Tilted Mondrian: Triage and Mitigation of Fine-grained Disparities in Conformal Prediction

Ziang Gao; Pengqi Liu; Archer Yi Yang; Mouloud Belbahri; Jesse C. Cresswell; Masoud Asgharian

stat.MLOct 17, 2026

Tilted Mondrian: Triage and Mitigation of Fine-grained Disparities in Conformal Prediction

Authors: Ziang Gao, Pengqi Liu, Archer Yi Yang, Mouloud Belbahri, Jesse C. Cresswell, Masoud Asgharian

Organizations: McGill University, Montreal, Canada · TD Insurance, Montreal, Canada · Layer 6 AI, Toronto, Canada

Abstract

Fairness in conformal prediction is often pursued through Mondrian calibration, which provides finite-sample coverage guarantees separately across groups. These guarantees, however, are coarse: substantial disparities can remain within finer strata. This coarse-to-fine fairness issue raises two questions: how severe can such disparities be, and how can they be mitigated? Aggregate disparities can depend on how groups are distributed across strata. For severity assessment, we introduce propensity standardization to examine the effect of population composition, and show how group composition can induce disagreement between aggregate and finer-stratum comparisons. For mitigation, we develop Tilted Mondrian to improve fairness compared to group-Mondrian at a finer level without additional model training. Tilted Mondrian retains groupwise coverage guarantees and mitigates coverage and prediction set size disparities within finer strata. Finally, we derive a method-agnostic bound that quantifies how much reduction in finer-stratum coverage disparity can be achieved.

Figures & tables

Figure 1: Top: Propensity standardization evaluates the aggregate–stratum set size disagreement. Bottom: Tilted Mondrian mitigates finer strata disparities and maintains groupwise coverage.

Figure 2: Tilted Mondrian mechanism for finer strata disparity mitigation.

Bias in Bios
Score	Reversals	Conflict strength	ESS
SIMPLE	$20\to 2$	$0.101\to 0.006$	$0.833$
APS	$41\to 7$	$0.120\to 0.012$	$0.833$
RAPS	$56\to 1$	$0.151\to 0.008$	$0.833$
SAPS	$38\to 0$	$0.131\to 0.005$	$0.833$

Table 1: Metrics before and after propensity standardization is applied. Top: Bias in Bios/ATUS across score functions

s(x,y)

. Bottom: Folktables with SIMPLE.

Experiment	Score	$L^{0}_{\max}\to L^{\theta}_{\max}$	$\max_{z}m_{z}^{\theta}$	Harm	$\Delta\|C\|$
Bias in Bios	SIMPLE	$0.161\to 0.090$	$0.071$ (surgeon)	$0.010$	$0.012$
Folktables occupation	SIMPLE	$0.037\to 0.014$	$0.024$ (Less than high school)	$0.010$	$0.007$
Folktables income	SIMPLE	$0.033\to 0.024$	$0.009$ (Bachelor)	$0.003$	$0.001$
ATUS	SIMPLE	$0.010\to 0.013$	$0.001$ (employed)	$0.009$	$0.000$

Table 2: Tilted Mondrian for coverage disparity mitigation.

Experiment	Score	$L^{0}_{\max,S}\to L^{\theta}_{\max,S}$	$\max_{z}m_{z,S}^{\theta}$	Harm	$\Delta\mathrm{Cov}(\mathrm{pp})$
Bias in Bios	SIMPLE	$0.065\to 0.063$	$0.005$ (psychologist)	$0.000$	$-0.041$
Folktables occupation	SIMPLE	$0.180\to 0.142$	$0.058$ (High school or GED)	$0.020$	$0.012$
Folktables income	SIMPLE	$0.188\to 0.146$	$0.041$ (Graduate or professional)	$0.003$	$0.007$
ATUS	SIMPLE	$0.240\to 0.238$	$0.002$ (not in labor force)	$0.002$	$0.000$

Table 3: Tilted Mondrian for set size disparity mitigation.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$X,Y,\mathcal{Y},K$	Input, label, label space, and number of classes
$s(x,y),\hat{q}_{\alpha}$	Nonconformity score and split-conformal threshold
$\mathcal{C}_{\alpha}(x),M_{\alpha}(X)$	Delivered prediction set and set size
$\boldsymbol{\alpha}$	Grid of nominal miscoverage levels used for curve reporting
$\alpha^{\star}$	Selected information efficiency anchor
$M_{\alpha}^{+}(X)$	Nonempty set-size version used in Proposition 7

Appendix

Table 4: Notation often used in the main text and appendices. Tildes denote quantities induced by the profile target distribution

Q_{H}

; they do not change the delivered prediction sets.

Setting	Year(s)	Train	Cal.	Prop./Tilt	Test	Total
Bias in Bios	all	127,023	79,390	79,390	31,756	317,559
Folktables occupation	2014	163,527	102,204	102,205	40,882	408,818
Folktables occupation	2015	162,130	101,331	101,332	40,533	405,326
Folktables occupation	2016–2018 each	164,195	102,622	102,622	41,049	410,488
Folktables income quintile	2014–2018 each	164,195	102,622	102,622	41,049	410,488
ATUS	all	71,356	44,683	44,655	17,783	178,477

Appendix

Table 5: Exact split sizes for the empirical reports. The split columns are training, calibration, propensity/tilt, and testing.

Setting	Threshold policy	Scores	$\alpha$ grid	Support rule
Bias in Bios	Gender-conditional thresholds	SIMPLE; APS; SAPS( $\lambda=0.2$ ); RAPS( $\lambda=0.2$ , $k_{\mathrm{reg}}=5$ )	0.020 to 0.100; step 0.005	At least 200 test observations per Gender group within each occupation stratum; the same supported grid is used for raw and standardized summaries.
Folktables occupation	Pooled threshold within year	SIMPLE	0.010 to 0.100; step 0.001	At least 200 test observations per group within each education stratum.
Folktables income quintile	Pooled threshold within year	SIMPLE	0.010 to 0.100; step 0.001	Same test education stratum support rule as Folktables occupation.
ATUS	Pooled threshold	SIMPLE primary; APS, RAPS( $\lambda=0.01$ , $k_{\mathrm{reg}}=5$ ), and SAPS( $\lambda=0.2$ ) checks	0.020 to 0.250; step 0.010	Prespecified employment stratum support rule.

Appendix

Table 6: Score policies, operating grids, and support rules. All raw-to-standardized arrows use the same test prediction sets.

Setting	Predictor and score	Profile $H$	Propensity model
Bias in Bios	DistilBERT top-10 occupation classifier; SIMPLE, APS, SAPS, and RAPS.	37-dimensional frozen probability profile.	Feed-forward neural-network propensity model with hidden layers $(128,64)$ and ReLU activations; fitted on the propensity split.
Folktables occupation	Histogram gradient boosting on ACS person records, one model per year; SIMPLE.	37-dimensional frozen softmax profile; Gender and education are excluded from $H$ .	Logistic propensity model with balanced class weights; fitted separately by year on the propensity split.
Folktables income quintile	Histogram gradient boosting for income quintiles, one model per year; SIMPLE.	22-dimensional frozen softmax profile; race and education are excluded from $H$ .	Logistic propensity model with balanced class weights; fitted separately by year on the propensity split.
ATUS	Regularized histogram gradient boosting on diary, time, calendar, demographic, region, and household features; SIMPLE with APS, RAPS, and SAPS checks.	Four probability features: $p_{\mathrm{top1}}$ , $p_{\mathrm{top2}}$ , margin, and entropy.	Logistic propensity model fitted on the propensity split.

Appendix

Table 7: Prediction profiles and propensity models for the standardized report. The reporting weight model enters only the curve summaries on the test split.

Dataset	Alignment loss	Squared composition residual $R^{2}$	Stratum-gap variance
Bias in Bios	$66.5$ – $72.6$	$98.7$ – $99.3$	$60.7$ – $69.9$
Folktables occupation	$85.1$ – $92.8$	$75.0$ – $94.2$	$85.9$ – $92.6$
Folktables income	$60.0$ – $72.5$	$88.8$ – $92.2$	$55.0$ – $69.0$
ATUS	$36.1$ – $57.0$	$59.7$ – $60.9$	$-88.8$ – $-5.9$

Appendix

Table 8: Relative reductions (%) in grid-averaged alignment loss and its two components. Ranges are across scores or years; negative values denote increases.

Measure	Value
Effective sample size fraction	$0.833$
Profile mean standardized mean difference	$0.0924\to 0.0133$
Profile maximum standardized mean difference	$0.4331\to 0.0458$

Appendix

Table 9: Bias in Bios balance evaluations for the standardized report on the test split.

Measure	Raw	Standardized
Reversals, summed	$930$	$614$
High conflict strength reversals	$400$	$1$
Conflict strength	$39.664$	$1.609$
Years with reversal count drop	$5/5$
Years with strength drop	$5/5$
Median cell total variation	$0.124$	$0.045$

Appendix

Table 10: Folktables occupation evaluation summary across 2014–2018.

Measure	Raw	Standardized
Reversals, summed	$922$	$717$
High conflict strength reversals	$85$	$0$
Conflict strength	$11.890$	$0.509$
Years with reversal count drop	$4/5$
Years with strength drop	$5/5$
Median cell total variation	$0.120$	$0.034$

Appendix

Table 11: Folktables income quintile evaluation summary across 2014–2018.

Measure	Raw	Standardized
SIMPLE reversals	$36$	$10$
SIMPLE conflict strength	$0.463730$	$0.002697$
Maximum profile standardized mean difference	$0.0817$	$0.0101$
Employment cell total variation	$0.142$	$0.091$
Effective sample size fraction	$0.981913$

Appendix

Table 12: ATUS evaluation details for the primary SIMPLE score on the test split.

Report	Target population	Weight behavior	Interpretation
Raw group curve	Group-specific $P(H\mid G=g)$	No reweighting	Observed-mixture burden delivered to each group.
Marginal-profile curve	Marginal distribution $P_{H}$	$1/e(H)$ and $1/\{1-e(H)\}$	Full-population standardized contrast; sensitive to weak support.
Standardized curve	Common standardized distribution $Q_{H}$	Bounded factors $1-e(H)$ and $e(H)$	Propensity standardized contrast; bounded and stable.
Uniform-cell curve	Selected reporting cells equally weighted	Cell weights chosen by design	Cell-equal sensitivity summary; can overweight rare or unstable cells.

Appendix

Table 13: Several population indices for the same delivered set sizes. The paper uses the raw and standardized reports because together they show the observed burden and the standardized burden.

Experiment	Group and intervention stratum
Bias in Bios	Gender and occupation, with $Z=Y$
Folktables occupation	Gender and education bucket
Folktables income	Race group and education bucket
ATUS	Gender and employment status

Appendix

Table 14: Intervention cells for Tilted Mondrian.

k_{g}^{0}=\sum_{i\in\mathcal{D}_{\mathrm{tilt}}:G_{i}=g}\mathbf{1}\{Y_{i}\in\mathcal{C}_{\alpha,0}(X_{i})\}.

Appendix

Algorithm 4 Tilted Mondrian CP with directional score shifting (detailed version of Algorithm 5.1 )

Experiment	Score	$L^{0}_{\max}\to L^{\theta}_{\max}$	$\Delta L_{\max}$	$\max_{z}m_{z}^{\theta}$	Harm	$\Delta\|C\|$
Bias in Bios	APS	$0.046{\to}0.043$	$0.003$	$0.008$ (nurse)	$0.0077$	$0.0414$
Bias in Bios ( $\lambda=0.01$ )	RAPS	$0.036{\to}0.036$	$0.000$	$0.006$ (nurse)	$0.0050$	$0.3389$
Bias in Bios	SAPS	$0.178{\to}0.083$	$0.095$	$0.173$ (surgeon)	$0.0082$	$0.0090$
Folktables occupation	APS	$0.105{\to}0.090$	$0.015$	$0.019$ (Bachelor)	$0.0098$	$-0.1008$
Folktables occupation	RAPS	$0.101{\to}0.084$	$0.017$	$0.017$ (Less than high school)	$0.0098$	$-0.0491$
Folktables occupation	SAPS	$0.028{\to}0.013$	$0.015$	$0.022$ (Bachelor)	$0.0100$	$0.0078$

Appendix

Table 15: Test APS, RAPS, and SAPS directional score intervention results. Columns follow Table 2 . Harm is the largest tilt split coverage loss among initially advantaged cells. Rounding values may conceal changes at higher precision.

Dataset	Score	Theorem 6 bound	Active term	Tilt split $\max_{z}m_{z}^{\theta}$	Test $\max_{z}m_{z}^{\theta}$
Bias in Bios	SIMPLE	0.156	gap $d_{z}$	0.125	0.071
	APS	0.028	$\epsilon$ -budget term	0.017	0.008
	RAPS ( $\lambda=0.01$ )	0.041	gap $d_{z}$	0.040	0.006
	SAPS	0.166	gap $d_{z}$	0.164	0.173
Folktables occupation	SIMPLE	0.037	gap $d_{z}$	0.022	0.024
	APS	0.039	$\epsilon$ -budget term	0.021	0.019

Appendix

Table 16: Tilt split result for Tilted Mondrian. The Theorem 6 bound is evaluated on the tilt split because the intervention feasibility constraints are checked there. The test column reports the corresponding achieved mitigation on the test split and is not used as a feasibility check.

Bias in Bios
Score	$\epsilon=.01$	$\epsilon=.02$	$\epsilon=.03$	$\epsilon=.04$	$\epsilon=.05$
SIMPLE	$0.1557/0.1245$	$0.1557/0.1197$	$0.1557/0.1268$	$0.1557/0.1341$	$0.1557/0.1363$
APS	$0.0276^{\dagger}/0.0169$	$0.0477/0.0169$	$0.0477/0.0169$	$0.0477/0.0169$	$0.0477/0.0169$
RAPS ( $\lambda=.01$ )	$0.0414/0.0399$	$0.0414/0.0399$	$0.0414/0.0399$	$0.0414/0.0399$	$0.0414/0.0399$
SAPS	$0.1656/0.1640$	$0.1656/0.1373$	$0.1656/0.1293$	$0.1656/0.1628$	$0.1656/0.1628$
Folktables occupation

Appendix

Table 17: Sensitivity to the harm budget

\epsilon

. Each entry reports

B_{\epsilon}/M_{\epsilon}

, where

B_{\epsilon}

is the Theorem 6 bound and

M_{\epsilon}=\max_{z}m_{z}^{\theta}

is the achieved tilt split mitigation. A

\dagger

indicates that the

\epsilon

-budget term is active in the bound; otherwise the original stratum gap

d_{z}

is active.

Score	Method	Bias in Bios	Folktables occupation	Folktables income	ATUS
SIMPLE	Tilted	$\mathbf{0.0436\pm 0.0032}$	$\mathbf{0.2579\pm 0.0009}$	$\mathbf{0.1915\pm 0.0009}$	$\mathbf{0.1390\pm 0.0007}$
	Cellwise	$0.1118\pm 0.0031$	$0.2722\pm 0.0007$	$0.1930\pm 0.0008$	$0.1536\pm 0.0011$
APS	Tilted	$\mathbf{0.3786\pm 0.0246}$	$\mathbf{0.2789\pm 0.0012}$	$\mathbf{0.2259\pm 0.0012}$	$0.1185\pm 0.0011$
	Cellwise	$0.4068\pm 0.0010$	$0.3005\pm 0.0012$	$0.2262\pm 0.0010$	$\mathbf{0.1133\pm 0.0015}$
RAPS	Tilted	$\mathbf{0.3036\pm 0.0153}$	$\mathbf{0.2391\pm 0.0012}$	$\mathbf{0.2259\pm 0.0011}$	$\mathbf{0.0998\pm 0.0009}$
	Cellwise	$0.3687\pm 0.0009$	$0.2619\pm 0.0014$	$0.2262\pm 0.0010$	$0.1041\pm 0.0016$

Appendix

Table 18: Set size dispersion under direct finer-cell calibration: Gini coefficient. For each resplit, let

S_{i}=|\mathcal{C}(X_{i})|

and

\bar{S}=n^{-1}\sum_{i}S_{i}

. We compute

G=(2n^{2}\bar{S})^{-1}\sum_{i,j}|S_{i}-S_{j}|

. Entries are mean

\pm

SD over the same 20 paired resplits. Lower is better; bold marks the lower value.

Score	Method	Bias in Bios	Folktables occupation	Folktables income	ATUS
SIMPLE	Tilted	$\mathbf{0.068\pm 0.007}$	$\mathbf{0.397\pm 0.001}$	$\mathbf{0.286\pm 0.002}$	$\mathbf{0.203\pm 0.001}$
	Cellwise	$0.209\pm 0.006$	$0.417\pm 0.002$	$0.289\pm 0.002$	$0.222\pm 0.002$
APS	Tilted	$\mathbf{0.564\pm 0.040}$	$\mathbf{0.411\pm 0.002}$	$\mathbf{0.350\pm 0.002}$	$0.172\pm 0.002$
	Cellwise	$0.609\pm 0.002$	$0.427\pm 0.002$	$0.352\pm 0.001$	$\mathbf{0.165\pm 0.002}$
RAPS	Tilted	$\mathbf{0.467\pm 0.028}$	$\mathbf{0.372\pm 0.002}$	$\mathbf{0.350\pm 0.002}$	$\mathbf{0.145\pm 0.002}$
	Cellwise	$0.557\pm 0.001$	$0.390\pm 0.002$	$0.352\pm 0.001$	$0.152\pm 0.003$

Appendix

Table 19: Set size dispersion under direct finer-cell calibration: relative mean absolute deviation. For each resplit, let

S_{i}=|\mathcal{C}(X_{i})|

and

\bar{S}=n^{-1}\sum_{i}S_{i}

. We compute

\mathrm{RMAD}=(n\bar{S})^{-1}\sum_{i}|S_{i}-\bar{S}|

. Entries are mean

\pm

SD over the same 20 paired resplits. Lower is better; bold marks the lower value.

\theta_{z}^{\prime+}(x)=\bigl[\theta_{z}^{+}(x)+\lambda^{+}h_{z,t}^{+}(x)\bigr]_{+},\qquad\theta_{z}^{\prime-}(x)=\bigl[\theta_{z}^{-}(x)+\lambda^{-}h_{z,t}^{-}(x)\bigr]_{+}.

Appendix

Algorithm 5 Boosted Tilted Mondrian

Bias in Bios
Method	Tilt split worst-gap reduction $\Delta L_{\max}$	Test worst-gap reduction $\Delta L_{\max}$	Tilt split maximum harm	Test mean set size change	Null interventions
TM (20)	$\phantom{-}0.0778\pm 0.0397$	$\phantom{-}0.0680\pm 0.0345$	$0.0090\pm 0.0020$	$\phantom{-}0.0038\pm 0.0034$	$0/20$
BTM (10)	$\phantom{-}0.0145\pm 0.0144$	$\phantom{-}0.0145\pm 0.0112$	$0.0041\pm 0.0027$	$\phantom{-}0.0002\pm 0.0003$	$0/10$
Folktables occupation
TM (20)	$\phantom{-}0.0231\pm 0.0025$	$\phantom{-}0.0209\pm 0.0060$	$0.0099\pm 0.0001$	$\phantom{-}0.0040\pm 0.0023$	$0/20$
BTM (10)	$\phantom{-}0.0099\pm 0.0049$	$\phantom{-}0.0075\pm 0.0055$	$0.0076\pm 0.0035$	$\phantom{-}0.0051\pm 0.0058$	$0/10$

Appendix

Table 20: SIMPLE robustness comparison between Tilted Mondrian and Boosted Tilted Mondrian. Tilted Mondrian (TM) is summarized over 20 resplits; Boosted Tilted Mondrian (BTM) over the ten shared resplits. For either method,

\Delta L_{\max}=L_{\max}^{0}-L_{\max}^{\mathrm{final}}

, where

L_{\max}

is the worst supported fine-grained coverage gap; positive values indicate improvement. Continuous entries are mean

\pm

SD. Null interventions count the unchanged solution.

	Tilt split final mean gap		Test final mean gap
Dataset	Tilted Mondrian	Boosted Tilted Mondrian	Tilted Mondrian	Boosted Tilted Mondrian
Bias in Bios	$\mathbf{0.03291}$	$0.05111$	$\mathbf{0.03353}$	$0.04918$
Folktables occupation	$\mathbf{0.00812}$	$0.01615$	$\mathbf{0.00994}$	$0.01845$
Folktables income	$\mathbf{0.00067}$	$0.00398$	$\mathbf{0.01127}$	$0.01132$
American Time Use Survey	$\mathbf{0.00576}$	$0.00641$	$0.01294$	$\mathbf{0.01047}$

Appendix

Table 21: Final mean fine-grained coverage gap on the SIMPLE resplits. Entries are averages across resplits; lower is better. This complements Table 20 , which reports worst-gap reduction, by measuring the average remaining disparity across strata.

Bias in Bios
Setting	Raw $\to$ standardized strength	Relative reduction median [P5, P95] (%)	Reversal D/T/I	Test ESS
SIMPLE	$0.118\pm 0.034{\to}0.006\pm 0.004$	$95.3\,[83.6,99.5]$	$49/0/1$	$0.824\pm 0.004$
APS	$0.152\pm 0.068{\to}0.012\pm 0.008$	$92.5\,[77.1,98.0]$	$48/1/1$	$0.824\pm 0.004$
SAPS	$0.151\pm 0.038{\to}0.006\pm 0.004$	$96.4\,[90.2,98.7]$	$50/0/0$	$0.824\pm 0.004$
RAPS	$0.156\pm 0.056{\to}0.010\pm 0.006$	$93.3\,[81.6,98.4]$	$49/1/0$	$0.824\pm 0.004$
Folktables occupation

Appendix

Table 22: Fixed model robustness of propensity standardization. Each row summarizes 50 calibration/propensity/test resplits. Conflict strength and normalized test effective sample size are reported as mean

\pm

SD; relative reduction is the median [P5, P95]. D/T/I counts resplits in which standardization decreases, ties, or increases the reversal count.

Bias in Bios
Score	$L^{0}_{\max}{\to}L^{\theta}_{\max}$	$\Delta L_{\max}$	Harm	$\Delta\|C\|$	P/T/N
SIMPLE	$0.172\pm 0.031{\to}0.104\pm 0.031$	$\phantom{-}0.0680\pm 0.0345$	$0.0090\pm 0.0020$	$\phantom{-}0.0038\pm 0.0034$	$20/0/0$
APS	$0.043\pm 0.007{\to}0.040\pm 0.005$	$\phantom{-}0.0025\pm 0.0030$	$0.0036\pm 0.0022$	$\phantom{-}0.2955\pm 0.2220$	$19/1/0$
RAPS	$0.041\pm 0.009{\to}0.034\pm 0.007$	$\phantom{-}0.0074\pm 0.0082$	$0.0046\pm 0.0012$	$\phantom{-}0.4799\pm 0.0959$	$12/7/1$
SAPS	$0.179\pm 0.034{\to}0.084\pm 0.016$	$\phantom{-}0.0951\pm 0.0328$	$0.0093\pm 0.0011$	$\phantom{-}0.0071\pm 0.0036$	$20/0/0$
Folktables occupation

Appendix

Table 23: Fixed-model robustness of Tilted Mondrian for coverage. Each row summarizes 20 tilt/calibration/test resplits; continuous entries are mean

\pm

SD.

L^{0}_{\max}

L^{\theta}_{\max}

\Delta L_{\max}

, and

\Delta|C|

are evaluated on the test split. Harm is the largest tilt split coverage loss among initially advantaged cells. P/T/N counts positive, numerically tied, or negative test worst-gap reductions. All 320 mitigation problems satisfy the prespecified feasibility constraints.

Bias in Bios
Score	$L^{0}_{\max,S}{\to}L^{\theta}_{\max,S}$	$\Delta L_{\max,S}$	Harm	$\Delta\mathrm{Cov.}$ (pp)	P/T/N
SIMPLE	$0.066\pm 0.024{\to}0.065\pm 0.025$	$\phantom{-}0.0013\pm 0.0039$	$0.0038\pm 0.0059$	$-0.001\pm 0.013$	$5/12/3$
APS	$1.269\pm 0.054{\to}0.514\pm 0.058$	$\phantom{-}0.7551\pm 0.0790$	$0.0061\pm 0.0064$	$-0.019\pm 0.092$	$20/0/0$
RAPS	$1.163\pm 0.041{\to}0.515\pm 0.050$	$\phantom{-}0.6480\pm 0.0573$	$0.0052\pm 0.0083$	$-0.028\pm 0.089$	$20/0/0$
SAPS	$0.052\pm 0.016{\to}0.054\pm 0.032$	$-0.0019\pm 0.0301$	$0.0018\pm 0.0022$	$-0.001\pm 0.021$	$11/5/4$
Folktables occupation

Appendix

Table 24: Fixed-model robustness of set-size-targeted Tilted Mondrian. Each row summarizes 20 tilt/calibration/test resplits; continuous entries are mean

\pm

SD. Set-size gaps and

\Delta\mathrm{Cov.}

are evaluated on the test split. Harm is the largest tilt split set-size increase among initially smaller set cells. P/T/N counts positive, tied, or negative test worst-gap reductions.

Bias in Bios
	Coverage-targeted mitigation		Set-size-targeted mitigation
Score	$\Delta\bar{M}$	$\max_{g,z}\|\Delta\bar{M}_{g,z}\|$	$\Delta\mathrm{Cov.}$ (pp)	$\max_{g,z}\|\Delta c_{g,z}\|$ (pp)
SIMPLE	$\phantom{-}0.0038\pm 0.0034$	$0.104\pm 0.068$	$-0.001\pm 0.013$	$0.352\pm 0.551$
APS	$\phantom{-}0.2955\pm 0.2220$	$0.887\pm 0.443$	$-0.019\pm 0.092$	$18.786\pm 10.999$
RAPS	$\phantom{-}0.4799\pm 0.0959$	$1.255\pm 0.266$	$-0.028\pm 0.089$	$12.836\pm 9.650$
SAPS	$\phantom{-}0.0071\pm 0.0036$	$0.137\pm 0.050$	$-0.001\pm 0.021$	$1.601\pm 3.629$

Appendix

Table 25: Cross-metric changes under the two Tilted Mondrian targets. Entries are mean

\pm

SD over 20 fixed-model resplits. Under coverage-targeted mitigation, we report the overall and maximum-cell set-size changes. Under set-size-targeted mitigation, we report the overall and maximum-cell coverage changes in percentage points.

Bias in Bios
	Coverage-targeted mitigation		Set-size-targeted mitigation
Score	median cell $\|\Delta\bar{M}\|$	max cell $\|\Delta\bar{M}\|$	median cell $\|\Delta\bar{M}\|$	max cell $\|\Delta\bar{M}\|$
SIMPLE	$0.0048\pm 0.0026$	$0.104\pm 0.068$	$0.0000\pm 0.0001$	$0.005\pm 0.007$
APS	$0.1407\pm 0.1041$	$0.887\pm 0.443$	$0.0720\pm 0.0391$	$1.191\pm 0.116$
RAPS	$0.1920\pm 0.1535$	$1.255\pm 0.266$	$0.0535\pm 0.0253$	$0.939\pm 0.117$
SAPS	$0.0074\pm 0.0035$	$0.137\pm 0.050$	$0.0007\pm 0.0011$	$0.016\pm 0.035$

Appendix

Table 26: Test set-size absolute change under both mitigation targets. For each resplit, we compute the median and maximum of

|\bar{M}^{\theta}_{g,z}-\bar{M}^{0}_{g,z}|

over cells; entries are mean

\pm

SD across 20 resplits.

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On the Burden of Achieving Fairness in Conformal Prediction
May 14, 2026Ziang Gao, Pengqi Liu, Archer Yi Yang +3Fair Conformal Prediction Conformal Prediction
McGill University Montreal, Canada · TD Insurance Montreal, Canada · Layer 6 AI Toronto, Canada
Fair Conformal Classification via Learning Representation-Based Groups
May 12, 2026Senrong Xu, Yanke Zhou, Yuhao Tan +5Fair Conformal Prediction Group-Conditional Conformal Prediction
State Key Lab of Novel Software Technology, Nanjing University, China · ETH Zürich · Birkbeck, University of London
Beyond Procedure: Substantive Fairness in Conformal Prediction
Feb 18, 2026Pengqi Liu, Zijun Yu, Mouloud Belbahri +3Algorithmic Fairness Conformal Prediction
McGill University, Montreal, Canada · TD Insurance, Montreal, Canada · UQAM, Montreal, Canada +1

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On the Burden of Achieving Fairness in Conformal Prediction

Ziang Gao, Pengqi Liu, Archer Yi Yang, Mouloud Belbahri+2

McGill University Montreal, Canada · TD Insurance Montreal, Canada · Layer 6 AI Toronto, Canada

Fair Conformal Prediction Conformal Prediction Group Fairness

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Conformal prediction is often calibrated with a single pooled threshold, but this can hide cross-group heterogeneity in score distributions and distort group-wise coverage. We study this phenomenon through the population score distributions underlying split conformal calibration. First, we derive a conservation law and lower bound showing that pooled calibration incurs irreducible group-wise coverage distortion at a scale set by cross-group quantile heterogeneity. Second, we demonstrate that the two leading fairness definitions for conformal prediction, Equalized Coverage and Equalized Set Size, are fundamentally in tension. Third, we quantify the cost of moving between policies which treat groups separately or pool them. Experiments on synthetic and real data confirm the same bidirectional trade-off after finite-sample calibration. Our results show that, for the policy families studied here, calibration choice does not remove cross-group heterogeneity; it determines whether the resulting distortion appears in the coverage or size dimension, providing a principled lens for analyzing fairness-oriented calibration choices in practice.

Abstract

Fair Conformal Prediction

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On the Burden of Achieving Fairness in Conformal Prediction

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Beyond Procedure: Substantive Fairness in Conformal Prediction