Title: Size-adaptive Hypothesis Testing for Fairness
Venue: NeurIPS 2025
Accepted: yes
Keywords: Fairness

Fairness Metrics

Hypothesis Testing

Bayesian Statistics
Abstract: Determining whether an algorithmic decision-making system discriminates against a specific demographic typically involves comparing a single point estimate of a fairness metric against a predefined threshold. This practice is statistically brittle: it ignores sampling error and treats small demographic subgroups the same as large ones. The problem intensifies in intersectional analyses, where multiple sensitive attributes are considered jointly, giving rise to a larger number of smaller groups. As these groups become more granular, the data representing them becomes too sparse for reliable estimation, and fairness metrics yield excessively wide confidence intervals, precluding meaningful conclusions about potential unfair treatments.

In this paper, we introduce a unified, size-adaptive, hypothesis‑testing framework that turns fairness assessment into an evidence‑based statistical decision.
Our contribution is twofold.  (i) For sufficiently large subgroups, we prove a Central‑Limit result for the statistical parity difference, leading to analytic confidence intervals and a Wald test whose type‑I (false positive) error is guaranteed at level $\alpha$.  (ii) For the long tail of small intersectional groups, we derive a fully Bayesian Dirichlet–multinomial estimator; Monte-Carlo credible intervals are calibrated for any sample size and naturally converge to Wald intervals as more data becomes available. 
We validate our approach empirically on benchmark datasets, demonstrating how our tests provide interpretable, statistically rigorous decisions under varying degrees of data availability and intersectionality.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose an adaptative way for testing whether a fairness criterion holds. Instead of using a pre-defined threshold, a formal hypothesis test is run, which adapts naturally to the number of samples available in different protected groups. The main focus of the paper is on demographic (statistical) parity, although some extensions are possible by applications of the Delta-method. Specifically, for relatively large samples an asymptotically valid Wald test is proposed, while for small samples a Bayesian approach with a conjugate Dirichlet prior is used.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- (S1) The notation is clear, nice, and easy to follow,\n- (S2) The paper is well-written, and the concepts are presented clearly,\n- (S3) A formal hypothesis test approach is adopted, whereas common approaches may use a pre-defined threshold.\n\nWeaknesses:\n\nWhile acknowledging the positives listed above, the major drawback is that all of the theory appearing in the paper is really well-known and widely used in the statistics literature, so there is no need for new theoretical developments (Theorem 1 follows from a variant of the CLT, Theorem 2 by the Delta method). In addition, the adaptation of the testing to the context of fairness does not require a huge amount of translation. Therefore, this paper presents a nice technical note, but does not reach the level expected from a NeurIPS contribution. Since the basic setting of demographic parity is mastered, one obvious way to improve is to study a more ambitious problem, such as conditional or causal variants of parity, which would likely provide more significant theoretical and methodological challenges.\n\nQUESTIONS:\n- (Q1) is there a need for incorporating uncertainty in the constructed classifier f? In the current treatment, f seems to be treated as if it is not derived based on data? \n- (Q2) Is there a way of performing a similar test using bootstrap, in a non-parametric fashion?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a new framework to measure fairness based on hypothesis testing instead of comparing a point estimate of the fairness metric with a predefined threshold. They provide two tests to detect statistical parity unfairness. These tests consider the uncertainty and group sizes due to finite test sets. The first test utilizes the confidence intervals based on asymptotic normal approximation of disparity; this test is valid for large sample sizes and based on CLT. When the sample sizes are small for subgroups, the authors propose another test that utilizes Bayesian approach. They propose a Dirichlet prior on the estimated values for the statistical parity, then get samples from the posterior and build an empirical confidence interval. The authors also demonstrate how the methodology can be generalized for other fairness notions, such as equal opportunity.  The numerical experiments cover 4 real datasets, and the proposed method is compared against the fixed threshold rule and the scaled(by the probability of the subgroup) threshold rule.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The idea of using hypothesis testing for unfairness detection is novel; the framework unifies both frequentist and Bayesian approaches and switches between them based on the test data availability.\n- The authors provide empirical results on four real-world datasets. The proposed method avoids false alarms and detects unfairness that threshold-based rules miss.\n- The paper is well-structured and easy to follow. The mathematical derivations are complete. \n\nWeaknesses\n\n- The technical contributions of the paper are limited; the contribution of the paper is mostly proposing hypothesis testing for fairness detection.\n- It is not clear how the small subgroup sizes affect the first proposed test. The variance estimate would be inflated due to small subgroups, probably, but a clear explanation of this could be beneficial before introducing the second test.\n- For the Bayesian approach, the authors do not report on the sensitivity of the prior selection. It could be interesting to see priors informed by the training set frequencies. \n- The experiments section can have better baselines, such as other confidence interval generation methods, i.e., bootstrap intervals available in FairLearn. The extension for the other fairness notions is explained in the paper, but the experiments section only focuses on statistical parity.\n\nQUESTIONS:\nPlease see the weaknesses section.\n\nLIMITATIONS:\nYes, authors addressed the limitations and broader impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a framework for group-size adaptive hypothesis test (SAFT) for auditing a fair classifier. When sufficient samples are available, they propose using a Wald test to get construct a statistically consistent test, and when there is a possibility of very few samples from certain groups, they propose using a Bayesian test with a Dirichlet prior. The advantage over other works lies in the fact that, depending on the subgroup sizes, the proposed testing framework can dynamically adapt.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. Solid theoretical results that tackle the said problem very well. The empirical results work very well for SP at least.\n2. The Bayesian framework to tackle the small sample regime is a good idea.\n\nWeaknesses:\n\n1. Empirical results for other fairness metrics: The authors admitted in their limitations that this is a pending line of work. Sparse subgroups will have a larger effect on conditional parity metrics like Equal opportunity and Equalized odds, and hence would have been the perfect demonstration of the proposed size-adaptive framework.\n2. Dependence on minimum support $\\tilde{n}$: The authors fix $\\tilde{n}=30$ for switching between the Bayesian inference test and the Wald test. Perhaps an ablation study on the sensitivity of this minimum support parameter can help the reader.\n3. While I understand that it is hard for the authors to theoretically validate the effectiveness of the Bayesian inference setup, it will be interesting to check how close it gets to the true hypothesis, perhaps with a simpler data distribution (eg. Gaussian).\n\nQUESTIONS:\nMy questions are around the three limitations posed previously. \n1. Empirical results for other fairness metrics: This can help nail down the effectiveness and \"size-adaptiveness\".\n2. Dependence on minimum support $\\tilde{n}$\n3. Some justification for how accurate the Bayesian testing framework is.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nWhen we look at intersections of protected classes (i.e. using an intersectional lens) then the size of the intersectional group is smaller and sometimes too small to allow for reliable testing of fairness measures such as disparate impact. The authors wish to find the granularity/resolution at which a given data set can be assessed for fairness/unfairness.\n\nAs I understand it, the goal is to provide different levels of \"fairness\" to intersectional groups based on their size. This can allow more fine-grained control of type I/type II error when testing/auditing for unfairness.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n* Proposes a framework for taking into account sample size when testing/auditing for fairness in small subgroups.\n* Extensive experiments help illustrate the results.\n\n**Weaknesses:**\n\n* The presentation of the results obscures the main technical tool, which is a straightforward result about the asymptotic normality of statistics calculated from plug-in estimates of discrete random variables.\n* The promised \"principled methodology\" also contains some arbitrary parameter choices.\n\nQUESTIONS:\n* One challenge in fairness research is that there are so many notions of fairness, each with their embedded assumptions: a nice exposition of this was given a while ago by Arvind Narayanan in a tutorial [21 fairness definitions and their politics](https://www.youtube.com/watch?v=jIXIuYdnyyk). How many of these (beyond the 3 mentioned in the paper) fall under Theorem 2?\n\n* One of the stated contributions is that many fairness measures have \"pitfalls\" with respect to intersectionality. However, I think many of the referenced papers also point out such issues with prior fairness measures. The fact that any threshold test will have false positives and false negatives is pretty obvious, so what do the authors mean by this contribution?\n\n* (line 47) Is the right question whether we *should* rigorously guarantee fairness or whether we *can* rigorously guarantee fairness?\n\n* I found the discussion and framing of this a bit confrontational with respect to prior work. Of course, checking for unfairness is a kind of (hypothesis) test: one computes a test statistic and compares it to a threshold. They claim prior measures *require* a practitioner to set a global threshold without looking at the particulars of the data set. From Figure 4 in the submitted manuscript, it seems that a linear correction for group size on $\\gamma$-SP would allow for a fixed threshold.\n\n* My understanding is that all of the distributions under consideration are PMFs or conditional PMFs. The estimators are plug-in estimates based on the empirical joint PMF and (as shown in the appendix). Obviously, you have to compute the covariances etc. to get a full characterization of the asymptotics, but from my reading Theorem 2 showing asymptotic normality is basically a kind of textbook application of the delta method: am I missing something here?\n\n* Is it really \"rigorous\" to set thresholds based on the thresholds given by the asymptotics? This is the large sample test and is the general recipe used in statistical practice but the type I and type II error guarantees are then only approximate, right? \n\n* For the Bayesian setting, I do like the idea of incorporating prior information into the prior. Is it possible to do an experiment exploring the impact of prior information/bias?\n\n* I feel like the word \"rigorous\" in the paper does a lot of heavy lifting. In Algorithm 1, what is the \"rigorous\" justification for $\\tilde{n} = 30$? Similarly, what $K$ was used for the Bayesian method?\n\n* Apart from conforming to arbitrary norms for error tradeoffs, why select $\\alpha = 0.05$? Shouldn't this problem be looked at in more general Neyman-Pearson tradeoff sense?\n\n\n**Smaller comments:**\n\nThere are some other works on addressing intersectionality in ML + fairness: the authors may be interested in seeing how those fit in within the landscape of the literature review:\n\n* Ghosh et al, [Characterizing Intersectional Group Fairness with Worst-Case Comparisons](https://proceedings.mlr.press/v142/ghosh21a.html), 2021.\n* Foulds et al, [Differential Fairness: An Intersectional Framework for Fair AI](https://www.mdpi.com/1099-4300/25/4/660), 2023. (Maybe subsumes reference [12]?)\n* Himmelreich et al. [The Intersectionality Problem for Algorithmic Fairness](https://arxiv.org/pdf/2411.02569), 2024. (This was posted close to the submission deadline, however).\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}