Algorithmic fairness with monotone likelihood ratios
Authors: Wes Camp
Venue: Accepted by TMLR ()

Abstract: 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.

Published in Transactions on Machine Learning Research (07/2025)
Algorithmic fairness with monotone likelihood ratios
Wes Camp
mesuccamppmmaillccom com
Optum
Reviewed on OpenReview: https: // openreview. net/ forum? id= Id=mtoWaOgIKy
Abstract
We show that inequalities of many commonly used fairness metrics (true/false posi- tive/ne negative rates, predicted positive/negatinn negative rates, and positive/negative ues) are guaranteed for groups with different outcome rates under a monotonically calibrated model whose risk distributions have monotone likelihood ratio, extending existing impossi- bility results. We further provide lower bounds on the FNR/FPF disparitics and PPR/ PPR/PNR /NNR disparities in the same setting, showing that either the FNR disparity or FPR disparity al least as large 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 met- rics have been demonstrated previously, we are unaware of any work that has demonstrated direct incompatibility of calibration with these individual equalitics, equivalence of these equalities, lower bounds for the disparity in these values under distributional assumptions about model's predictions.
negative predictive val-
1 Introduction
In 2016, ProPublica 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 assist judges in making pre-trial release and sentencing decisions. Using publicly available data the algorithm's '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, mcaning 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 satisfics 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 Dntiiivee 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 ansitive/negattinn class (equality of average scores for members in the positive/negative /ngatiiv class). Plciss et al. (2017) demonstratod additional conflict between error rate equality and well-calibration by showing that well-calibration is compatible with only 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 pos- itive/ negative rate oquality. Many authors have identified both technical and social issucs resulting from reliance on these metrics (Corbett-Davics al., 2023; Chohlas- Wood et al., 2023; Pierson, 2024), including "the problem of infra-margen "arginatriinaltty" (Ayres, 2002; Simoiu et al., 2017), the sensitivity of these metrics to

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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 exam- ple, many popular algorithmic fairness toolkits, including Microsoft's ' Pairlearn (Bird et al., 2020), IBM's AI Fairness 360 (Bellamy et al., 2019), and Acquitas (Salciro et al., 2019), calculate ratios of these met- rics across groups as part of their fairness reports, with some identifying the ratios as the disparily in the corresponding metric. Some of these further instruct users that these disparities and/or the disparities predicted wositive/negative rates must be between 80% and 125% in order for the model to be fair, range derived from disparate impact law in process that has been deemed "epistemic trespassing" (Watkins & Chen, 2024). Tf 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 ex- amine 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 a (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-famílies) and sums, if the distributions are log-concave (sce Shaked & Shanthikumar (2007) for details of these and other properties). Gaebler & Goel (2025) considered distri- butions 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 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, visit frequency across groups 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 distri- butions, 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 a model is monotonically calibrated for two groups and the risk distributions of the groups under the model satisfy the MIRP, then for any non-trivial dccision threshold, the group with the higher outcome rate will have:
larger predicted positive rate (equivalently, smaller predicted negative rate),
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:
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the disparity in predicted positive rates is greater than the disparity in positive outcome rates,
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 uequul, and even approximate equalized odds (equal false negative and false positíve rates) and approximate demographic parity cannot be satisfied
between groups with large diffcrences in outcome rates. We conclude with an example how these results guaranlee large disparilies in the error rales studied the COMPAS dataset, and discuss the dangerous consequences of reliance on these metrics for detecting bias.
2
Main Results
In what follows discuss model M intended to predict the probability of binary outcome, but 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 model defined on groups P and Q and used to predict the probability of a binary outcome O(r): PUQ + (0, }} for E PUQ. M is calibrated for groups P and Q if for each predicted value Pr(O(p:) pt E
x
x) Pr(O(g)) = 1 q E M()) x)
and M is well-caliben cell-calibrated for P and Q if for each predicted value x
Pr(O(p.) pt E P,P( x) = Pr(O(u)) | a Q, M(qi) = = =
The lerm 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 incrcasing outcome rates. This avoids degenerale cases where predictions do not correspond to outcomes in any natural way. Such cases rarely occur in practice, and most models satisfying (not necessarily vell-Jccaibration for groups will also satisfy this stronger property.
Definition 2. Let M be model defined on groups P and Q and used to predict the probability of a binary outcome 0(r): : P UUQ + 1) } for r E PUQ. M is monotonically calibrated for groups P and Q if it calibrated for those groups, and the function e: Range(M) - [o, 1] given by
c(x) Pr(O(p.)
P P. P(M( x) Pr(O(g)) = 1 1. E M( x)
is increasing and not almost everywhere constant.
For what follows we treat a model M as real-valued random variable on probability space PUQ, and assume the cumulative distribution functions Fr(x) = Pr(M(p:) pt E P) and Fo(x)) Pr(M(g)) (MM()))((M(()) S 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
p(zo) < p(x1) We denote
(MLRP) with dominating, denoted p 2r 9, if for almost every pair xo < Iı, ((x0) ((x1)
>r q if b but q Eir or cquivalently p Eir and p(x) is not constant almost everywhere. q(x)
Note that if p Zir g. then p first-order stochastically dominates g. See Shaked & Shanthikumar (2007) for more general properties of distríbutions satisfying the MIRP.
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Lemma 1.
Let p(x) and q(x) be the probabilty density functions of the predictions of a model M groups P and Q; and let O(P) = Pr(O(p:) = pi E P) and O(Q) Pr(O(g)) = qa E Q) demote the binary outcome rates for groups P and Q. If M is monotonically calibrated for groups P and Q (via function c(z)) and p and q satisfy the MLRP (with either p Eir 9 or q Eir p) then O(P) > ((Q) iff p > q.
Proof. (+) Letting Co = =inf, c(x) 2 0 and using that c(x) and p(z) ((()) are increasing and not almost everywhere constant, we must have both c(a) > cn and p(x) > q(x) on some set of non-zero measure, so we have O(P) ((Q) c(x)(((x) (z)))dx> > co(p(z) ((x))dr 0.
(+) From the proof of the direction above, if g >o p then O(Q) > O(P), so we must have that Zir q- If p(x)/a(z) is constant almost everywhere then using that / p(x) = /a(x) 1 we obtain
O(P) = = do))p()) = f (()))(()==(()))) thus p 1r q-
Definition 4. The predicted posilive rule PPRMM, of model M with decision threshold L (so that prediction less than t is considered a predicted negative, and prediction at least t is considered predicted positive) on group P is the proportion of P with predicted value at Icast t. The predicted negative rate P.....P = PPRMM is the proportion of P with predicted value less than L.
The predicted positive rate defined above is sometimes called selection rale, admission rale, or various other terms, depending on the application of the model under discussion. Equality these rates across groups is also know by various terms including demogruphic parily, group fairness, and statistionl parily. 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 model M two groups P and Q. Let be decision threshold for the predictions of M. If >ir then PPRM.. S PPRMa, (with equality only if both are 0 or 1) and PN PNRMA.P S PNRMM. (with equality only if both are 0 or 1).
Proof. The desired inequality is:
PPRMLQ = / g(x)dxx < a22t
/ p(x)ddx PPRMA.
.22t
which follows directly from the first-order stochastic dominance of p over g. To obtain strictness of the incquality when both are not or 1, note that if Jaz: g(x)dx= = Saz1 p(x)dr, integrating both sides of the MLRP inequality p(z0)g(21) I p( ((())((Z)) for zo S S In over the region 2 I gives g(x) 2 p(x) on
x < t. But since [ q(x)d== f p(z)dr 1, we have [
{ p(x), implying ((x) p(x) Juce q(z)dr Jeet
almost everywhere on Integrating both sides of ((1)) S p(z1))((z)) p(a) (xx) on 2 1, which then implies q(x) p(x) almost everywhere on > t, contradicting that
over the region To < gives
p 1> q.
As the PPR for any group must approach 1 as the decision threshold t decreases, the PPR disparity across groups is necessarily small for sufficiently small t. Similarly, the PNR disparity is necessarily small for sulficiently large 4. However, the next lemmn shows that the MLRP and monotonic calibration guarantee that for a non-trivial decision threshold, one of these disparities is at least as large as the positive outcome rate disparity (for PPR) or negative outcome rate disparity (for PNR).
Lemma 3. Let p(x) and q(x) be the probabilty density functions of the predictions of a model M on two groups and Q. If M is monotorically calibrated for groups and Q (by a function c(x)) and p > q. then for any decision threshold t where the ratãos below are defined, we have either
PPRMA.P 2 O(P)
PPRMIQ ((Q)
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PN RMM 1 0(Q)
PN PN O(P)
Proof. Using that M is monotonically calibrated the first inequality is
J(a p(x)dx / (c)))(()dd [ alx)dx / ccx)p(z)ddx
1220
and substituting Sazt p(z)dz= 1 P(x)do and 1n<r9(z)) - -Jaz ((z) the second inequality becomes
f q(z)dr I c(x)p())dx+ / p(x)dx+ + / c(z))(z)dz 221
122
(ii)
p(z)dr clo)g(z)de + g(z)drx c(z)p(z)de.
J p(z)dr
From the MLRP,
J2 (())dr is increasing in t (by the mediant incquality), so if (i) holds for a given to. it
holds for t (to, 1]. Similarly (ii) holds for to, it holds for e [0, to). Let to be the minimum t such that
p(z)dx= c(()))( c(x)p(z)dz. At tu, both (i) and (ii) reduce to c(z)g(x)dr 2 ff q(z)dr. Writing
#2t
#2ta
f 2200
(()dd / (1- (1-c(z))(z)dr++ + / c(z)g(x)dz,
x2t0
(i) and (ii) further reduce to [ cx))(z)dz 2
(1 (1-c(z))((z)dr. We have
veto
2210
p(x)dr= c(t)p(a)dr +
(1 - (1-c(z))p(z)dd= = c(x)p(x)dd
3210
82t0
x210
SO
[ dx)p(x)dr= = Jazt (1- (1-c(x))((zddz.
Now, if / (())(())d
(1 (1-c(s))(()ddz, there must be li 2 lo where the set.
zzto
x20
(t](K<A & p(t))a(t) 2 (tt) or (t> t p())g(1)) 2 p(1))(()))
has full measure and / c(x)g(x)dr < ff (1 (1- (z)))(((())dz. Note that if p(t1) = 0 for all such 1214
22h h, then either O(P) 0, contradicting that > g. O(P) = 1, contradicting that the ratio (L-0())((--0)) O(P)) is defined, so t can be chosen such that p(t)) > 0. But using the MLRP at &
and that c(z)p(x)dz
(1 c(z))p(z)dz we have:
224
Jecta (l))(x))((t)dr 2 c(x)p(=)a(t)ddz
Jucu
Ja26 (dd)))))))((()d 2 f Jazn (1 (=))))()))((()dd
Thus (i) and (ii) must hold to, and we therefore have (ii) on [0, tul and (i) on [to 1), as required.
We note that the lower bounds provided in Lemma are reached in the discrete case when c(x: takes only values O and 1, these bounds tight in general.
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Definition 5. The false negative rate FN R of a model M with decision threshold t group P is the proportion of the positive outcomes in P that have predicted value less than t. The false posilive rule FPRM&P of a model M with decision threshold t on a group P is the proportion of the negative outcomes in P that have predicted value at least
The following lemma demonstrates that inequality of these rates across groups is an immediate consequence of monotonic calibration and (strict dominance under) the MLRP.
Lemma 4. Let p(x:) and q(z) be the probabilty density functions of the predictions of model M on two groups P and Q. Let & be a decision threshold for the predictions of M. If M is monotonically calibrated for groups and Q (by a function c(z)) and >ir q, then FNR.P & FNN RM,.Q (with equality only if both are or 1) and FPRM& S FPRMM.P (with equality only if both are 0 or 1).
Proof. Using that M is monotonically calibrated on P and Q the first desired inequality becomes:
FNRMAPP Sece (())(()d)
f((x)p(z)dx S Jaci (()g(()d) FNRM.Q [((x))(2)d
or equivalently
[ d())(()d / ((o))(a)do [ do))(t)d f clu)p(o)de
Jace
[ c(v))(x))d from both sides gives Jzzt
Subtracting
c(x)p(x)dr
Juct
f (())(())dr [ d(r)g(x)dx S / 121
Jaze c(x)g(z)dr / d())p(xdrr
121
J.SCe (zo)p(2o)dro
c(a))((x)dd J c(zo)g(zo)ddoo 112t
/ c(x1)((Z1)dz
21
Jn 212t
(xo))(z1)p(ooddde )dr S S c(vo)c(r ((o J
which follows directly from the MLRP. To obtain strictness of the inequality, observe that if p(x0)g(21) = p(z1)q(Z0) almost everywhere on the rectangle [zo < t] x [ 2 t] (which nontrivial if both FNR values are not or 1), then integrating both sides with respect z6 over xo < f and using the stochastic dominance of p over q gives p(x1) q(x1) almost everywhere on æ L, and integrating both sides with respect to æ over 21 2 t gives p(an) = ((x)) almost everywhere zo contradicting that p > q.
The second desired incquality follows in an identical manner, by reducing
FPRM.Q = Sazel (())) (1=)(=-(=)))( c()))p(F)dr
f(1 - (lz=c(z)) S [11-cr)))(=zdd = FPRMAP
to
1
-1- J (1 ((200/ &
2011 212
(1 -1- c(xo))( -1c))))(((()))((mm dr
20St J12t
As with PPR and PNR disparitics, FNR and FPR disparitics are necessarily small for sufficiently small and large decision thresholds, respectively. The next lemma shows that they are also subject the same pairwise lower bounds, under the ongoing assumptions.

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Lemma 5. Let p(x) and q(z) be the probobilty density functions of the predictions of model M two groups P and Q. IJ the model M is monotonically colùbrated for groups P and Q (by a function c(x) and b> 4, then for any decision threshold t where the rutios below are defined, we have either
FN z O(P)
ENRMAP 0(Q)
or
FPRMAP 1 1- 0QQ
FPRMIQ
O(P)
Proof. Observe that whenever
Juct c(z)g(0)d 121
c(x)p(x)dr; we have
FNRMAQ Sect (()) J c(x)p(x)dr O(P)
FNRMAP Sact
c(x))(x)) O(Q)
So if the first inequality in the lemma does not hold, we have S c(z)g(x)dz jzet
c(z)p(r)dr, from which
it follows that q almost everywhere in some neighborhood of t (else p(x) S q(x) for all x < t from the (1 - c(o))g(z)do,
{(1 (1 c(x))p(x)dr 2
MLRP). Similarly, whenever
J22t
#2
FPRMAP L222 - (1-(1)))))(aa S c(x)) (())d z O(P)
0(Q)
FPRM&Q (1= dx [
So if the second inequality in the lemma does not hold,
(1 (1-c(z)) c(=)) < [ (1- (1=(x)))())dd giving q > p almost everywhere in some neighborhood of t.
Thus at least one of the incqualities holds.
Both lower bounds in Lemma 5 can be reached simultancously in the discrete case if p(x)) and q(x)) disagree only when c(æ) {0, meaning that these bounds are also tight in general. Intuitively, the FNR incquality will fail to hold as / 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 Seee (())g(z)dr
ratio of FNRs is Jeer (()))(x)d higher than the ratio of the outcome rates, and this factor is just the ratio
of false negative prevalences within the two populations. Similarly, the relationship of FPRs to the ratio of non-outeome rates is determined by the ratio of false positive prevalences within the two populations. These relationships follow directly from the definitions of FNR and FPR, and Lemma 5 simply shows that, under the given assumptions, at any (non-trivial) threshold the group with a highor outcome rate has either a lower false negative prevalence or higher false positive prevalence. Note that, following the proof, for a threshold where q() pt) (and where is not in the zero- zero-measure measure set where the MLRP relationship is not satisfied), both incqualities must hold, and in most cases they will both tend to hold in some neighborhood of such threshold.
Definition 6. The positive predictive value PPVM.. of a model M with decision threshold t a group P is the proportion of the predicted posilives (i.e. with predicted value at least in P that have positive outcome. The negative predictive value NPVM.. of a model M with decision threshold on a group P is the proportion of the predicted negatives (i.e. with predicted value less than 1) in P that have negative outcome.
The next lemma shows that these metrics are also unequal across groups under the MLRP and monotonic calibration assumptions, except when the outcome rate or likelihood ratio is (almost everywhere) constant above or below the decision threshold. With any guarantee of strictness in these functions (e.g. strict MLRP and well-calibration), inequality in these metries is similarly impossible non-trivial decision thresholds.
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Lemma 6. Let p(x) and q(z) be the probabilty density functions of the predictions of model M two groups P and Q- Let t be decision threshold for the prediclions of M. If M is monolonically calibrated for groups P and Q (by function c(x)) and p b q, then for any t where both sides of the inequality are defined,
PPVM&Q S PPVMM.P, with equality only if p(x)/q(x) or d() is constant almost everywhere on x 2 I and
NPVMM.P S NPVM.. with equality only if p(x)/q(x) or c(x) is constand almost everywhere
x< <tt.
Proof. Using that M is monotonically calibrated on P and Q the first desired inequality becomes:
Sazi c(z)q(x)dz Jazi ((x)dr
Jazi (z)p(()d PPVM. P())d
PPVM1Q
or equivalently
/ c(z)g(x)dr / p(z)dxx S a2t
/ c(z)p(z)dz / q(z)dr
222
#22
/ c(21)((1)ddo 2120
/ g(zo)
p(ro)doo S ((11)p(11ddr, 1022
112
102t
J2 J ((x))((zr)((x))ddm
c(x))p(r ((lo)lla)a
2o2t
11 2o2t
da)))((z))(zz)) ((z1))((r)) ddndrr 11 2o2t
If p(x)//(2) c(x) is constant almost everywhere for 2 t the right hand side is otherwise, by the MLRP have that p( ((a1)4(=0) )((o)) 2 0 for almost every pair zo and p(a (r()))())) (()))(())))))) < 0 for almost every < zo. Viewing the integral as nonnegative and nonpositive component across these two regions:
[ I
0
Ja121 21
and interchanging and xp in the second integral we obtain
((o))p((21)(00))0ppl )p(rr))ddddda)
2121 212000120021
21 2120006
which follows immediately using that c(x) is increasing and nonconstant. The NPV inequality follows in identical fashion.
The following theorem unifies the lemmas proven previously, showing that strict inequality in outcome rates or any of the examined metrics implics all of the incqualities or pairs of incqualities given by the carlier lemmas.
Theorem 1. Let p(æ) and q(x) be the probabilly 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 ralios in (7)-(10) all exist. If M. is monotonically calibrated for groups P and Q and and satisfy the MLRP (with either p Zir q or q ir p), then any of (1)-(4 or strict inequality in (5) or (6) implies all of (1)-(6):
O(Q) <PP)
2. PPRMAQ < PPRMS. (equivalently PNRMP < PNRMA
3. FNRMAP FNRMAQ (equivalently TPRMAQ TPRMAr))

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4. FPRMAQ FPRM, (equivalently TN RM..P < TNRMAQ)
PPVMLQ S PPVM.P
NPVMAP & NPVMAQQ
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):
PPRMAP O(P)
2
PPRMEQ ((Q)
PNRM.LQ 1 QQQ)
8.
PNRMAP 1 1- O(P)
FNRMAQ O(P)
FNRMAP (
FPRMAP 0
10.
FPRM O(P)
Proof. Inequality in any of (6)-(6) implies that p and q not equal almost everywhere, and the direction
of the incquality then implics that p Irr q. This then implics (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
0.8
Recidivism Rate
0.6
Race
Black
White
0.4
0.2
2
8
9
10
COMPAS Decile Risk Score
Figure 1: Calibration of the COMPAS risk score
Using ProPublica'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 cali- brated for the two groups as well. addition, viewing the histogram of risk scores across races (Figure 2), we would expect that the distríbutions of risk scores of these two groups would satisfy the MLRP. Letting p(x)

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307
000
200
COMPAS Decile Risk Score
COMPAS Declie Score
Figure 2: Histogram of the COMPAS risk
Figure 3: Ratio of likelihood functions for the COMPAS risk scores
scores by
3.0
2.5
(NR(W))NR(B)
FPR(B)/PR(W)
O(B)/O(W)
2.0
(1-0(W))(--((8)
PPR(BYPPR(W)
PNR(W)PNR(B)
1.5
1.0
2
3
9
10
COMPAS Decile Risk Score
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 g(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 and 5 across the possible decision thresholds. The ProPublica analysis used decile scores of 5 higher (corresponding to "Medium" or "High" risk) a decision threshold, and because p(5) and q(5) are approximately equal, from the proof of Lemina 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 disparitics in false ncgative/positive rates are guarantced due the shapes of the risk dis- tributions of White and Black defendants, it possible that these differences in distribution, instead of the behavior of the model itself, provide evidence of systemic unfairness? Consider hypothetical county
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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 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 marginaatgyyylty (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, guarantoe higher false negative rate (i.c. 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 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 sesitivvenegativee negative rates to reveal unfairness in an algorithm 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 imme- diate 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 pos- itive/negative rates or used to help set expected ranges for the disparities in these metrics. Nor should they be viewed 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/negativg rate parity, even though disparitics in these metries 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. Tf the model's 's intended use to offer, say, a screening for those at highest risk of a certain medical condition, and the model designer determines that false negative rale parity is 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, disparitics 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
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inequities is not to mitigate error rate difference via intentional mis-calibration enforce metric parity at the cost of model efficacy (Pfohl et al., 2021). Instead, developers should seek to understand the sources of these differences and ensure their models are ultimately used in ways that close gaps instead of widening them, a process which cannot be reduced to computation and comparison standard algorithmic fairness metrics.
5
Acknowledgments
The author would like to thank Michael Northington V for many helpful discussions during the drafting of this paper and the anonymous reviewers at TMLR for their comments and feedback. The views expressed in this paper are those of the author and not those of his employer.
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13


Reviews:
[{"review_id": "ZvWU3KMK4R", "reviewer": "Reviewer_YLGZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n- Addresses widespread misinterpretations of fairness metrics by practitioners and policymakers.\n- Moves beyond existing impossibility results by analyzing metric disparities under distributional assumptions rather than just constraints.\n- The COMPAS analysis is a compelling illustration that ties the theory back to real-world consequences.\n\nWeaknesses: \n- MLRP is a strong assumption that may not hold in many practical cases; more discussion on limitations would be helpful.\n- Theoretical results are well-developed, but only a single empirical case study is presented.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a new theoretical framework for understanding disparities in common algorithmic fairness metrics under realistic modeling assumptions. Specifically, the authors define monotonic calibration, which is a strengthening of traditional calibration that ensures predictions increase with outcome likelihood. They analyze its implications when combined with the monotone likelihood ratio property (MLRP) on group-wise risk distributions. Under these assumptions, they prove that disparities in widely used fairness metrics, such as false positive and negative rates, are not only inevitable when outcome rates differ, but also satisfy tight lower bounds. The authors also provide equivalence relationships among these metric disparities and illustrate their theoretical findings with an empirical analysis of the COMPAS dataset.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "-  Provide examples or discussion on when MLRP is realistic in applied settings.\n-  Supplement the COMPAS case study with one or two more datasets.", "broader_impact_concerns": "There are no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "o4hg7ULbIR", "reviewer": "Reviewer_mCHv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The introduction of monotonic calibration is both intuitive and practically relevant. The findings on the compatibility of common algorithmic fairness metrics under the assumptions are interesting.\n- The mathematical proofs are rigorous and detailed.\n- The authors provide a unifying theorem (Theorem 1) that links disparate fairness metrics under MLRP, capturing a broad class of findings.\n\nWeaknesses:\n\n- MLRP is a strong assumption; many real-world models (especially deep learning models) may not satisfy it. Some discussion is provided, but empirical sensitivity analysis is lacking. While the COMPAS case helps, there’s no empirical test across multiple domains to validate how commonly these theoretical conditions apply.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the compatibility of common algorithmic fairness metrics—such as false positive rate (FPR), false negative rate (FNR), positive/negative predictive value, and selection rate—under the assumptions that: the model is monotonically calibrated, and the group-wise score distributions satisfy the Monotone Likelihood Ratio Property (MLRP).The authors mathematically prove that fairness metric parity is generally impossible across groups with different base outcome rates, even when models are calibrated. They apply this to the COMPAS dataset to show that apparent bias in false positive/negative rates can emerge purely from group-level distribution differences, not from algorithmic unfairness.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "A discussion of the strong assumption of MLRP would strengthen the paper. Addressing its practical relevance could enhance the clarity of the contribution. Nonetheless, I find the core findings interesting, and I am inclined to recommend acceptance.", "broader_impact_concerns": "The paper currently lacks a Broader Impact Statement—though touched upon in the conclusion—are not sufficiently addressed. For example, the work can be misunderstood as a justification for ignoring fairness disparities; the paper critiques popular fairness metrics but offers no alternative criteria.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "4cMokjqrWO", "reviewer": "Reviewer_hQ3n", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths  \n- The results are elegant: a fairly simple pair of assumptions on the model yields a variety of novel inequalities.\n- Existing work has mainly focused on the impossibility of simultaneously satisfying different definitions of fairness; by establishing _bounds_ this work shows the impossibility of simultaneously _approximately_ satisfying different definitions of fairness.\n- The proofs appear to be correct, non-trivial, and well-presented.\n- The COMPAS case study helps contextualize the theoretical findings, and shows that the assumptions can hold in natural settings.\n\n\n### Weaknesses  \n1. Strength of MLRP assumptions:\n\t- The paper only references one paper that uses MLRP in the fairness context. While others certainly do so (for example, Chemla and Hennessey [1] and recent work by Chiang et al. [2], it would be helpful to cite other work, as the MLRP assumption is quite strong. \n\t- Moreover, my sense is that MLRP is more often assumed as a property of the distribution of outcomes rather than the distribution of the model (the paper by Gaebler and Goel cited by the authors does impose this assumption on the risk distribution but my understanding is they assume the model's risk distribution *is* the conditional outcome distribution). If my understanding is incorrect, please correction me! Otherwise, it would be helpful to either identify work that imposes the MLRP assumption on the model's risk distribution, or else further justify why the MLRP should hold for the model's risk scores from first principles. \n\t- This is especially important for the paper's conclusions, which uses the phrases \"in many cases\" and \"rarely\". Without a more solid understanding of the reasonableness of MLRP for the model's risk scores, it is difficult to evaluate these claims.\n2. It would be helpful to include more text between theorems to interpret the results in plain language.\n\n\n### Minor\n3. The paper includes formal definitions of calibration and predicted positive rate, but is missing formal definitions of PPV and FPR.\n\n\n### References\n[1] Gilles Chemla and Christopher A. Hennessy. \"Controls, belief updating, and bias in medical RCTs.\" Journal of Economic Theory, 184:104929, 2019. https://www.sciencedirect.com/science/article/pii/S0022053119300808.\n\n[2]  Erica Chiang, Divya Shanmugam, Ashley N. Beecy, Gabriel Sayer, Deborah Estrin, Nikhil Garg, Emma Pierson. \"Learning\nDisease Progression Models That Capture Health Disparities\". AHLI Conference on Health, Inference, and Learning (CHIL), 2025.  https://www.arxiv.org/pdf/2412.16406", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper explores the implication of a common fairness criterion for models (monotone calibration, that a model's risk scores correspond to outcomes in the same way across groups) in a decision-making setting with two groups, when the model satisfies the monotone likelihood ratio principle (or MLRP, that risk scores are more likely to be higher for one group than the other). The authors show that these assumptions imply that (a) except in trivial settings, there is a disparity between outcome rates for the two groups and (b) this disparity is a lower bound for the disparity many classic notions of fairness. This implies that in calibrated models that satisfy the MLRP, even approximately guaranteeing other definitions of fairness is fraught because of underlying differences in the outcome rates. The authors then show that in the classic COMPAS dataset, the model's risk scores satisfy the two assumptions (MLRP and monotone calibration).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "In light of the weaknesses identified above, it would be helpful if the authors could\n1. further justify the strong MLRP assumption on the model's risk distribution through related work or first principles arguments\n2. consider including more text between theorems to improve readability\n\nPossibly I missed this, but from reading the introduction it is unclear that well-calibration is a special case of monotone calibration (although it's clear from the formal definitions). I think this contributes to the strength of your findings, so it could be worth emphasizing this!", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)