# algorithmic_fairness_with_monotone_likelihood_ratios__275fa019.pdf Published in Transactions on Machine Learning Research (07/2025) Algorithmic fairness with monotone likelihood ratios Wes Camp wesacamp@gmail.com Optum Reviewed on Open Review: https: // openreview. net/ forum? id= mto Wa0g IKy We show that inequalities of many commonly used fairness metrics (true/false positive/negative rates, predicted positive/negative rates, and positive/negative predictive values) are guaranteed for groups with different outcome rates under a monotonically calibrated model whose risk distributions have a monotone likelihood ratio, extending existing impossibility results. We further provide lower bounds on the FNR/FPR disparities and PPR/PNR disparities in the same setting, showing that either the FNR disparity or FPR disparity is at least as large as the positive outcome rate disparity (for FNR disparity) or negative outcome rate disparity (for FPR disparity), and either the PPR disparity or PNR disparity is at least as large as the positive outcome rate disparity (for PPR disparity) or negative outcome rate disparity (for PNR disparity). While incompatibilities of some combinations of these metrics have been demonstrated previously, we are unaware of any work that has demonstrated direct incompatibility of calibration with these individual equalities, equivalence of these inequalities, or lower bounds for the disparity in these values under distributional assumptions about a model s predictions. 1 Introduction In 2016, Pro Publica published an analysis (Angwin et al., 2016) of the COMPAS recidivism prediction algorithm, which assesses a defendant s risk of committing another crime and is used to assist judges in making pre-trial release and sentencing decisions. Using publicly available data on the algorithm s predictions and outcomes in Broward County, Florida, they determined that the false positive rate for Black defendants was significantly higher than for White defendants, i.e. a non-recidivating Black defendant was almost twice as likely to be asssessed as high-risk than a non-recidivating White defendant. The false negative rate for White defendants was also significantly higher than for Black defendants, meaning a recidivating White defendant was much more likely to be incorrectly labeled as low risk than a recidivating Black defendant. Based on these results, they concluded that the algorithm was biased against Black defendants. Northpointe, the developer of the COMPAS software, rebutted that the algorithm is fair because it satisfies predictive parity (equal positive predictive values) across the two groups (Dieterich et al., 2016). Shortly after, both Chouldechova (2017) and Kleinberg et al. (2017) established that these notions of fairness (equal false positive/negative rates and predictive parity) are impossible to satisfy simultaneously, except in trivial cases. Kleinberg et al. (2017) further prove the impossibility of well-calibration (scores equaling the outcome probabilities across groups) and balance for the positive/negative class (equality of average scores for members in the positive/negative class). Pleiss et al. (2017) demonstrated additional conflict between error rate equality and well-calibration by showing that well-calibration is compatible with only a single error rate constraint, but achieving this compatibility essentially requires randomizing predictions from an existing classifier, an intuitively disturbing procedure. While well-calibration is clearly a desirable condition for both the fairness and effectiveness of such an algorithm (Pleiss et al., 2017; Corbett-Davies et al., 2023), the same is not necessarily true of false positive/negative rate equality. Many authors have identified both technical and social issues resulting from reliance on these metrics (Corbett-Davies et al., 2023; Chohlas-Wood et al., 2023; Pierson, 2024), including the problem of infra-marginality (Ayres, 2002; Simoiu et al., 2017), the sensitivity of these metrics to Published in Transactions on Machine Learning Research (07/2025) model behavior and population distributions far away from the decision margin. Nonetheless, they are still commonly relied upon by machine learning practitioners for detecting potential bias in models. For example, many popular algorithmic fairness toolkits, including Microsoft s Fairlearn (Bird et al., 2020), IBM s AI Fairness 360 (Bellamy et al., 2019), and Aequitas (Saleiro et al., 2019), calculate ratios of these metrics across groups as part of their fairness reports, with some identifying the ratios as the disparity in the corresponding metric. Some of these further instruct users that these disparities and/or the disparities in predicted positive/negative rates must be between 80% and 125% in order for the model to be fair, a range derived from disparate impact law in a process that has been deemed epistemic trespassing (Watkins & Chen, 2024). If the toolkit or end user determines that a given disparity between two groups is sufficiently large, mitigation algorithms (such as those described by Agarwal et al. (2018) and Hardt et al. (2016)) are provided that can adjust the model s predictions or choose group-specific thresholds to achieve a specified level of parity, at the cost of (potentially substantial) performance degradation at both the population and group level (Pfohl et al., 2021). When an algorithm satisfies calibration across two groups (where predictions correspond to equal outcome rates across groups, but not necessarily the exact outcome probabilities), differences in these metrics are direct consequences of differences in group risk distributions under the algorithm. In what follows, we examine scenarios where these two risk distributions satisfy the monotone likelihood ratio property (MLRP), a frequently considered property in statistical problems that is satisfied by many commonly encountered families of distributions, including the exponential, binomial, Poisson, or normal with a fixed σ (Karlin & Rubin, 1956). Any two beta or gamma distributions that cross exactly once also satisfy the MLRP (Gaebler & Goel, 2025), and the property is preserved under monotone transformations (e.g. the above statements also hold for the corresponding log-families) and sums, if the distributions are log-concave (see Shaked & Shanthikumar (2007) for details of these and other properties). Gaebler & Goel (2025) considered distributions satisfying the MLRP in the context of testing outcomes for fairness, and they provide empirical evidence that the property holds in domains including recidivism, loan approvals, police searches, and law school admissions, by demonstrating that predictive models developed on data from those domains have group risk distributions satisfying the property. Chiang et al. (2025) proposed a method for developing disease progression models that account for systemic disparities, using the MLRP to reason about the effects of differences in initial severity, rates of progression, or visit frequency across groups on model outcomes. The property has also been studied in contexts such as measuring treatment effects in randomized control trials (Chemla & Hennessy, 2019) and identifying racial disparities in police searches (Anwar & Fang, 2006; Feigenberg & Miller, 2022). While the MLRP is often assumed as a property of the underlying risk distributions, an assumption that is difficult or impossible to verify in many situations, our results only require that the risk distributions under the examined model satisfy the property. This has the advantages of being both easily verifiable and at least equally reasonable, as a sufficiently appropriate model will approximate the true risk distributions. The contributions of this paper are as follows: we introduce a slightly stronger version of calibration (but still significantly weaker than well-calibration) called monotonic calibration, which requires increasing model predictions to correspond to increasing outcome probabilities, but not the exact outcome probabilities as required by well-calibration. We then show that if a model is monotonically calibrated for two groups and the risk distributions of the groups under the model satisfy the MLRP, then for any non-trivial decision threshold, the group with the higher outcome rate will have: a larger predicted positive rate (equivalently, a smaller predicted negative rate), a larger false positive rate and smaller false negative rate, a larger positive predictive value (unless the likelihood ratio or outcome probabilities are constant above the decision threshold, in which case the PPVs are equal), and a smaller negative predictive value (unless the likelihood ratio or outcome probabilities are constant below the decision threshold, in which case the NPVs are equal). We further show that either: Published in Transactions on Machine Learning Research (07/2025) the disparity in predicted positive rates is greater than the disparity in positive outcome rates, or the disparity in predicted negative rates is greater than the disparity in negative outcome rates and also that either the disparity in false negative rates is greater than the disparity in positive outcome rates, or the disparity in false positive rates is greater than the disparity in negative outcome rates. In particular, for many commonly encountered distributions of model predictions, calibration is incompatible with equality of any of these metrics when outcome rates are unequal, and even approximate equalized odds (equal false negative and false positive rates) and approximate demographic parity cannot be satisfied between groups with large differences in outcome rates. We conclude with an example of how these results guarantee large disparities in the error rates studied in the COMPAS dataset, and discuss the dangerous consequences of reliance on these metrics for detecting bias. 2 Main Results In what follows we discuss a model M intended to predict the probability of a binary outcome, but we do not require that M take values only in [0, 1], so M may be a more general risk score (as in the COMPAS example). Definition 1. Let M be a model defined on groups P and Q and used to predict the probability of a binary outcome O(r) : P Q {0, 1} for r P Q. M is calibrated for groups P and Q if for each predicted value x Pr(O(pi) = 1 | pi P, M(pi) = x) = Pr(O(qi) = 1 | qi Q, M(qi) = x) and M is well-calibrated for P and Q if for each predicted value x Pr(O(pi) = 1 | pi P, M(pi) = x) = Pr(O(qi) = 1 | qi Q, M(qi) = x) = x. The term calibration is often used to mean well-calibration as given above. We distinguish between the two properties here because well-calibration is not necessary for the results that follow; however, we do require a slightly stronger version of calibration, that ensures that increasing predictions correspond to increasing outcome rates. This avoids degenerate cases where predictions do not correspond to outcomes in any natural way. Such cases rarely occur in practice, and most models satisfying (not necessarily well-)calibration for groups will also satisfy this stronger property. Definition 2. Let M be a model defined on groups P and Q and used to predict the probability of a binary outcome O(r) : P Q {0, 1} for r P Q. M is monotonically calibrated for groups P and Q if it is calibrated for those groups, and the function c : Range(M) [0, 1] given by c(x) = Pr(O(pi) = 1 | pi P, M(pi) = x) = Pr(O(qi) = 1 | qi Q, M(qi) = x) is increasing and not almost everywhere constant. For what follows we treat a model M as a real-valued random variable on a probability space P Q, and assume the cumulative distribution functions FP (x) = Pr(M(pi) x | pi P) and FQ(x) = Pr(M(qi) x | qi Q) are absolutely continuous with associated density functions p(x) and q(x). Definition 3. Two probability density functions p(x) and q(x) satisfy the monotone likelihood ratio property (MLRP) with p dominating, denoted p lr q, if for almost every pair x0 < x1, p(x0) q(x0) p(x1) q(x1). We denote p lr q if p lr q but q lr p, or equivalently if p lr q and p(x) q(x) is not constant almost everywhere. Note that if p lr q, then p first-order stochastically dominates q. See Shaked & Shanthikumar (2007) for more general properties of distributions satisfying the MLRP. Published in Transactions on Machine Learning Research (07/2025) Lemma 1. Let p(x) and q(x) be the probabilty density functions of the predictions of a model M on groups P and Q, and let O(P) = Pr(O(pi) = 1 | pi P) and O(Q) = Pr(O(qi) = 1 | qi Q) denote the binary outcome rates for groups P and Q. If M is monotonically calibrated for groups P and Q (via function c(x)) and p and q satisfy the MLRP (with either p lr q or q lr p) then O(P) > O(Q) iff p lr q. Proof. ( ) Letting c0 = infx c(x) 0 and using that c(x) and p(x)/q(x) are increasing and not almost everywhere constant, we must have both c(x) > c0 and p(x) > q(x) on some set of non-zero measure, so we have O(P) O(Q) = Z c(x)(p(x) q(x))dx > Z c0(p(x) q(x))dx = 0. ( ) From the proof of the direction above, if q lr p then O(Q) > O(P), so we must have that p lr q. If p(x)/q(x) is constant almost everywhere then using that Z p(x) = Z q(x) = 1 we obtain O(P) = Z c(x)p(x) = Z c(x)q(x) = O(Q), thus p lr q. Definition 4. The predicted positive rate PPRM,t,P of a model M with decision threshold t (so that a prediction less than t is considered a predicted negative, and a prediction at least t is considered a predicted positive) on a group P is the proportion of P with predicted value at least t. The predicted negative rate PNRM,t,P = 1 PPRM,t,P is the proportion of P with predicted value less than t. The predicted positive rate defined above is sometimes called selection rate, admission rate, or various other terms, depending on the application of the model under discussion. Equality of these rates across groups is also know by various terms including demographic parity, group fairness, and statistical parity. The following lemma demonstrates that inequality of these rates is an immediate consequence of (strict dominance under) the MLRP, without any assumptions about the calibration of the model. Lemma 2. Let p(x) and q(x) be the probabilty density functions of the predictions of a model M on two groups P and Q. Let t be a decision threshold for the predictions of M. If p lr q then PPRM,t,Q PPRM,t,P (with equality only if both are 0 or 1) and PNRM,t,P PNRM,t,Q (with equality only if both are 0 or 1). Proof. The desired inequality is: PPRM,t,Q = Z x t q(x)dx Z x t p(x)dx = PPRM,t,P which follows directly from the first-order stochastic dominance of p over q. To obtain strictness of the inequality when both are not 0 or 1, note that if R x t q(x)dx = R x t p(x)dx, integrating both sides of the MLRP inequality p(x0)q(x1) p(x1)q(x0) for x0 t x1 over the region x1 t gives q(x) p(x) on x < t. But since Z q(x)dx = Z p(x)dx = 1, we have Z x t1 & p(t)q(t1) p(t1)q(t))} has full measure and Z x 0. But using the MLRP at t1 x q almost everywhere in some neighborhood of t (else p(x) q(x) for all x < t from the MLRP). Similarly, whenever Z x t (1 c(x))p(x)dx Z x t (1 c(x))q(x)dx, FPRM,t,P FPRM,t,Q = x t(1 c(x))p(x)dx R x(1 c(x))q(x)dx R x t(1 c(x))q(x)dx R x(1 c(x))p(x)dx 1 O(Q) So if the second inequality in the lemma does not hold, Z x t (1 c(x))p(x)dx < Z x t (1 c(x))q(x)dx, giving q > p almost everywhere in some neighborhood of t. Thus at least one of the inequalities holds. Both lower bounds in Lemma 5 can be reached simultaneously in the discrete case if p(x) and q(x) disagree only when c(x) {0, 1}, meaning that these bounds are also tight in general. Intuitively, the FNR inequality will fail to hold as t becomes sufficiently large and FNR values approach 1, and the FPR inequality will fail to hold as t becomes sufficiently small and FPR values approach 1. From the first line in the proof, the ratio of FNRs is x x0 and p(x1)q(x0) q(x1)p(x0) 0 for almost every x1 < x0. Viewing the integral as a nonnegative and nonpositive component across these two regions: x1>x0 t c(x1)(p(x1)q(x0) q(x1)p(x0))dx0dx1+ Z x0>x1 c(x1)(p(x1)q(x0) q(x1)p(x0))dx0dx1 and interchanging x1 and x0 in the second integral we obtain x1>x0 t c(x1)(p(x1)q(x0) q(x1)p(x0))dx0dx1 Z x1>x0 t c(x0)(p(x1)q(x0) q(x1)p(x0))dx0dx1 which follows immediately using that c(x) is increasing and nonconstant. The NPV inequality follows in an identical fashion. The following theorem unifies the lemmas proven previously, showing that strict inequality in outcome rates or any of the examined metrics implies all of the inequalities or pairs of inequalities given by the earlier lemmas. Theorem 1. Let p(x) and q(x) be the probabilty density functions of the predictions of a model M on two groups P and Q. Let t be a decision threshold for the predictions of M where the ratios in (7)-(10) all exist. If M is monotonically calibrated for groups P and Q and p and q satisfy the MLRP (with either p lr q or q lr p), then any of (1)-(4) or strict inequality in (5) or (6) implies all of (1)-(6): 1. O(Q) < O(P) 2. PPRM,t,Q < PPRM,t,P (equivalently PNRM,t,P < PNRM,t,Q) 3. FNRM,t,P < FNRM,t,Q (equivalently TPRM,t,Q < TPRM,t,P ) Published in Transactions on Machine Learning Research (07/2025) 4. FPRM,t,Q < FPRM,t,P (equivalently TNRM,t,P < TNRM,t,Q) 5. PPVM,t,Q PPVM,t,P 6. NPVM,t,P NPVM,t,Q. Any of (1)-(4) or strict inequality in (5) or (6) further implies at least one of (7) or (8), and at least one of (9) and (10): 7. PPRM,t,P PPRM,t,Q O(P) 8. PNRM,t,Q PNRM,t,P 1 O(Q) 9. FNRM,t,Q FNRM,t,P O(P) 10. FPRM,t,P FPRM,t,Q 1 O(Q) Proof. Inequality in any of (2)-(6) implies that p and q are not equal almost everywhere, and the direction of the inequality then implies that p lr q. This then implies (1)-(6) from Lemmas 1, 2, 4 and 6, and further implies at least one of (7) and (8) from Lemma 3 and at least one of (9) and (10) from Lemma 5. 3 COMPAS Analysis 1 2 3 4 5 6 7 8 9 10 COMPAS Decile Risk Score Recidivism Rate Figure 1: Calibration of the COMPAS risk score Using Pro Publica s COMPAS data (Larson et al., 2016), we observe first that the algorithm is approximately calibrated for the groups of Black and White defendants (Figure 1), and approximately monotonically calibrated for the two groups as well. In addition, viewing the histogram of risk scores across races (Figure 2), we would expect that the distributions of risk scores of these two groups would satisfy the MLRP. Letting p(x) Published in Transactions on Machine Learning Research (07/2025) Black White 1 2 3 4 5 6 7 8 9 10 COMPAS Decile Risk Score Figure 2: Histogram of the COMPAS risk scores by race 1 2 3 4 5 6 7 8 9 10 COMPAS Decile Risk Score Figure 3: Ratio of likelihood functions for the COMPAS risk scores 1 2 3 4 5 6 7 8 9 10 COMPAS Decile Risk Score FNR(W)/FNR(B) FPR(B)/FPR(W) (1 O(W))/(1 O(B)) PPR(B)/PPR(W) PNR(W)/PNR(B) Figure 4: FPR, FNR, PPR, and PNR disparities across different decision thresholds for the COMPAS risk scores denote the pdf of risk scores for Black defendants and q(x) denote the pdf of risk scores for White defendants, we confirm that p(x)/q(x) is increasing as required (Figure 3). Figure 4 provides the ratios compared in Lemmas 3 and 5 across the possible decision thresholds. The Pro Publica analysis used decile scores of 5 or higher (corresponding to Medium or High risk) as a decision threshold, and because p(5) and q(5) are approximately equal, from the proof of Lemma 5, we can expect both inequalities from Lemma 5 to hold, instead of just one of them. We observe that both inequalities from Lemma 3 hold at this threshold as well. If significant disparities in false negative/positive rates are guaranteed due to the shapes of the risk distributions of White and Black defendants, is it possible that these differences in distribution, instead of the behavior of the model itself, provide evidence of systemic unfairness? Consider a hypothetical county Published in Transactions on Machine Learning Research (07/2025) with identical distributions of Black and White defendants found in the COMPAS data, except where Black residents are far more likely to be arrested for minor offenses, resulting in the addition of many more lower-risk Black defendants and approximately equalizing the distributions of risk, and therefore the false positive/negative rates, for the resulting Black and White defendants (Corbett-Davies & Goel, 2018). If additionally, White residents are far more likely to not be arrested for minor offenses, we could see the distributions in Figure 2 flip; as a result, in this hypothetical county, the COMPAS algorithm would have a far higher false positive rate for White defendants than for Black defendants. But this would be due to explicit systemic unfairness changing the relative distributions of lowest-risk defendants, where the model s predictions likely have no meaningful impact on the resulting pre-trial decisions, an example of the problem of infra-marginality (Ayres, 2002; Simoiu et al., 2017). Similar results arise in other contexts, such as the loan approval data examined by Gaebler & Goel (2025). They found that, for a well-calibrated model predicting loan repayment, risk distributions for Black and White borrowers satisfy the MLRP, which would, by Theorem 1, guarantee a higher false negative rate (i.e. more denied loans for applicants who would actually repay) and lower false positive rate (i.e. fewer approved loans for applicants who would not repay) for the group with the lowest repayment rate. A bank that explicitly discriminates against Black borrowers by holding them to a much higher standard for acceptance would see a significantly higher repayment rate for those borrowers, and fail the traditional outcome test for discrimination. However, in a setting where data on false negatives is available (e.g. collected from other banks who granted loans to rejected applicants), the model predicting repayment evaluated on that bank s outcomes would have a higher false negative rate for White borrowers, leading one to incorrectly believe the model and/or bank is somehow unfair to White borrowers. Reliance on false positive/negative rates to reveal unfairness in an algorithm or underlying institution can mask, or even further exacerbate, systemic bias. 4 Conclusion We have shown that, in many cases, inequalities of commonly used algorithmic fairness metrics are immediate consequences of differences in outcome rates between groups, and large disparities in at least some of these metrics are guaranteed for groups with large differences in outcome rates. We emphasize that these results should not be the basis for testing group distributions for the MLRP prior to comparing false positive/negative rates or used to help set expected ranges for the disparities in these metrics. Nor should they be viewed as justification for not assessing algorithms for appropriate performance or potential harms across groups, in the numerous ways such harms can arise (Suresh & Guttag, 2021). Instead, they should be taken as further evidence that the specific metrics discussed here, as well as many other metrics commonly used to assess model performance, are highly sensitive to differences in distribution. In contexts where such differences are expected, the ability of these metrics to quantify unfairness seems extremely limited, if not non-existent. While it is straightforward to understand how enforcing demographic parity between groups ignores that different groups may have different outcomes, the same clarity does not appear to exist for false positive/negative rate parity, even though disparities in these metrics are very often due to the same phenomena. Attempts to enforce parity of these metrics will result in sub-optimal decision-making which may harm members of all groups involved (Corbett-Davies et al., 2023), and may actually widen existing disparities. For example, comparing the predictions of a model on two groups under the assumptions of Theorem 1, a group with a significantly lower outcome rate will necessarily have a significantly higher false negative rate for a sufficiently low decision threshold. If the model s intended use is to offer, say, a screening for those at highest risk of a certain medical condition, and the model designer determines that false negative rate parity is a necessary goal and mitigates the disparity in the optimal model, the group at highest risk of the condition will be most negatively impacted by the change to the model (i.e most under-screened relative to their actual need). The determination that such a model is always unfair to the lowest-risk group, and increasingly unfair as the risk of this group decreases, is clearly inconsistent with any reasonable notion of fairness. In the presence of sufficient calibration, disparities in these metrics are direct consequences of differences in group distributions of risk, which are most commonly symptoms of actual differences between groups. Even when these distributional differences are the direct result of systemic inequities, the solution to these Published in Transactions on Machine Learning Research (07/2025) inequities is not to mitigate an error rate difference via intentional mis-calibration or enforce metric parity at the cost of model efficacy (Pfohl et al., 2021). 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