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The Role of Uncertainty in Assessing the Fairness of Machine Learning Models

Francesca Panero, Ernst C. Wit, Marco Scutari

arXiv:2609.07959v1stat.MLcs.LGstat.ME

TL;DR

Fairness risk assessment requires accounting for uncertainty in model selection and estimation, but the literature has largely focused on point estimates. The paper discusses frequentist and Bayesian approaches to quantify uncertainty; its analyses show that a target unfairness level does not guarantee that level, so compliance is probabilistic.

  • Problem

    Fairness risk assessment requires uncertainty from model selection and parameter estimation, but the literature largely focuses on point estimates.

    Source: §1 · p. 2

  • Method

    The paper discusses frequentist bootstrap resampling and Bayesian hierarchical modelling as approaches to uncertainty quantification for fair machine learning.

    Source: §1 · p. 2, §4 · p. 19

  • Results

    A model constrained to a target unfairness level yields a distribution of unfairness levels rather than a guaranteed target, making compliance probabilistic; no simulated bootstrap method dominates in both regression settings.

    Source: §4 · pp. 20–21

  • Takeaways & Limitations

    To meet an unfairness threshold with high probability, set the target below it by a data-dependent amount that can be quantified.

    Source: §4 · p. 20

    The Bayesian proposal relies on frequentist estimates to select λ, though it could be extended to a stand-alone fully Bayesian model.

    Source: §4 · p. 21

Abstract

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Machine learning models are widely used in clinical applications, social media, law enforcement and critical infrastructure. Verifying whether their outputs are biased against disadvantaged groups or individuals is crucial to ensuring they are fair and allowing their use in such settings. A rigorous risk assessment of possible fairness violations requires quantifying the uncertainty associated with selecting and estimating such models. Yet, this is rarely done in the literature, which focuses on identifying a single model with a suitable trade-off between predictive accuracy and fairness. In this paper, we move beyond point estimation and discuss frequentist and Bayesian approaches to uncertainty quantification for fair machine learning, with practical examples and implications for simulated and real data.

1. Introduction

The introduction frames fairness as a risk-assessment problem: excluding all unfair model outputs may affect practical suitability, while uncertainty in model selection and estimation remains underexplored. It outlines frequentist and Bayesian approaches and discusses how uncertainty varies across fairness and processing choices.

1. Introduction

  • Fairness assessment identifies and seeks to eliminate algorithmic bias against disadvantaged groups, but unfair outputs cannot be excluded entirely without potentially affecting practical suitability.The paper frames fairness as a risk to assess rather than an outcome that can be guaranteed without trade-offs.Source: §1 · p. 1
  • Fairness risk assessment requires uncertainty quantification for model selection, parameter estimation, and data noise, yet the literature mainly produces point estimates.The paper presents bootstrap resampling and hierarchical modelling as representative frequentist and Bayesian approaches, not universal recommendations.Source: §1 · p. 2

1.1. Background and notation

  • In-processing formulates learning as loss minimization subject to a user-defined unfairness bound, trading predictive accuracy against fairness.The bound r is a user-defined unfairness level; the paper also describes an equivalent penalized-loss formulation.Source: §1.1 · pp. 3–4
  • Pre-processing transforms the data without access to the model, while post-processing selects a prediction transformation; both require tuning that adds to model-selection uncertainty.The approaches use tuning parameters to balance predictive accuracy and fairness.Source: §1.1 · p. 4

1.2. Implications of pre-, in- and post-processing approaches in uncertainty quantification

  • Predictive uncertainty can be decomposed into irreducible aleatoric uncertainty and epistemic components from model estimation, model selection, and sampling variability.For variance, the decomposition reduces to the law of total variance applied at each level.Source: §1.2 · pp. 4–5
  • Post-processing leaves model-selection and estimation uncertainty unquantified, while separate auxiliary tuning in pre- and post-processing makes overall calibration difficult.The paper therefore argues that accurate uncertainty quantification, and thus risk assessment, is more accurately done with in-processing.Source: §1.2 · pp. 5–6

2. Methods

The methods quantify fairness uncertainty with frequentist bootstrap procedures and Bayesian models that examine posterior distributions of parameters and fairness measures. They address parameter uncertainty and the risk of fairness violations while making explicit the assumptions and trade-offs of each approach.

2.1. Frequentist uncertainty quantification of fairness

  • When sensitive-attribute coefficient intervals include zero, the model is considered fair in the sense that those effects are not statistically significant; large samples still call for practical-significance checks.The farther an interval lies from zero, the stronger the associated sensitive-attribute effect.Source: §2.1 · p. 7
  • In penalized models, residual resampling loses its efficiency advantage as residuals become non-orthogonal to predictors, and the fairness penalty may produce artificially narrow confidence intervals.Residual resampling is more efficient than predictor resampling in non-penalized linear regression.Source: §2.1 · p. 7
  • Bootstrap methods construct confidence intervals for model parameters and assess the risk of fairness violations, including variability across training-validation splits and parameter estimates.The methods also assess the joint distribution of empirical fairness and predictive accuracy, or summarize risk using expected loss over validation sets.Source: §2.1 · pp. 6–8

2.2. Bayesian uncertainty quantification of fairness

  • The Bayesian approach studies posterior distributions of model parameters and fairness measures; residualizing predictors against sensitive attributes separates their contributions in the regression.The residualized predictors are the part of X uncorrelated with S, and the response is modeled using those predictors together with S.Source: §2.2 · p. 8
  • A ridge prior with penalty λ shrinks sensitive-attribute coefficients toward zero as λ increases, reducing model dependence on sensitive attributes.The prior for β is a Zellner g-prior, while the block structure gives mutually independent Gaussian priors.Source: §2.2 · pp. 8–9
  • The fairness constraint caps the share of fitted-predictor variance attributable to sensitive attributes; at r = 0, the predictor is uncorrelated with S, equivalent to statistical parity when (S, X) are normally distributed.Outside that distributional setting, the correlation-based fairness definition is weaker than statistical parity.Source: §2.2 · p. 9
  • The Bayesian construction extends to binary responses with a Bernoulli likelihood and uses NUTS for inference when the posterior is not available in closed form.The fairness target sets λ empirically; posterior distributions across values of r can inform the target fairness level and its confidence.Source: §2.2 · pp. 9–10

3. Results

The simulations show that bootstrap coverage and interval width vary by regression setting, while Bayesian posterior uncertainty can extend beyond fairness targets. Real-data analyses show that threshold compliance depends on the data and that intervals help interpret models but do not select the fairness target.

3.2. Frequentist: the bootstrap

  • Double bootstrap coverage is closer to nominal than nonparametric coverage for linear regression, while residual bootstrap coverage is exactly one across the considered settings.Nonparametric coverage is 0.89–0.94; double bootstrap reaches 0.97 at n ∈ {200, 500}.Source: §3.2 · pp. 11–12, Figure 1 · p. 11
  • In linear regression, double-bootstrap intervals are about 60% longer than the reference, whereas residual-bootstrap intervals are about 20 times longer and become more inflated as the target unfairness decreases.Nonparametric intervals have nearly the reference length; their undercoverage arises from location rather than width.Source: §3.2 · pp. 12–13, Figure 2 · p. 12
  • For logistic regression, nonparametric bootstrap is preferable on both coverage and interval length: its coverage is 0.94–0.96, while double bootstrap undercovers at 0.90–0.93 and gives longer intervals.Source: Figure 3 · p. 12, §3.2 · p. 13, Figure 4 · p. 13

3.1. Simulated data

  • Simulations use corr(Si, Xj) values of 0.1 and 0.3, targets r ∈ {0.05, 0.1}, and sample sizes n ∈ {100, 500}; responses are Gaussian for linear regression and Bernoulli for logistic regression.Source: §3.1 · pp. 10–11, §3.3 · p. 13

3.3. Bayesian: studies a posteriori

  • As correlation between sensitive and other predictors increases, posterior unfairness variance increases because the fairness constraint caps sensitive-attribute signal and more signal moves into residual variance.The authors hypothesize that the sensitive attributes increasingly proxy legitimate covariates as correlation grows.Source: §3.3 · p. 14
  • Bayesian fairness posteriors can reach less-fair values than requested with probability greater than 0.5, and higher correlation can place probability mass on values up to 50% more unfair than the target.Source: §3.3 · p. 15

3.4. Real data

  • In the real-data analyses, intervals distinguish neighboring fairness targets for LSAC but not COMPAS, and nonparametric, double-bootstrap, and Bayesian intervals can give different significance judgments.The intervals help interpret effects at a chosen target; they do not generally support selecting that target.Source: §3.4 · p. 17
  • The spread around a target is wider for COMPAS than LSAC, so the same target leaves different probabilities of exceeding a fairness threshold.To meet a threshold with high probability, the target must be set below it, with the required margin depending on the data.Source: §3.4 · p. 18

4. Conclusions

The paper argues that point estimates are insufficient for assessing fairness risk and presents frequentist and Bayesian uncertainty approaches for in-processing models. Its conclusions emphasize setting compliance targets with uncertainty in mind and accounting for uncertainty in model selection.

4. Conclusions

  • Fairness is commonly framed as constrained optimization yielding a single point estimate, which the paper argues is inadequate for risk assessment in sensitive settings.The conclusion situates risk assessment within legal frameworks requiring identification and containment of biases.Source: §4 · pp. 18–19
  • The paper illustrates one frequentist and one Bayesian in-processing approach: the former addresses typical procedure outcomes, while the latter addresses compliance of the observed model.The authors exclude post-processing because it cannot account for model-selection and parameter-estimation uncertainty, and pre-processing because it increases uncertainty.Source: §4 · p. 19
  • To meet a fairness threshold with high probability, set the target below it; LSAC and COMPAS analyses quantify the needed distance and examine fairness–accuracy uncertainty.The LSAC plot uses R2 and the COMPAS plot uses F1; both scale fairness by target r, while the COMPAS figure also displays intervals for sensitive attributes.Source: Figure 9 · p. 19, Figure 10 · p. 19, §4 · pp. 20–21
  • No uncertainty-quantification method dominated across the simulated settings, and fairness penalties shifted uncertainty from parameter estimation to model selection rather than reducing it.Double bootstrap was closest to nominal coverage for linear regression, while the first two methods reversed ranking for logistic regression.Source: §4 · p. 21
  • The Bayesian proposal is not fully Bayesian because it relies on frequentist estimates to select λ, and the approach assumes all paths are inadmissible in causal fairness settings.The authors describe the analyses as a preliminary step and identify uncertainty quantification in causal settings as future work.Source: §4 · p. 21

5. Data availability

Section 3.1 uses simulated data, while Section 3.4 uses open-source data available through the fairml R package or code repository.

5. Data availability

  • Section 3.1 uses simulated data; Section 3.4 uses open-source data available in the fairml R package or code repository, and the code is attached for future open-source release.Source: §5 · p. 21

8. Funding

F.P. acknowledges the Department of Informatics at Università della Svizzera Italiana for granting her a 2025 Visiting Lecturer position to work on the project.

8. Funding

  • F.P. thanks USI’s Department of Informatics for granting her a 2025 Visiting Lecturer position to work on the project.Source: §8 · p. 22
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