes-Optimal Fair Classification with Multiple Sensitive Features

Yi Yang, Yinghui Huang, Xiangyu Chang

TL;DR

This work addresses fairness in binary classification when multiple sensitive features are present by studying Bayes-optimal fair classifiers under approximate fairness, modeled via Mean Difference (MD) and Mean Ratio (MR). It proves that the Bayes-optimal fair classifier can be expressed as an instance-dependent threshold on the posterior $\eta(x)=P(Y=1|X=x)$ with corrections that are weighted sums of group-membership probabilities, applicable to both attribute-aware and attribute-blind settings and to composite notions like Equalized Odds. The authors provide two practical learning paradigms—in-processing, via a fair cost-sensitive objective, and post-processing, via a plug-in thresholding approach—that realize the Bayes-optimal fair classifier from finite data. Empirical results on Adult and COMPAS demonstrate favorable accuracy-fairness trade-offs under DP, EO, PE, and AP (including the new AP case) with MD and MR, highlighting the practical impact of enabling fair decisions when multiple sensitive attributes define intersectional groups.

Abstract

Existing theoretical work on Bayes-optimal fair classifiers usually considers a single (binary) sensitive feature. In practice, individuals are often defined by multiple sensitive features. In this paper, we characterize the Bayes-optimal fair classifier for multiple sensitive features under general approximate fairness measures, including mean difference and mean ratio. We show that these approximate measures for existing group fairness notions, including Demographic Parity, Equal Opportunity, Predictive Equality, and Accuracy Parity, are linear transformations of selection rates for specific groups defined by both labels and sensitive features. We then characterize that Bayes-optimal fair classifiers for multiple sensitive features become instance-dependent thresholding rules that rely on a weighted sum of these group membership probabilities. Our framework applies to both attribute-aware and attribute-blind settings and can accommodate composite fairness notions like Equalized Odds. Building on this, we propose two practical algorithms for Bayes-optimal fair classification via in-processing and post-processing. We show empirically that our methods compare favorably to existing methods.

Bayes-Optimal Fair Classification with Multiple Sensitive Features

TL;DR

This work addresses fairness in binary classification when multiple sensitive features are present by studying Bayes-optimal fair classifiers under approximate fairness, modeled via Mean Difference (MD) and Mean Ratio (MR). It proves that the Bayes-optimal fair classifier can be expressed as an instance-dependent threshold on the posterior with corrections that are weighted sums of group-membership probabilities, applicable to both attribute-aware and attribute-blind settings and to composite notions like Equalized Odds. The authors provide two practical learning paradigms—in-processing, via a fair cost-sensitive objective, and post-processing, via a plug-in thresholding approach—that realize the Bayes-optimal fair classifier from finite data. Empirical results on Adult and COMPAS demonstrate favorable accuracy-fairness trade-offs under DP, EO, PE, and AP (including the new AP case) with MD and MR, highlighting the practical impact of enabling fair decisions when multiple sensitive attributes define intersectional groups.

Abstract

Existing theoretical work on Bayes-optimal fair classifiers usually considers a single (binary) sensitive feature. In practice, individuals are often defined by multiple sensitive features. In this paper, we characterize the Bayes-optimal fair classifier for multiple sensitive features under general approximate fairness measures, including mean difference and mean ratio. We show that these approximate measures for existing group fairness notions, including Demographic Parity, Equal Opportunity, Predictive Equality, and Accuracy Parity, are linear transformations of selection rates for specific groups defined by both labels and sensitive features. We then characterize that Bayes-optimal fair classifiers for multiple sensitive features become instance-dependent thresholding rules that rely on a weighted sum of these group membership probabilities. Our framework applies to both attribute-aware and attribute-blind settings and can accommodate composite fairness notions like Equalized Odds. Building on this, we propose two practical algorithms for Bayes-optimal fair classification via in-processing and post-processing. We show empirically that our methods compare favorably to existing methods.
Paper Structure (44 sections, 16 theorems, 85 equations, 8 figures, 7 tables, 2 algorithms)

This paper contains 44 sections, 16 theorems, 85 equations, 8 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Binary Classification
  4. Fairness-Aware Learning in Binary Classification
  5. General Approximate Fairness Measures
  6. Mean Difference
  7. Mean Ratio
  8. Bayes-Optimal Fair Classifiers
  9. Mean Difference
  10. Mean Ratio
  11. Algorithms
  12. In-Processing-Based Bayes-Optimal Fair Classification
  13. Post-Processing-Based Bayes-Optimal Fair Classification
  14. Experiments
  15. Conclusion
  16. ...and 29 more sections

Key Result

Lemma 1

For any randomized classifier $f$, any $\delta\in[0,1]$, and the group fairness notions in Table tab:genernal_fair_measure, $\mathrm{MD}(f)\leq \delta \Leftrightarrow R^\mathrm{MD}_{m}(f) \in [-\delta, \delta]$ for all $m\in [M]$, where Here, values of $a_{m}$, $b_{m}^{y}$, and $c_{m}^{y}$ depend on the chosen fairness notion and are as defined in Table tab:genernal_fair_measure.

Figures (8)

  • Figure 1: Trade-offs between accuracy and fairness using MD. The prefix 'Multi-' represents the case of multiple sensitive features, and '(aware)' in classifier names indicates the attribute-aware setting. (uncal): uncalibrated; (cal): calibrated.
  • Figure 2: Trade-offs between accuracy and fairness using MD. The prefix 'Multi-' represents the case of multiple sensitive features, and '(aware)' in classifier names indicates the attribute-aware setting. (uncal): uncalibrated; (cal): calibrated.
  • Figure 3: Trade-offs between accuracy and fairness under PE and AP on two datasets. The '(aware)' in the name of classifiers indicates the attribute-aware setting.
  • Figure 4: Trade-offs between accuracy and fairness on COMPAS, using MR measure. The '(aware)' in the name of classifiers indicates the attribute-aware setting.
  • Figure 5: Trade-offs between accuracy and fairness on Adult, using MR measure. The '(aware)' in the name of classifiers indicates the attribute-aware setting.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Definition 1: Cost-Sensitive Risk
  • Definition 2: Demographic Parity (DP)
  • Definition 3: Equal Opportunity (EO)
  • Definition 4: Predictive Equality (PE)
  • Definition 5: Accuracy Parity (AP)
  • Definition 6: Mean Difference
  • Lemma 1
  • Definition 7: Mean Ratio
  • Lemma 2
  • Lemma 3
  • ...and 25 more