When Machine Learning Gets Personal: Evaluating Prediction and Explanation
Authors: Louisa Cornelis
Guillermo Bernardez
Haewon Jeong
Nina Miolane
Venue: ICLR 2026 Poster (2026)
Keywords: Fairness
explainability
personalization
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: In high-stakes domains like healthcare, users often expect that sharing personal information with machine learning systems will yield tangible benefits, such as more accurate diagnoses and clearer explanations of contributing factors. However, the validity of this assumption remains largely unexplored. We propose a unified framework to quantify how personalizing a model influences both prediction and explanation. We show that its impacts on prediction and explanation can diverge: a model may become more or less explainable even when prediction is unchanged. For practical settings, we study a standard hypothesis test for detecting personalization effects on demographic groups. We derive a finite-sample lower bound on its probability of error as a function of group sizes, number of personal attributes, and desired benefit from personalization. This provides actionable insights, such as which dataset characteristics are necessary to test an effect, or the maximum effect that can be tested given a dataset. We apply our framework to real-world tabular datasets using feature-attribution methods, uncovering scenarios where effects are fundamentally untestable due to the dataset statistics. Our results highlight the need for joint evaluation of prediction and explanation in personalized models and the importance of designing models and datasets with sufficient information for such evaluation.

Published as a conference paper at ICLR 2026
WHEN MACHINE LEARNING GETS PERSONAL: EVALUATING PREDICTION AND EXPLANATION
Louisa Cornelis Guillermo Bernárdez' Haewon Jeong'
Nina Miolane'
University of California, Santa Barbara, USA
"Correspondence to: Louisacornelistucsb..
ABSTRACT
In high-stakes domains like healthcare, users often expect that sharing personal information with machine learning systems will yield tangible benefits, such as more accurate diagnoses and clearer explanations of contributing factors. However, the validity of this assumption remains largely unexplored. We propose unified framework to quantify how personalizing model influences both prediction and explanation. We show that its impacts on prediction and explanation can diverge: a model may become more less explainable even when prediction is unchanged. For practical settings, we study a standard hypothesis test for detecting personalization effects on demographic groups. We derive finite-sample lower bound on its probability of error as a function of group sizes, number of personal attributes, and desired benefit from personalization. This provides actionable insights, such as which dataset characteristics are necessary to test effect, or the maximum effect that be tested given dataset. We apply our framework to real-world tabular datasets using feature-attribution methods, uncovering scenarios where effects are fundamentally untestable due the dataset statistics. Our results highlight the need for joint evaluation of prediction and explanation in personalized models and the importance of designing models and datasets with sufficient information for such evaluation.
INTRODUCTION
In critical domains like healthcare and education, machine learning models are increasingly per- sonalized by incorporating input attributes that encode personal characteristics. These attributes can be sensitive and linked to historical bias, such as sex or race, or costly, for example requiring expert-admministered medical assessments. When users provide personal attributes to model, they implicitly expect improved predictions, but does personalization consistently meet that expectation?
Personalization can indeed enhance predictive accuracy. For instance, cardiovascular risk prediction models often perform better when including sex (Paulus et al.) [2016] Huang et al.] [2024] Mosca et al.) 2011) and race (Paulus al., 2018). This is because men, women, and different racial groups exhibit different heart discase patterns. For example, hypertension is more common in African American populations (Flack et al. 2003). Hence, personalization enhances clinical predictions by capturing meaningful biological and sociocultural variation.
However, personalization can also pose risks. Including sensitive attributes such race, gender, or age can amplify biases in machine learning and perpetuate damaging inequality. For example, Obermeyer al. (2019) showed that a health algorithm relying on health care costs, an attribute shaped by racial inequities, systematically underestimated illness in Black patients compared to equally sick white patients. This reduced their access to extra care by over half.
Generally, personalization may benefit overall accuracy while harming specific groups, making such risks harder to detect. In sleep apnea classification, adding age and sex improved overall performance but increased errors for older women and younger men (Suriyakumar et al. 2023). Similar group disparities have been observed in explainable machine learning, where some users receive less faithful or reliable explanations than others (Balagopalan al.) 2022 Dai et al. 2022). These gaps matter: when explanations are less faithful to the true model logic, they can give users an inaccurate picture of how the model makes decisions, leading misplaced trust missed warning signs. Dai et al.

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illustrate this in healthcare setting where explanations for men correctly reveal the model's reliance on a spurious feature, helping doctors override bad predictions, while explanations for women hide the spurious reasoning and instead highlight clinically relevant cues, causing doctors to trust incorrect predictions and resulting in higher misdiagnosis rates. However, these studies did not examine whether personalization itself contributes to explanation disparities, making it critical to assess whether model personalization may reduce explanation quality for some users, Hence, before personalizing a model, practitioners must consider if delivers consistent gains across demographic groups in both prediction and explanation- see Figure !
(2022)
This showcases the need for quantitative framework rigor- ously and fairly assess the benefits and risks of personalization. We focus on two key goals of machine learning models in high-stakes settings like healthcare: (i) making accurate pre- dictions and (ii) providing explanations for them. Our central question is: how reliably can we evaluate whether personal- ization improves prediction accuracy and explanation quality, both overall and across groups?
Does personalization benefit. prediction & explainability?
(x(
Prediction Accuracy
Personalize (+sex)
Personalize (-9ex)
Contributions. We propose a comprehensive study of the im- pact of personalization for prediction accuracy and explanation quality in machine learning models. Specifically:
hy (X]
1. We show that even when personalization does not improve prediction, can enhance or degrade explainability in terms of how sufficient and comprehensive an explanation is. This highlights the need to evaluate both independently to ensure fairness in settings where accuracy and interpretability are critical (Section 4
Explanation Quality
Figure 1: Impact of personalization on prediction and explanation: some groups benefit, others are harmed. ho is a generic model, h, is a person- alized model that takes an additional group attribute, S.
2.
We derive distribution-aware limits on when personaliza- tion cannot be reliably tested, showing how many attributes or samples are needed in finite datasets. Our theory extends prior work beyond binary classifica- tion general supervised learning, revealing key differences between evaluating prediction and explanation in classification versus regression (Section 5)
3. We apply our proposed framework to real-world tabular datasets on classification and regression tasks, finding empirical evidence that personalization can affect explanation and prediction differently (Table We illustrate how group-level gains from personalization are fundamentally untestable, thereby precluding statistical justification across different scenarios (Section 6
Overall, offer cautionary perspective on the promise of personalized medicine and the personal- ization of machine learning in other critical domains. Even when personalizing a machine learning model could be beneficial, it might be impossible reliably prove it it-thus limiting its practical use.
RELATED WORK
Studies that investigate how personalizing machine learning models influences group outcomes (Suriyakumar al. 2023) are limited to narrow subset of performance measures and do not address explanation quality as described next. Extended related works are in Appendix A
Theory. Few works theoretically characterize the impact of personalization. Monteiro Paes al. (2022) define the Benefit of Personalization (BoP) as the smallest gain in performance that any group can expect from personalization. While the definition of BoP applies to any supervised learning task and "performance" measure, the theory supporting its use is confined to binary performance measures, such as accuracy in binary classification (0/1 loss) or false negative and positive rates (Bernoulli variables). Hence, it does not extend to continuous metrics like regression accuracy or explanation quality for regression and fails to provide a complete framework. Moreover, the thcorems make unrealistic assumptions about dataset statistics (e.g.. demographic groups of equal size) that further restrict their applicability in real-world settings. The general impact of personalization therefore remains theoretically uncharacterized.
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Table 1: Costs of model h. for group used to evaluate the impact of personalization data
(X, Y) where X = X for a generic model hu. X SX for personalized model Ag. while
X denotes the input when removing the most important features and X, its complement
(see Section 4). Personalization benefits group s E S if C(ho, s) C(hp, > 0 and harms if
C(ha, 8) Chp,, 8) < O. Incomprchensiveness is abbreviated as Incomp.
C(h, 8)
Classification
Regression
Explain Predict
Loss
Pr(M(X) +
E [| Y| |S=
Evaluation metric
AUC(M(X), Y S
[P(M()) Y S
Sufficiency
Pr(h(X) / h(X,) |S =)
E +11(X)) A(X,)]l |
Incomp.
-Pr (M(X) M(X,)) 5 5=s)
-B =[|[(8) - ((X,))
Empirical Evidence. While the impact of personalization on explanation quality has never been measured, a few empirical studies have evaluated the fairness of explanations.
Specifically, [Blagooalaan et al. (2022) train a human-interpretable model to imitate black-box model, and quantify explanation quality using fidelity, defined as agreement with the black-box predictions. They find that fidelity (and thus explanation quality and reliability) varies across groups, but their experiments are restricted to binary classifiers, and to fidelity as the only explanation method. By contrast, Dai et al. (2022) evaluate various post hoc explanation methods across different evaluation metrics. They show that explanations can vary in quality across demographic groups, leading to fairness concerns, though their experiments are also restricted to binary classifiers. Neither work considers regression tasks or examines how personalization would affect differences in explanation quality across groups. These constraints limit the practical relevance of existing empirical results, as real-world scenarios do not always align with such settings.
Link to Fairness. Fairness in machine learning aims mitigate biased outcomes affecting individuals or groups (Mehrabi et al. 2022). Past works have defined individual faireess, which seeks similar performance for similar individuals (Dwork al. 20IT) group fairness (Dwork & IIvento 2019)
Hardt et al. 2016). which seeks similar performance across different groups. Within this literature, most methods, metrics, and analyses are intended for classification tasks (Pessach Shmueli 2022).
As for the fair regression literature, authors focus designing fair learning methods (Hebert- [Hlebert-Johnson
ct al. 2018 Berk et al. 2017] Fukuchi et al., 2013; Pérez-Suay et al.) 2017, Calders et al. 2013) 3),
such multicalibaation defining fairness criteria for regression tasks (uursyy & Kakadiaris 20022]
Agarral et al. 2019). By contrast, our approach does not require equal performance across individuals or groups. Instead, we study relaxed fairness notion: ensuring that no group is systematically harmed by personalization. We propose a framework to evaluate whether this weaker fairness criterion is satisfied, both theoretically and empirically, rather than proposing corrective algorithms.
BACKGROUND: BENEFIT OF PERSONALIZATION FRAMEWORK
Let X, y denote, respectively, the input feature, group attribute, and outcome spaces. A personal- ized model hpp X' x S + y' aims to predict outcome variable y y using both input feature vector x E X' and a vector of group attributes s E S. In contrast, a generic model fo : X - y does not use group attributes. We consider that a fixed data distribution P Px,s, is given, and that ho and hp are trained to minimize a loss over a training dataset Dtrain-
Cost. We first evaluate how model h (generic personalized) performs for given group.
Definition 3.1 (Expected Group Cost). The expected cost of model h for the group S measured by the cost function cost is defined as: C(h, s) A Eclcost (h, X. Y) S s), where X X for generic model heg and X (X, S) for personalized model hpp
In what follows, we use cost and expected cost interchangeably, with the convention that lower cost means better performance. In practice, the cost is evaluated over a set, D, that is independent from the train set. Costs of interest are shown in Table top rows focus prediction accuracy (loss and evalu- ation metrics), while bottom ones address explanation quality (sufficiency and incomprehensiveness)... As explanation metries are less common than accuracy metrics, we review them next.
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Cost for Explainability. We assume access an auxiliary explanation method that assigns impor- tance scores to input features e.ggu..eeeeg.......... based on the magnitude of input gradients. Then, the explanation quality metric measures whether the features with the highest importance scores actually meaning- ful (see) Nauta et al. 2023) for review). We use sufficiency and incomprehensivenes. as explanation quality metrics illustrate our framework and apply our framework using Integrated Gradients, DeepLIFT and Shapley Value Sampling. These metrics quantify the change in prediction when the most important features are removed or retained. For comprehensive discussion of our rationale in selecting these metrics, see Appendix A We emphasize that importance is defined relative to the explanation method, not to any ground truth. This is by design: the goal is not to assume a known set of truly important features, but to assess how well given explanation method identifies features that meaningfully affect the model's prediction.
Benefit of Personalization. We can quantify the impact of personalized model in terms of the benefit of personalization, defined next:
Definition 3.2 (Group Benefit of Personalization (G-BoP) (Monteiro Paes et al. from personalizing a model can be measured by G-BoP hp. s) Choo, s) ()), com- paring the costs of the generic ho and personalized models hi for group E S. By convention, G-BoP > 0 if the personalized model performs better than the generic one.
2022 The gain
We use G-BoP p and G-BoPx x to refer to G-BoP for prediction and explanation respectively see Appendix B Table 3 for concrete examples. For example, for prediction evaluation in regression [[),p(X, [X, s) Y SS $), and for incom- prehensiveness E [| hp X s) hy (X, , s s(,) S - [l (X) ho ,)) S s]. To evaluate whether all groups benefit from personalization, or if any are harmed, we use the following definition as our final assessment metric:
using MSE GoBoP is: E
(X) - Y S s] E
Definition 3.3 (Benefit of Personalization (BoP) (Monteiro Paes et al. 2022)). The BoP is defined as: ly: +)), ; i.e., the minimum group BoP value across groups S to capture the worst group improvement, or degradation, resulting from personalization.
7 (ho, hp) min.es (G-BoP (Ginnes(G-BoPo//
A positive indicates that all groups receive better performance with respect to the cost function. Contrary to this, a negative y reflects that at least one group is disadvantaged by personalization. negative, the practitioner might want to reconsider the use of personalized
When y is small attributes in terms of fairness with respect to all groups. When y is used to evaluate improvement in prediction and explanation, it is referred to 7/P and 7x. respectively.
Remark. The definitions of G-BoP and y were originally introduced in Monteiro Paes et al. (2022). While formally applicable to any cost function, these definitions have only been studied and used with binary costs-Such as 0-1 classification loss or false positive/negative rates-due to theoretical gap that prevents their use with continuous costs, including an analysis of prediction and explanation for regression tasks. Since a holistic analysis of prediction and explanation across machine learning tasks is our primary focus, addressing this gap is central to our contribution in Section
4 IMPACT OF PERSONALIZATION ON PREDICTION AND EXPLAINABILITY
This section provides the first formal analysis showing that personalization's 's effect on prediction does not determine its effect on explainability, highlighting the need to evaluate both. A common intuition machine learning is that if personalization improves prediction, it should also improve the quality of explanations derived from the model. This intuition is reflected in the XAI literature, where evaluation practices often conflate model accuracy with explanation correctness. For example, recent survey notes that "these commentaries relate to the inherent coupling of evaluating the black box' predictive accuracy with explanation quality. As pointed out by Robnik-Šikonn and Bohanee Soabnik-Sikonjj & Bohanec (018), the correctness of an explanation and the accuracy of the predictive model may be orthogonal" (Nauta et al. (2023). This assumption also appears implicitly in many high-stakes applications, where explanations from high-performing models are used to draw insights about real-world structure (Elmarakeby et al.) 2021 Chereda et al. 2021). Despite its prevalence, this presumed connection between predictive performance and explanation quality has not been formally analyzed in the context of personalization.
Theorems 4. and 4.2 prove that prediction gains and explanation gains can diverge, demonstrating that gains prediction performance (measured by BoPp) and gains in explanation quality (measured
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by BoPx) need not align. Theorem 4.4 where the two align. Though idealized, this boundary case clarifies when practitioners can trust prediction and explanation to align. Proofs are in Appendix C
provides partial converse, identifying an additive setting
No Prediction Benefit Does Not Imply No Explainability Benefit. The following theorem shows that a personalized model may match generic model in aceuracy. yet offer better explanation. Thus, focusing only on prediction can overlook significant interpretability gains.
Theorem 4.1. There exists data distribution Px.S,y Y.Y. such that the Bayes optimal classifiers ho and hp satisfy 7p (ho, hp) = 0 (with "P measured by 0-1 loss) and 7x (ho, hp) > 0 (with "IX measured by sufficiency and incomprehensiveness).
Example 1. We illustrate Theorem [.] with real-world example. Consider a model with many imput features that are partially redundant, for instance, loan approval model that uses credit score, income, and debt-to-income ratio. Adding a personal feature that is highly correlated with existing features may not change the predictions. However, it can alter the explanation if that feature is the most direct or informative input. For example, adding a binary feature like "pre-approved by another bank", which is strongly correlated with existing features, may leave predictions unchanged, but explainer might now assign most importance to this new feature because it provides a clearer justification. Figure [5] illustrates the construction behind the proof for sufficiency, where both generie hg and personalized h, models predict perfectly (left side), yet only keeping the most important feature for each (right side) shows that the personalized model is more explainable. For this distribution, G-BoP p(ho, hp. s) = 0 and G-BoP x (ho. hp.s)) s) > 0 for each group s. so all groups are impacted similarly by personalization. Figure Gillustrate the proof for incomprehensiveness.
No Prediction Harm Does Not Imply No Explainability Harm. A personalized model may match a generic model in accuracy yet offer worse explanations. Thus, focusing only on predictive performance can obscure significant harms to explainability.
Theorem 4.2. There exists data distribution Px.s.y such that the Bayes optimal classifiers ho and hip satisfy 7PP = 0 (with 7p measured by 0-1 loss) and 7x < (with "xx measured by incomprehensiveness/
Example 2. To illustrate Theorem consider a pneumonia detection model using chest X-ray findings that perfectly predict outcomes. Adding white blood cell count leaves accuracy unchanged, but the personalized model now splits importance between X-ray findings and white blood cell count. The explanation is worse because it's now split across two features, making it less clear which feature drives the decision, even though the X-ray alone was already perfectly predictive.
Additionally, Theorem 4.3 proves this phenomenon for both sufficiency and incomprehensiveness by showing how personalization might not alter predictive accuracy across groups but it might affect explainability differently for different groups
Theorem 4.3. There exists data distribution Px,s,x Y such that the Bayes optimal classifiers ho and hn satisfy G-BoP p (ho, hp. s) = O (measured by 0-1 loss) for all groups s, but some groups have G-BoP x (ho, hpp 8) while others have G-BoP x (ha, hpp 8) (measured by sufficiency and incomprehensiveness)..
Figure 7 illustrates the proof for sufficiency, where both generic ho and personalized hp models predict perfectly (left), yet only keeping the most important feature for each (right) shows that the personalized model is more explainable for the group (s' 1, 0), and less explainable for group (s'=0,8 8 = 1). Figure iillustrates the proof for incomprehensiveness.
and 4.3 show that knowing Yp 0 provides information about 7xx This motivates the need to evaluate both prediction and explainability, as we offer to do in Section 5
Together, Theorems 4.1
No Explainability Benefit Can Imply No Prediction Benefit. We now ask the converse: can lack of explainability benefit imply no predictive benefit? We show this is true for a simple additive model, as long as two notions of explainability measures -sufficiency and incomprehensiveness- do not any benefit.
Theorem 4.4. Assume that hg and hpp are Bayes optimal regressors and Px Y follows additive model, i.e. Y = a X, + + axX, + a141S1 +... + a Sx + c, where X....., Xx, and S1.... .Sk are independent, and e is independent random noise. Then, for s S we have G-BoP tha hp. s) = G-BoP incomp (hn. hpp 5) = 0, then G-BoP (ho, hp. s) = 0. Consequently, if. for all groups 8, G-BoP sug ha, hpp 5) G-BoP meomp (ompphhn (bo, hpp 8) = 0, then 7pp 0.
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This theorem demonstrates that under an additive model, if there is no benefit in explanation quality, then there is also no benefit in prediction accuracy. Figure 9 illlustraees this proof. Additionally, we get the following corollary:
Corollary 4.5. Under the assumptions of Theorem O, then it also holds that G-BoP sug(ho hpp s) quently, if "/p * O, then there exists group G-BoP incoopp (ho, lip 8) +
4.4 if ffr s S, we have G-BoP (ho, hp. s) # F
O or G-BoP incop (h,, up. 8) # O. Conse- S such that G-BoP sug (ho, hps s) * 0 or
This theorem shows that if personalization affects prediction, it must also affect explanation for at least one explainability measure and one demographic group. This result establishes rare direct link between explanation and prediction, in a simplified linear setting. Proving this for general models remains an open question.
TESTING PERSONALIZATIONNSS S IMPACT ON PREDICTION AND EXPLANATION
Having emphasized the importance of evaluating both prediction and explainability, we now introduce methodology to assess them in practice. The true BoP y, defined over the whole data distribution, inaccessible and needs be estimated from finite samples. Then, its estimate is positive, one must consider whether the true 7 is also likely to be positive. In scenarios where personalization incurs a price-such as requesting sensitive user information-ove should determine how large ? This section analyzes the
must be to ensure that the true benefit exceeds desired threshold validity of BoP hypothesis testing and provides guidelines for its application.
5.1 VALIDITY OF HYPOTHESIS TESTS
Hypothesis Tests. Given an audit dataset D with & binary group attributes, we want to know whether personalization improves each group by at least > O. We formalize the null and the alternative hypotheses using a standard framework for the BoP (Monteiro Paes et al. 2022):
Ho (hos hpi D) S + Personalized hp does not bring any gain for at least one group, H1 (ho, hp: D) > + Personalized hp yields at least improvement for all groups.
Importantly, Ho and H1 are not complementary, because we want to reject the null in favor of the alternative if the impact is both positive and practically meaningful, i.e., 2 e. With these hypotheses, we ask: can we rule out that there is no harm and assert meaningful benefit of at least e?
The improvement e in cost function units and represents the improvement for the group that benefits the least from the personalized model. The value is domain-specific and should be chosen by the practitioner. For example, in healthcare, if personalization requires time-intensive and sensitive inputs-like mental health assessments-it may only be justified if improves diagnostic accuracy by at least a few points, making a clinically and ethically meaningful threshold. In such cases, becomes threshold for balancing speed and clinical value.
Once is chosen, the practitioner may run the hypothesis test by computing the estimate & on D and follow the rule: 7 2 E = Reject Ho: Conclude that personalization yields least e improvement for all groups. We note that different testing strategies could also be used. To capture this generality, we define a decision function
(ho, hy. D,c) + {o. 1}, where v indicates rejection of Ho. In our case, (ho, hp. D) (6 2 c). Regardless of its specific form, our goal is to assess the validity
of any test aiming to evaluate the impact of personalization 7.
Invalidity of the Tests: Probability of Error. of a test in terms of its probability of error, defined as: Pa (Pr(Rejecting Ho is true) + Pr(Failing to reject Hol Hy is true)). The probability of er- ror is a composite measure that weights Type and Type II errors, under the assumption that Ho and H₁ are equiprobable with probability 1/2.
We quantify the (in)validity
We propose to derive a minimax lower bound on the probability of error Pe. This involves considering the worst-case data distributions that maximize Pe and the best possible decision function v that minimizes it. Notably, a high lower bound guarantees a high error probability for any test with Ho and Hn the BoP, flagging settings where testing the impact of personalization is unreliable.

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Clessfeation
Categorical iBoP
0.25
Regression
Laplace iBoP
Gaussian iBoP
Dutzset size
Dataeel size
Datases size
N
.01
.01
0.0
.07
d
10 Number of srtbiuts
20
10
20
10
20
.71
Number of attributes
Number of attributes k
Figure 2: Testing personalization for prediction and explanation depends learning task. Lower bound on the probability of eтror Pe with respect to number of personal attributes k. for dataset sizes N 10a 10S, and 10a with 0.01. In classification (orange), the bound fixed by the categorical nature of the individual BoP (iBoP) and is identical for prediction and explanation. In regression (green and blue), Pe depends on the spread of individual BoPs te--paaameteiized by variance o² (Gaussian) or scale b (Laplace). Smaller variance or scale allows more attributes before testing becomes unreliable (Pa 0.25). Computed for m = [N/d] samples per group with d 2 groups.
Theorem 5.1. Consider k binary group altributes, S a 13;,, that specify d - |S| 2 groups, each containing my individuals, j= =1,... d. Let Ho (resp. H1) denote the data distributions under which the generic model hi (resp. the personalized model htp performs better, i.e., 7 S (resp. E). Denote H' (resp. H₂) the data distributions that are mixtures of components in Ho (resp. H1). Then, there exists P E Hl' (resp. Q E Hi), for which the individual benefit of personalization
B = cost(ho, X, Y) cost p X, follows a probability density (resp. V¹ for one group), where E, (B] and Epe [B] e, such that:
my 1
(B) pBB
min max Pe
Ey
(I)
POH AEH
2
2vd
The proof for Theorem 5.1 is in Appendix D.1 We note that the lower bound cannot exceed 1 1/2, which is expected. random guessing would yield P. 1/2. Consequently, the optimal probability of error (the minimum over all decision rules v must lie in the interval [0, 1/2].
binary hypothesis test with equal priors, a decision rule corresponding to
Crucially, this lower bound can be tailored the practitioner's specific case, i.e., the distribution of the individual benefit B under Ho and Hj. For example, if B is known observed to follow Laplace distribution with scale the practitioner should choose p Laplace(0.bl b) and p (aplace(e, b). Figure 3 shows the expression of the lower bound for the Laplace distribution (proof provided in Appendix [.8]. If none of these standard distributions provided in the appendix are good match for the BoP distribution, Theorem remains valid for any distribution, as long as its probability density function is known. such cases, practitioners may use flexible density estimation tools, such as normalizing flows, approximate the PDF from data and apply Theorem directly, The next corollary expresses it for distributions in the exponential family, which we use to find a bound for when B follows Gaussian distribution (see Appendix D.5).
Corollary 5.2. The lower bound in Th. 5.1 for distributions P.P' in the exponen- tial family (parameter 6, moment generating function M) minl maxp,e P. PEHHi
(2(200) m; -1] with A0 = 0- (A
-1
2vd
à
2
The proof for Corollary 5.2 in Appendix D.2 These results generalize and tighten an existing bound for categorical distribution only (Monteiro Paes et al. 2022) (see Appendix D.3) and provide the first general framework to evaluate the (in)validity of hypothesis tests on personalization for prediction and explanation, and across supervised machine learning tasks.
Remark. These bounds apply to any metric that be formulated as evaluation score and used to compare model performance across subgroups (across classification and regression)-ice our statistical testing tools are not tied any particular explainability or performance measure method.
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Experimental Design: Group Attributes, Sample Size, and Detectable Gain. We investigate how probability error depends on the dataset, and how determines the practitioner's ability to test the impact of personalization. For example, with a fixed number of individuals N, larger number of personal attributes & increases the number of groups d = 2 reducing the number of samples per group, which increases the risk of error. Accordingly, if the practitioner commits fixed & to test a desired gain e (resp. fixed k and N), they need a minimum group size m to keep the error bound below desired level, shown next.
Corollary 5.3. To ensure min max P. I for chosen threshold v, equal group sizes must satisfy
1o2(122"1-20)++1 for a categorical BoP. log(1++2)
2 Mmmu where: Mminn
Momin = a² log (1
for Gaussian BoP of variance o², and Muiin =
log [ (1+2) 2 (1 - (1-20)²) for a Laplace BoP of scale b.
App. E provides practitioners with another dataset-specific feasibility check: Corollary E. bounds the maximum number of attributes that can be used before the lower bound error exceeds 25%.
5.2 PRACTICAL CONSIDERATIONS WHEN TESTING PREDICTION AND EXPLANATION
We examine how the lower bound in Theorem 5.1 depends on the distribution of individual BoPs B, and how this determines the practitioner's ability to test for prediction or explanation gains.
Testing Prediction and Explanation in Classification Tasks. When the task is classification with 0-1 loss, the individual BoPs follow categorical distributions with values in { 0, 1}:
Bp (X) Y) (X, Y). Bx (ho(X) * ho
(X, S) + h (X,, Sy))
for prediction and explanation (e.g., sufficiency). respectively -see costs in Table 1 In this setting, the lower bound in Theorem i. identical for prediction and explanation (see Figure bottom): either both are testable, neither is.
Figure Sggure[show, shows the lower bound on the probability of error Pe as a function of k. for typical dataset sizes in medical settings N E {10?, 10a, 10a). In classification (orange curves), even small number of personal attributes l leads to high error lower bounds. For instance, at N = 100 and & = 1, the bound already exceeds 40%, making reliable testing impossible for both prediction and explanation.
Testing Prediction and Explanation in Regression Tasks. In regression, the situation more nuanced. For instance, with MSE loss, have continuously valued individual BoP random variables:
Bp (X) - (X. S) Y Bx /ha (X) ho (X -1,, -ho (x,,, S
for prediction and explanation, respectively. Suppose these follow Laplace distributions with scales by and bx. Then, the lower bounds will differ for prediction and explanation (Figure 3 bottom): one could be testable while the other is not, highlighting an asymmetry absent in the classification case.
As illustrated in Figure smaller scale values (b) allow for a larger number of personal attributes kmax to be tested without theoretical barriers. Unlike classification, there is no proof that regression tasks cannot support reliable testing of personalization for dataset sizes encountered in medical settings N E (102, 100, 104), even with many personal attributes k.
6 CASE STUDIES: EVALUATING PERSONALIZATION ON REAL DATASETS
We illustrate how to use our results to investigate the impact of personalization on prediction and explanation, to reveal the many cases where reliable testing is in fact impossible. This section focuses real-world healthcare scenario, while other scenarios are provided in Appendix G Remark. Across these hypothesis tests we always evaluate if there is a benefit of personalization, i.c.
y > 0, but interested practitioners may want to evaluate whether existing machine learning model could harm one group. In that case the hypothesis test should be flipped, i.e. 7 < c < O.
Healtheare Scenario. Consider MIMIC-III (Medical Information Mart for Intensive Care) (Johnson ct al.) 2016), a dataset of patients admitted to critical care units at a large tertiary hospital conttinning vital signs, medications, lab results, diagnoses, imaging reports, and outcomes such as length of stay.
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1
Set variables
Set min benefit. of personalization needed justify collecting personal attributes Decide consider the test unreliable if its probubility of error provably above
2
Train
Train generic model ho and personalized model hg Compute estimate of benefit of personalization (BoP) 5
3
Test BoP
If f > c rejoct mull hypothesis, conclude that there is a bencfit of personalization Else: camnot reject mull hypothesis, cannot conclude
4
Audit test
If lower bound on the probability of error above test is decmed mureliable Elsc: cannot conclude on whether probability error above v
Get lower bornd
1
Prediction & Explanation
Classification
Histogram of iBoP
min max P. PycRI
2
2
(1
Prediction Explanation
Regression;
Laplace Fit
Data
min
2
1
my
Figure 3: Summary of the steps to test BoP for prediction and explanation.
Suppose that practitioner has developed deep learning model to predict a patient's length of stay (regression) or whether the length of stay exceeds days (classification) - see details in Appendix F.] They are wondering whether their model should be personalized by including (or not) two personal attributes: Age x Race E {18 - 45, 45+} x [White(W), Non WNW). However, they are concerned this could disadvantage some groups, not only by reducing prediction accuracy but also by limiting the ability to uncover factors that explain critical care duration. We provide a step-by-step procedure to our framework to evaluate the benefit of personalization (summarized in Figure 3).
① Select e and v. The practitioner first chooses the minimum improvement they expect from personalization for prediction and 6x for explanation (e.g., 6p cx 0.002). They then set tolerance threshold e for the probability of error beyond which they will not trust the hypothesis test (e.g., 25%).
② Report empirical benefits of personalization. The practi- tioner trains ho and hp (with ad- ditional attributes age and race) and reports empirical personal- ization benefits in Table (0-1 loss for classification, MSE for regression). They utilize the Integrated Gradients explainer method and evaluate it using the sufficiency and incomprehensive- ness metrics. Across tasks, some groups seem show benefits for prediction but harm for explana- tion, and vice versa. This should not be surprising given the re- sults of Section 4 which show that prediction and explanation gains can diverge.
(A) Classification
(B) Regression
Lower Bound P.
0.5
00202
40%0 PITO
40% PITOO
0.002
25% error
25% OTTOr
0.25
6. Minimum BoP required (10
10.
10. Minimum BoP required (10 ")
Prediction, Explanation Prediction ..... w...Incomp.... Sufficiency
Figure 4: Lower bound of Pa vs. on MIMIC-III: classification (A) and regression with Laplace (green) and Gaussian (blue) mod- els for the individual BoPs (B). At the minimum BoP set in this case study 0.002), testing personalization for prediction and explanation is impossible for classification (same for sufficiency for regression) as Pa 2 40% regardless of the hypothesis test.
3 Perform hypothesis test. The practitioner assesses whether f exceeds ep or ex. It does for all metrics with a positive 7, hence they can reject the null hypothesis for these cases.
Assess reliability of the results. Next, the practitioner assesses whether the empirical results are statistically meaningful using the framework from Section [3] For the classification model, the lower bound on the probability of error exceeds 40% (Figure 4 = 0.002), indicating that it is not even possible to test whether personalization helps or harms performance. As a result, the practitioner would likely retain the generic classifier. For the regression model, they examine the distributions of individual BoPs, Bp and Bx (Figure bottom, and Appendix E.I). Sufficiency is best fit by Gaussians with varying variances; prediction and incomprehensiveness align with Laplace distributions of different scales. The corresponding lower bounds error exceed 40%

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Table 2: Benefits of personalization (C(h)) Chpp)) ) on the MIMIC-III test set for predicting length of stay (LOS)): regression or classification (LOS 3 days). Incomprehensiveensss is abbreviated as incomp. and population as pop. Values that are worsened by hp are colored red.
Classification
Regression
Group
Prediction
Incomp.
Sufficiency
Prediction
Incomp.
Sufficiency
White, 45+
8443
0.0063
-0.0226
0.0053
8379
0.0021
-0.0906
0.1914
White, 18-45
1146
0.0044
0.0489
0.0244
1197
0.0023
0.1219
0.2223
NonWhite, 45+
3052
-0,0026
-0.0023
0.0029
3044
0,0108
-0,0501
0,3494
NonWhite, 18-45
696
-000216
0.0560
0.0072
717
0.0212
0.0441
0,3293
All Pop.
13337
0.0026
-0.0077
0.0065
13337
0.0051
-0.0550
0,2376
Minimal BoP
13337
-0,0216
-0,0226
0.0029
13337
0,0021
-0,0906
0,1914
for aafficiency-makiiee it untestable- fall below 5% for prediction and incomprehensiveness (Figure 4 E 0.002). Now, we provide insights that were gained from applying our framework to this scenario, and others in Appendix G
Insight: A high empirical benefit of personalization can be misleading. As shown in Table and lower bounded in ④ the regression experiment reports the largest apparent benefit for sufficiency (9 = 0.1914), yet the data did not permit a valid test, making the result inconclusive. Prediction showed a much smaller benefit (f- = 0.0021), but our analysis found no barriers to testing, and the null was rejected. This shows that large & does not guarantee a valid conclusion; empirical values must be paired with our framework to assess validity.
The choice of improvement threshold is key. Increasing e reduces the lower bound on the probability of eπror Pe. making hypothesis testing potentially less unreliable (Figure g. but also raises the bar for rejecting the null, requiring a larger 5. Thus, e trades off test validity against ability to detect effects. In real- world applications, reflects the minimum performance gain a practitioner needs to justify collecting costly personal data (e.g.. genetic markers). For instance, c = 0.002 in length-of-stay prediction accuracy may be worthwhile. Converting this back the original units, the improvement is approximately 0.06 days per patient. Although small at the individual level, across 100 patients per day this amounts to cumulative gain of about 6 patient-days of hospital stay estimated more accurately each day. This illustrates that should be set based on practical value, not statistical convenience, even though higher e values tend to make hypothesis tests more reliable.
Insight:
Insight: Results do not depend on the explanation method. Table reportss results with Integrated Gradients (Sundararajan et al.) 2017p. Since our framework applies to any explanation method, we test whether this choice affects the evaluation of the impact of personalization. Appendix G analyzes Shapley Value Sampling (Strumbelj & Kononenko 2010) and DecpLIFT (Shrikumar et al. 2017). finding substantial agreement across the methods- though effect sizes differ.
Insight: Personalization is hard to evaluate across medical datasets. To show the practicality of the
framework, we also include experiments on the Heart Dataset (Janosi et al. 1989) and the MIMIC- III Kidney injury cohort (Suriykkumar et al.) 2023), again utilizing range of explanation methods (see App. G Using the same as above, no test is valid for the V.S explainer on the Heart dataset (see caption of Table D. This shows the difficulty of reliably evaluating personalization. Generally, this analysis points to a limitation of personalized medicine and healthcare: while personalization may yield improvements, demonstrating them reliably can be infeasible-restrieting restricting applicability.
CONCLUDING REMARKS
We present unified framework for evaluating the benefits of personalization with respect both prediction accuracy and explanation quality, facilitating nuanced decisions regarding the use of personal attributes. Our analysis shows that in many practical settings, particularly classification tasks, the statistical conditions required to validate personalization often unmet. As result, even when personalization shows empirical gains, meaningful validation may not be feasible.
Limitations & Future Work. While we relax several assumptions relative to prior work, our theoretical results still rely on assumptions not always met in practice; further reducing them remains an important direction. Additionally, while focused explanation quality due to its importance in clinical adoption, our results in Section SaationnSexxten extenn to other goals. Future work can build on this framework to evaluate additional desiderata such as fairness, robustness, and uncertainty calibration.
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ACKNOWLEDGMENTS
Nina Miolane acknowledges support from NSF Grant 2313150 and the NSF CAREER Award 240158. Guillermo Bernárdez acknowledges support from the Chan Zuckerberg Initiative, the UC Noyce Foundation, and Arlequin AI. Louisa Cornelis acknowledges support from the Chan Zuckerberg Initiative, the UC Noyce Foundation, and the NSF Graduate Research Fellowship Program (NSF GRFP) Grant 2139319.
REFERENCES
Alekh Agarwal, Miroslav Dudík, and Zhiwei Steven Wu. Fair regression: Quantitative definitions and reduction-based algorithmss, 2019. URL https://arxiv.. org/ abs 1
Subhan Ali, Filza Akhlaq. Ali Shariq Imran, Zenun Kastrati, Sher Muhammad Daudpota, and Muhammad Moosa. The enlightening role of explainable artificial intelligence in medical health-
care domains: A systematic literature review. Computers in Biology and Medicine, 166: 166:107555, 2023. ISSN 0010-4825.
doi:
107555. URL https
www. www.scciencedirect com
Aparna Balagopalan, Haoran Zhang, Kimia Hamidieh, Thomas Hartvigsen, Frank Rudzicz, and Marzyeh Ghassemi. The road explainability is paved with bias: Measuring the fairness of explanations. In 2022 ACM Conference on Faimees, Accountability, and Transparency, FAceT '22. ACM, June 2022. doi: 10. 10.1145/3511146.333311 URL http: http:://dx.oo www.dooi.ooo http://ddd.oo..g/0...11111111
3531146. 3533179
Richard Berk, Hoda Heidari, Shahin Jabbari, Matthew Joseph, Michael Kearns, Jamie Morgenstern, Seth Neel, and Aaron Roth. A convex framework for fair regression, 2017. URL https
//arxiv.org/ abs aaaaaba11100 1706602400
Toon Calders, Asim Karim, Faisal Kamiran, Wasif Ali, and Xiangliang Zhang. Controlling attribute effect in linear regression. 2013 IEEE 13th International Conference Data Mining, pp. 71-80, 2013. /tttps:///attt
semant wiscchiar........c.oooccc..c...e org /Copg//ooppssi:ie 1664417888
Tirtha Chanda, Katja Hauser, Sarah Hobelsberger, Tabea-Clan Clara Bucher, Carina Nogueira Garcia, Christoph Wies, Harald Kittler, Philipp Tschandl, Cristian Devarrete-Dechele Sebastian Podlipnik, Emmanouil Chousakos, Iva Crnaric, Jovana Majstorovic, Linda Alhajwan, Tanya Foreman, Sandra Peternel, Sergei Sarap, irem Özdemir, Raymond I.. Barnhill, Mar Llamas-Velasco, Gabriela Poch, Sören Korsing, Wiebke Sondermann, Frank Friedrich Gellrich, Markus V. Heppt. Michael Erd- mann, Sebastian Haferkamp. Konstantin Drexler, Matthias Goebeler, Bastian Schilling, Jochen Utikal, Kamran Ghoreschi, Stefan Fröhling, Eva Krieghoff- Thiem, Alexandris Dimitrios, Amr Mohammad Ammar, Ana Sanader Vučemilović, Andrea Miyuki Yoshimura, Andzelka Ilieva, Anja Gesierich, Antonia Reimer-Taachhnnreed Kolios, Arturs Kalva, Arzu Ferhatosmanoglu, Aude Beyens, Claudia Pföhler, Dilara Ilhan Erdil, Dobrila Jovanovic, Emoke Racz, Falk G. Bechara, Federico Vaccaro, Florentia Dimitriou, Gunel Rasulova, Hulya Cenk, Irem Yanatma, Isabel Kolm, Isabelle Hoorens, Iskra Petrovska Sheshova, Ivana Jocic, Jana Knuever, Janik Fleißner, Janis Raphael Thamm, Johan Dahlberg, Juan José Lluch- Galcerá, Juan Sebastián Andreani Figueroa, Julia Holzgruber, Julia Welzel, Katerina Damevska, Kristine Elisabeth Mayer, Lara Valeska Maul, Laura Garzona-Navas, Laura Isabell Bley, Lau- renz Schmitt, Lena Reipen, Lidia Shafik, Lidija Petrovska, Linda Golle, Luise Jopen, Magda Gogilidze, Maria Rosa Burg, Martha Alejandra Morales-Sánchez, Martyna Slawinska, Miriam Mengoni, Miroslav Dragolov, Nicolás Iglesias-Pena, Nina Booken, Nkechi Anne Enechukwu, Oana-Diana Persa, Olumayowa Abimbola Oninla, Panagiota Theofilogiannakou, Paula Kage, Roque Rafael Oliveira Neto, Rosario Peralta. Rym Afiouni, Sandra Schuh, Saskia Schnabl-Scheu, Seçil Vural, Sharon Hudson, Sonia Rodriguez Saa, Sören Hartmann, Stefana Damevska, Stefanie Finck, Stephan Alexander Braun, Tim Hartmann, Tobias Welponer, Tomica Sotirovski, Vanda Verena Gerlinde Frings, Viktor Simeonovski. Zorica
Alexander Salava, Alexander
Antonios G. A.
Bondare-Ansberge Verena Ahlgrimm-SSest Zafirovik, Tatjana Maul, Saskia Lehr, Marion Wobser, Dirk Debus, Hassan Riad, Manuel P. Pereira, Zsuzsanna Lengyel, Alise Balcere, Amalia Tsakiri, Ralph P. Braun, and Titus Brinker. Dermatologistlikke explainable ai enhances trust and confidence in diagnosing melanoma. Nature Communicationss 15(1), January 2024. ISSN 2041- doi: 10. 10.1038//41467-022 -23-0955--------444-- URL http::/dx. doi /tt.://....o.....11///11111..111

---

Published as a conference paper at ICLR 2026
Hryhorii Chereda, Annalen Bleckmann, Katharina Menck, et al. Explaining decisions of graph convolutional neural networks: -stient-specific molecular subnetworks responsible for metastasis prediction in breast cancer. Genome Medicine, 13(1):42, 2021. doi: 10. 10.11886413333--0--00084 URL http:::::/ddo...orggg.
186/ 00010/11111300000000-553333333 00845--0777
Jessica Dai, Sohini Upadhyay, Ulrich Aivodji, Stephen H. Bach, and Himabindu Lakkaraju. Fairness via explanation quality: Evaluating disparities in the quality of post hoc explanations. In Pro-
ceedings of the 2022 AAAI/ACM Conference on AI, Ethics, and Society, AIES '22. pp. 203-214
ACM, July 2022, doi: 10. 10.1 10.1145/35144944333433 URL wtl.http:// http::/dxx..x.e http::/dxddoii..oooo wwwp://wo...ogg....ooo /
351 5514094.00 3534159
Sanjoy Dasgupta, Nave Frost, and Michal Moshkovitz. Framework for evaluating faithfulness of local explanations, 2022. URL
Jay DoYoung, Sarthak Jain, Nazneen Fatema Rajani, Eric Lehman, Caiming Xiong, Richard Socher, and Byron C. Wallace. ERASER: A benchmark to evaluate rationalized NLP models. In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 4443-4458 Online, July 2020. Association for Computational Linguistics. doi: 10. 18653/v1/2020 aww.://....mmmii..44..m URL https
7/aclantholoo howanohooooyy. wwt.oorggg00/ 2020. -cw.mall.lnaammm.mmm. 408
Yuhan Du, Anna Markella Antoniadi, Catherine McNestry, Fionnuala M. McAuliffe, and Catherine Mooney. The role of xai in advice-taking from a clinical decision support system: A comparative user study of feature contribution-based and example-based explanations. Applied Sciences, 12 (20), 2022. ISSN 2477 doi: (0.3390/app122003333, URL https: / /wwww//ww.../..// mdpi com/
1077-3417/12/20/10
Cynthia Dwork and Christina Ilvento. Fairness under composition. Schloss Dagstuhl Leibniz- Zentrum für Informatik, 2019. doi: 104230LLPPCSS URL https: //drops/
dagstuhl /ent iinttiiiiinn eeseeeeeeeeee http:///cmmeeett.13//////1/
I ITCS 2019.33
Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Rich Zemel. Fairness through awareness, 2011. URL
3913
Haitham A. Elmarakeby, Jaeil Hwang. Rami Arafeh, et al. Biologically informed deep neu- ral network for prostate cancer discovery. Nature, 598(7880):348 352, 2021. doi: 10. s415 0111566001-03922-12
uRL.https://deot http:/diio..gg//////// org/ http:/110.1113/11115811 020011886--01-0222-2222
John M Flack, Keith C Ferdinand, and Samar Nasser. Epidemiology of hypertension and cardio- vascular disease in african americans. The Journal of Clinical Hypertension, 5(1):5-11,200
Kazuto Fukuchi, Toshihiro Kamishima, and Jun Sakuma. Prediction with masel-based neutral- ity. In ECML/PKDD, 2013. URL /ttpps://appp
6964544
Susanne Gaube, Harini Suresh, Martina Raue, Eva Lermer, Timo K. Koch, Matthias F, C. Hude- cek, Alun D. Ackery, Samir C. Grover, Joseph Coughlin, Dieter Frey, Felipe C. Kitamura, Marzyeh Ghassemi, and Errol Colak. Non-task expert physicians benefit from correct ex- plainable Al advice when reviewing X-rays. Scientific Reports, 13:1383, January 2023. doi: 00.1038/541588-023 11-5588022-2863--2 28633-..
Furkan Gursoy and Ioannis A. Kakadiaris. Error parity faimess: Testing for group fairness in regression tasks, 2022. utLptts: http::///wwwwwwwwo/bb/////////// www.oosg..b....o..oo.. httpg////aa/////022223/
Moritz Hardt, Eric Price, and Nathan Srebro. Equality of opportunity in supervised learning, 2016. URL htt.:pps://arriv..orrraaa///eeee
Ursula Hebert-Johnson, Michael Kim, Omer Reingold, and Guy Rothblum. Multicalibration: Cal- masses. In Jennifer Dy and Andreas Krause (eds.). Proceedings the 35th International Conference Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 1949-1948. PMLR, 10-15 Jul 2018. URL https:://rrceeeh ml www.:rpreees.eeeee http:://.ppcoccoosssssrrsrressrr johnson 8 htm
ibration for the CCompuationally--dentffaaal
12

---

Published as a conference paper at ICLR 2026
Bi Huang, Mayank Dalakoti, and Gregory Y H Lip. How far from accurate sex-specific risk prediction of cardiovascular disease? One size may not fit all. Cardiovascular Research, 120(11):1237-1238) 06 2024. ISSN 0008-6363,
doi: 10.1093/cwr/evae133. U
7/doi.orge
1093/
cvr cvr 2tt.0//////////13//3333 cvae135
Alon Jacovi and Yoav Goldberg. Towards faithfully interpretable NLP systems: How should we define and evaluate faithfulness? In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault (eds.), Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 4198-4205, Online, July 2020. Association for Computational Linguistics. doi: 10.18653/v1/ 2020.acl-main.333 ht..tttsss //acc antho 1 weny.org/20220 2020, acl -ma maiin.3. 3
Andras Janosi, William Steinbrunn, Matthias Pfisterer, and Robert Detrano. Heart Disease. UCI Machine Learning Repository, 1989. DOI: https://bbiorrg/002444/2////
Alistair E. E.W. Johnson, Tom J. Pollard, Lu Shen, Li-wei H. Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G. Mark. Mimic-iii, freely accessible critical care database. Scientific data, 3()::660035-660035 2016. ISSN 2052-4463.
Lucien Le Cam. Asymptotic methods in statistical decision theory. Springer Science Business Media, 2012.
Qing Lyu, Marianna Apidianaki, and Chris Callison-Burch. Towards faithful model explanation in nlp: A survey, 2024.
wwwpa/wwwwwww...///a/baaa// 12326
11326
Ninarch Mchrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan. A survey on bias and fairness in machine learning, 2022. URL https: h//p//////// 1908
09635
Lucas Monteiro Paes, Carol Long, Berk Ustun, and Flavio Calmon. On the epistemic limits of personalized prediction. Advances in Neural Information Processing Systems, 35:1979-1991, 2022.
A. Moreno-Sánchez. Improvement of a prediction model for heart failure survival through explainable artificial intelligence. Frontiers in Cardiovascular Medicine, 10: 1219586,, 2023. doi: 10.3389/fcvm.0222 2023
httL:tttts///oo..rrrtt
3389/33893300 2023. 1219586
Lori Mosca, Elizabeth Barrett- Conne,,, and Nanette Wenger. Sex/gender differences in cardiovascular disease prevention what a difference a decade makes. Circulation, 124:2145-54, 11 2011 doi:
10.1 10.116//CICCULLATIOONAAA 110.96892.
Meike Nauta, Jan Trienes, Shreyasi Pathak, Elisa Nguyen, Michelle Peters, Yasmin Schmitt, Jörg Schlötterer, Maurice van Keulen, and Christin Seifert. From anecdotal evidence to quantitative evaluation methods: systematic review on evaluating explainable ai. ACM Computing Surveys, 55(13s): July 2023, ISSN 1557-7341. doi: 10. 145/3583558. URL http:/ /tt://xx.. doi
org/10.1145/3588 /5854445/555588.
Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 366((466):447-453) October 2019.
ISSN 1095-9203. doi: 10. 2224 URL http::/dkx doi. http:///ddd..oi.oo//111.1.11.1
scie scwence. aax2342)
Jessica Paulus, Benjamin Wessler, Christine Lundquist, Lana Yh, Gowri Raman, Jennifer Lutz, and David Kent. Field synopsis of sex in clinical prediction models for cardiovascular disease. Circula- tion: Cardiovascular Quality and Outcomes, 9:88-S15, 02 2016. doi: 10. 10.166/CCRCOOUTCMMES 115.002473.
Jessica Paulus, Benjamin Wessler, Christine Lundquist, and David Kent. Effects of race are rarely included in clinical prediction models for cardiovascular disease. Journal of General Internal Medicine, 33, 05 2018. doi: 10. 1606-018-47755
Dana Pessach and Erez Shmueli. A review fairesss in machine learning. ACM Comput. Surv., 55 (3), feb 2022. ISSN 0360-0300. doi: 10/1445/349 0494672
//dots wtt.d//.oog/...11....11
13

---

Published as a conference paper at ICLR 2026
Adrián Pérez-Suay, Valero Laparra, Gonzalo Mateo- Careoa, Jordi Muñoz-Mari, Luis Gómez-Chova. and Gustau Camps- Valls. Fair kernel learning, 2017. URL http:ttss////x..rooppp
05578
M. Robnik-Sikonja and Marko Bohanec. Perturbation-bassd explanations of prediction models. In Human and Machine Learning, 2018. URL https://apa www.nticscheaa...c...o arr.iiaarr.or...
CorpusID: 52198297
Ahmed M. Salih, Iaaria Boscolo Galazzo, Polyxeni Gkontra, Elisa Rauseo, Aaron Mark Lee, Karim Lekadir, Petia Radeva, Steffen E. Petersen, and Gloria Menegaz. A review of evaluation approaches for explainable ai with applications in cardiology. Artificial Intelligence Review, 57, 2024. URL /ap..ppi.../// semant htt.://o.......oooooooo.aa.a///.
Sofia Serrano and Noah A. Smith. Is attention interpretable? In Anna Korhonen, David Traum, and Lluís Mårquez (eds.), Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. pp. 2931-2951, Florence, Italy, July 2019. Association for Computational Linguistics. doi: 10.18653/v 1/169/////P19-... 1282. hts:t httpp//psse//llaaaoooooooo....//
Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. CoRR, abs/1 abss//744.0285,, 2017. http://///arxiv abs/1704 02685
Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps, 2014. URL/attps://axsvv.rrg/
abs/1312 6034
Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viégas, and Martin Wattenberg. Smoothgrad: removing noise by adding noise, 2017. http:ttts://axiv..ooo http://////aa...oo////////6././ 03825
Erik Štrumbelj and Igor Kononenko. An efficient explanation of individual classifications using game theory. Journal of Machine Learning Research, 11(1):1-18, 2010. URL htLp::t:////m.r.or
html
Mukund Sundararajan, Ankur Taly, and Qigi Yan. Axiomatic attribution for deep networks, 2017. https://////
abs 703.. 01365
Vinith M. Suriyakumar, Marzyeh Ghassemi, and Berk Ustun. When personalization harms: Recon- sidering the use of group attributes in prediction, 2023. URL httpps://arxiv.org//aa
Berk Ustun, Alexander Spangher, and Yang Liu. Actionable recourse in linear classification. In Proceedings of the Conference on Fairness, Accountability, and Transparency, FAT* '19, pp. 10-19, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450361255 255. doi: 10. 10.1145/328756003287666 /Rt wtg. 1145/ 328 60. 328758888
Fan Yin. Zhouxing Shi, Cho-Jui Hsich, and Kai-Wei Chang. On the sensitivity and stability of model interpretations in nip, 2022. URI.
org/ abs 2 08782
Bin Yu. Assouad, fano, and le cam. In Festschrift for Lucien Le Cam: research papers in probability and statistics, pp. 423 433-435. Springer, 1997.
Hao Yuan, Haiyang Yu, Shurui Gui, and Shuiwang Ji. Explainabiliity in graph neural networks: taxonomic survey, 2022. URL/tttss://axxiie hattps////aaxvv..rrr/////2222211
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A EXTENDED RELATED WORKS
We provide additional extended works about explainability methods and fairness of recourse below.
Explainability Typical approaches model explanation involve measuring how much each input feature contributes to the model's ' output, highlighting important inputs to promote user trust. This process often involves using gradients or hidden feature maps to estimate the importance of inputs (Simonyan et al. 2014; Smmikov et al.) 2017, Sundararajan et al. 2017 Yuan et al. 2022). For instance, gradient- based methods use backpropagation to compute the gradient of the output with respect to inputs, with higher gradients indicating greater importance (Sundararajan et al. 2017 Yuan et al. 2022). We focus on feature-attribution explanations as they remain the most widely used form of post hoc interpretability in practice (Nauta et al., (023). To reflect range of underlying assumptions, we employ three distinct and widely adopted explainers: Integrated Gradients traadient-based DeepLIFT (backpropagatioo--asell) and Shapley value sampling (perturbation-basee)
The quality of these explanations is often evaluated using the principle of faithfunness (Lyu et al. 2024
Dasgupta et al. 2022 Jacovi & Goldberg 2020), which measures how accurately expla explanation represents the reasoning of the underly Miinrllinggnn model. Two key aspects of faithfulness are sufficiency and comprehensiveness YDung al. 2020; Yin et al. (022); the former assesses whether the inputs deemed important are adequate for the model's prediction, and the latter examines if these features capture the essence of the model's pecision-making process. We selected these metrics as they are widely-adopted, model-agnostic measures that directly assess explanation faithfulness through standard perturbation-base evaluation (Serrano & Smith 2019), aligning with established principles of correctness and completeness in the explainability literature (Nauta et al. (023).
Explanations in Practice: Medical Domain Explainable Al methods are widely deployed in the medical domain, and clinicians routinely interact with explanations when interpreting Al outputs. A recent review identified 454 medical Al articles published between 2018-2022, with 93 analyzed in depth, showing extensive use of explainable AI techniques across diagnostic and clinical decision- support applications (Ali et al. 2023; Salih et al., 2024). A growing body of work shows that explainable AI is already shaping consequential medical decisions across multiple clinical domains. In obstetrics, explainable decision-support systems for gestational diabetes significantly influence clinicians' choices and advice-taking behavior, demonstrating that explanations directly affect medical judgment (Du et al.] 2022). In dermatology, domain- specific explanations increase diagnostic accuracy, confidence, and trust, highlighting clinicians' willingness adopt explainable Al systems in practice (Chanda et al. 2024). In radiology, physicians achieve their highest diagnostic accuracy when receiving Al advice paired with visual explanatory annotations, with non-experts benefiting most from explainable guidance (Gaube et al. 2023). In cardiology, explainable AI methods are used to select and justify heart-failure survival prediction models, with explainability explicitly enabling clinicians to understand model reasoning and make more informed treatment decisions MMoreno-Snnchez 2023). Together, these studies demonstrate that explanations influence diagnosis, trust, and decision pathways in real clinical cettronments-udderssoono the importance of evaluating whether explanations faithfully reflect model behavior.
Fairness of Recourse A related line of work examines fairness of algorithmic recourse, which studies whether diffcrent demographic groups face unequal effort to obtain favorable outcomes from predictive model. |stun al. (2019) show that recourse burden can vary sharply across groups, even when recommended actions look formally identical, either because the recourse itself differs or effort required to carry it out is unequal. This line of work demonstrates that fair prediction does not guarantee fair recourse. Our framework offers a complementary perspective: instead of analyzing post-hoc interventions, we study when personalization produces unequal benefits or harms across groups in prediction and explanation. Like the recourse literature, our results highlight that different desiderata, here, prediction benefit and explanation benefit, can diverge and therefore must be evaluated jointly.
because the
B BOP
In the following table, we show how these abstract definitions can be used to measure BoP for both predictions and explanations, each across both classification and regression tasks. The empirical
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population and group BoP are defined as: BhP hp) C(ho) (hpp) and BoP Ap: 6) C(ha, s) - Chpp s), respectively.
Table Formal definitions of the benefit of personalization for prediction and explanation metrics, evaluated for subgroup s. The generic model foe takes input X, while the personalized model Ap takes input (X, S); this corresponds to the quantity previously denoted as X when referring to an unspecified model h in Table For the explanation metries, X denotes the input obtained when removing the most important Teatures, and X denotes the complementary set of features that kept. Likewise, J and sy denote the removed and retained subsets of the personalized attribute 8. Higher BoP values in each row indicate greater benefit of personalization for subgroup
Evaluation Type
Benefit of personalization group
Predict (Clssilliaatoon U-1 loss)
Pr[iin (X) Y 15=
Pr.h.
X.
Y
Predict (Reyression, MSE)
Pr[ha X] [X) Y S
15
-
[
X
Y
S-a
Explain (Sufficiency, classitication, lOSS)
X
Prih.
K
Explain (Sufficiency, regression, MSE)
E
huf Xx,
S=a
(x ar]"" 1
Explain (incompcchonsvveeesss, classification, 0-1 loss)
Pr (ba
hal x. S=
hat
S=
Explain (Incomprchensvvenesss regression, MSE)
[hp(
[o(XX
JolX, 5
C COMPARISON BoP FOR PREDICTION AND BoP FOR EXPLAINABILITY PROOFS
In this section, we present the full proofs comparing the impact of personalization on prediction accuracy versus explanation quality, highlighting situations under which their effects diverge align.
C.1 PROOF FOR THEOREM 4.1
We provide the proof for theorem for two metrics of explanation quality: sufficiency and incomprehensiveness, from Table The proof for sufficiency illustrated in Figure The proof for incomprehensivess is illustrated in Figure
Proof. Let X = X2) where X, and X2 are independent and each follows Unif(- 1, 4 Let us define one binary personal attribute E {0. 1] as S 1(1, + X₂ 0) and assume that we seek to predict = S. Then, (x = [X, + X² > 0) and hp (x) = 1(S > 0) are the generic and personalized classifiers of interest.
Prediction. Both classifiers achieve perfect accuracy. Therefore, BoP hp)) = 0.
In particular, they also achieve perfect accuracy when restrict the input X any subgroup, subgroup s = 0 or subgroup = 1, such that:
G-BoP p.oo hp. = 0) G-BoP p(o, hp = 1) = BoP pho. hp) = 0,
7r(ho, hp) min G-BoP G-BooPr(ho.. hpp 0. sE(0,1)
Explanation (sufficiency). We now test sufficiency by evaluating the accuracy of classifiers using only the important feature.
For model hg. its important feature set Jo is either {X1} (X2). Without loss of generality, let Jo = ). For the personalized model, Jp = [S}.
For sufficiency, we compute:
Pr(ho(X) + ho())))) = -Pr(X, + X2 < 0|1 0)Pr(X, > 0)
+ Pr + X > 0 S 0)Pr(X $0)
(2)
4'
where the computation per group also gives:
(X) 7 ho ho( )1s 0) = Pr(ho(X) ho(Xio) = =1)
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Prediction Accuracy (0-1 Loss)
Explanation Quality Sufficiency
Both generic and personalized classifier have perfect accuracy
Only keep the most important feature of the original classifiers (hg X1, hp S)
decision boundary
X
X
X
X1
j=0
X²
X
X
hg = 1(x1 + X2 > 0) hp =
hg = 1(x > 0)
hp S
C(ho, s) = Chhp, 8)
C(ho, 8)
C(hpr 8)
0 0
1/4 1/4
0
== 0 8
8 =
G-BoP P
0
+ 7p = 0
G-BoP suff
1/4
1/4
Yx -
0
0 ==1
Figure 5: Comparing generic model (ho) and a personalized model (h₂) on prediction and expla- nation (sufficiency). Top-left: The generic model ho uses both X₃ and X2 for predictions, with its decision boundary defined by X1 + X₂ 0. The personalized model, hp. has access to the group attribute S (defined as i (X,
X2 > 0)), and its prediction rule is to output S. Bottom-left: Since both classifiers achieve perfect accuracy (on both groups of Personalization (G - BoPp) is 0 on both groups, and thus: 7p = 0. Top-right: In the sufficiency evaluation, where only the most important feature is kept, htp achieves perfect prediction since it relies solely on S, reaching a sufficiency cost of 0 for each group. In contrast, hg. using only X1, now makes prediction errors and has worst sufficiency cost of for each group. Bottom-right: Since the personalized model has better sufficiency than the generic model, the G-BoP is positive and equal to for both groups, and hence 7z though prediction accuracy remains the same.
O and s = 1), the Group Benefit
0. Hence, personalization can enhance explainability even
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Prediction Accuracy (0-1 Loss) Both generic and personalized classifier have perfect accuracy
Explanation Quality- Incomp.
Remove the most important features
(ho X2, h2 X1, X2)
y
decision boundary
X
X2>>
X
i
)=0
X
X2
X 2
ho 1(X, X2 0) hp = S
hg
1(2 >
hp random guess
C(ho, s) = Chpp 8)
C(ho, 8)
C(hp, 8)
0
0
1/4
1/4
1/2
1/2
8 0
8
8
0
= 8
+ G-BoPP
0
7 7p = 0
G-BoP incomp +
1 1/4 /
1/4
7x >
8
= 0
Figure 6: Comparing generic model (ho) and personalized model (hp) on prediction and explana- tion (incomprehensiveness).. Both achieve perfect accuracy, but fy relies solely S 1 X2 >
0). yielding higher incomprehensiveness. Hence, personalization can improve explainability even when accuracy is unchanged: here, TP 0 and 7x
On the other hand, the sufficiency for hp is
Pr(hp(X,S) * (X,, Ss,)) O,
as Jp (S} sufficient to make a prediction for hy. The computation per group also gives 0, since the model makes perfect predictions independently of the value taken by S.
Thus, BoPx x in terms sufficiency is also H. Computing this quantity per group gives:
1
G-BoPx(hoo, hp, 0) = G-BoP ( hp, =
4'
= %oul Tho hpp)
min .E(0.1)
G-BoP ( Np: 8) 4'
(3)
Explanation (Incomprehensiveness))) Incomprehensivennss the opposite of comprehensiveness. For clarity, provide the computations for comprehensiveness first.
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Comprehensiveness of ho is Pr(ho(X) ho(X).o))
Pr((1 X2 < 0|2 0)pr(X2 > 00)
Pr(X1 + X2 0 S 0)Pr(X2 5
(4)
1
= Pr((1 + X2 s 0/2₂ > 0) + Pr(X1 + X2 > 0 S
0)
2
2
= Pr(Xi + X, < 0| > 0) (due to symmetry of the distribution)
Pr(xi, x2 drr > 0)
2
Pr(x2)
Pr(x1)dx(de
12=0
-15-22
2
Pr(x2) ) - I2 + . doz
2==0
1
2
22
2
22
2
Hence, incomprehensiveness of hg is 1
Computing this quantity per group gives, by symmetry of the problem:
Pr(ho(X) * ho ho()) 0) Pr(ho(X) * ho (J)) 8 = 1)
1 Pr(ho(X) * ho ho(
2
(5)
4"
Hence, incomprehensiveness per group is also - 4
For hp. comprehensiveness is:
Pr(hp(X.S) + ( S1,,)) 2'
as without S, hp can only make random guess. Hence, incomprehensiveness for each group is -1
Computing this quantity per group also gives since hp makes random guess independently the subgroup considered:
(x( # hx(.)),))) 0) Pr(h X (x,))) Pr(hp (X) hp (X{,,))
(6)
2
while the incomprehensiveness per group is therefore - 4
Hence, BoPx in terms of incomprchensivenes is }.
Computing this quantity per group gives:
1
G-BoP x hp,, 0) GG-BoXx - 4'
Tincomp (incomp(hho111 hppp) = min G-BoPx hpp 4'
7
sE(0,1)
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C.2 PROOF FOR THEOREM 4.2
We provide the proof for Theorem 4.2 for explainability incomprehensiveness.
Proof. Let = (X) where X follows Unif(-2, 2. 2). Define one binary personal attribute & {0, as S = X and assume that the true label that we seek to predict is Y classifiers of interest as:
X > 0. We define the
1 (X + S). hx(X) = 1(X 02, -
Prediction. Both ho and ht are perfectly aligned with the ground truth and yield ý = Y. Therefore, they achieve perfect accuracy. In particular, they also achieve perfect accuracy when we restrict the input X to any subgroup, subgroup = 0 or subgroup = 1, such that:
G-BoP ( hp, s 0) G-BoP pho,, hps s = 1) BoP Boorr(oo (ho. hp)
+ 7p(oo hp) = min G-BoP G-BoPp((ho, (ho, hp = O. eE(0.1)
Therefore, BoP CeePp(to.. hp)) = O.
Explanation (sufficiency). For ho. the most important feature is X, while for hp, the most important feature is S.
We now test sufficiency by evaluating the accuracy of classifiers using only the important feature.
For hg. keeping X results in the original predictor. Therefore, prediction does not change at all and the feature is maximally sufficient for both groups (G-BoP suff = for s = 0 and 1, hence 7x 0.
For hy. keeping does not change the prediction output because X 0 X 0. Therefore, prediction does not change at all and the feature is maximally sufficient for both groups (G-Bol suff 0 for 0 and 1, hence 7x
Therefore, BoPx = 0 for sufficiency.
Explanation (Incomprchensiveness) In this setting, we evaluate incomprchensiveness by measur- ing the degradation in model predictions when the most important feature is removed.
Removing X from ho: For hg. incomprchensiveness. is:
Pr(ho + h2) 2'
as without X, fy can only make random guess. Hence, incomprehensiveness for each group and 7x
Removing S from hp: For hp, we compute:
Pr(h, # (x)) = Pr(X S50 > 0)
+ Pr x Pr(X <0)
1
(8)
where the computation per group also gives:
(X, + (x)lss 0)
+ hx(X)ss 1) 4'
Hence, 7x
Therefore, BoP-X
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Prediction Accuracy (0-1 Loss)
Explanation Quality Sufficiency
Both generic and personalized models have perfect classification accuracy
We consider models that only keep the
most important feature of the original classifier X₂
X2
X
X
X
y
G ==
ho x = 1
X2 > 0) 1 0) hp X, =
hu X 0
hp X, S) =
Classifiers with only the most important feature
Explanation:
(hg : X1, h2 S)
X most important feature for ho
S most important feature for hp
model decision boundary
8
1
05 08=1
C(ho,
8)
Chh,, 8)
C(ho, 8)
C(h hhp.
0
0
0
0
0
0
1
0
0
G- C-BoPPo suff'
G-BoP
0
7x < 0
0
+ YP 0
0
0
Figure 7: Comparing generic model (ho) and a personalized model (h₂,) on prediction and expla- nation (sufficiency). Top-left: The generic model hn uses both X, and X, for predictions with its decision boundary defined by 1 (X1 + X₂ > 0) 1(X, < 0). The personalized model, hp instead predicts using the binary group attributes
0, and s' E O, 1 via the rule s s'. Bottom-left: Both classifiers achieve perfect accuracy across all four groups, hence ?/P = Top-right: Sufficiency evaluation reveals a difference in explanation quality. For hu, keeping only the top feature X1 results in a constant prediction circle). For hys keeping only S yields h, S, which fails to recover the true Y for the group (s = 1, = 0) (blue circles). Bottom-right: Thus, the G-BoP is positive for & = s' for 1, 0, yielding 7x 0. This shows that even with identical predictive performance, the models rely on different features, and personalization can reduce sufficiency- based explainability for some groups.
(X{)) = 0, causing an error for the group s = s' = (orange
but negative
C.3 PROOF FOR THEOREM
We provide the proof for Theorem 4.3 for two measures of explainability evaluation: sufficiency and incomprehensiveness, as illustrated in Figure 7 and Figure 8 Figure 7 illustrates the proof for sufficiency, where both generic hg and personalized hp models predict perfectly (left), yet only keeping the most important feature for each (right) shows that the personalized model is more explainable for the group (s' = 1, s = 0), and less explainable for group (s' =,, 1). Figure illustrates the proof for incomprehensiveness.
Proof. Let = (X1, X2) where X, and X2 independent and follow Unif(-1,,1) 1). Define two binary personal attributes s E {0. 1} and s' E {0, 1} such that the true label that we seek to predict is Y S- S'. We define the classifiers of interest as:
(x = 1x1, X2 > 0) 1(X2 < 0), (x =
Prediction. Both ho and htp are perfectly aligned with the ground truth and yield g = Y. Therefore, they achieve perfect accuracy. In particular, this holds for both values of S and S':
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G-BoPp
8'
0
0
0
Such that we get:
Tp hp) min G-BoP G-BoPp(ho,he phho, 0. e,5'((
Explanation (sufficiency). For hos the most important feature is X1, while for hp, the most important feature S.
We now test sufficiency by evaluating the accuracy of classifiers using only the important feature.
For hy. keeping only X1 results in constant predictor ho (x1) = 0. This fails to recover when 1 and 5' (red orange dot), leading an error for the subgroup (s 1, 1), while the three other subgroups still enjoy perfect prediction.
For hp, keeping only S yields (S) = S, which fails to recover ý when = 1 and s' = 0 (red blue circles) but still correctly predicts for the other three subgroups.
Combining per-group values gives: such that we get:
G- C--oosane suff
8
$
8 =
0
11
8'
0
1
7x (ho, hp)
min G-BoP G-BoPsun(ho, shor hp, 1. SE (0,1)
Explanation (incomprehensiveness) In this setting, we evaluate incomprehensiveness by measur- ing the degradation in model predictions when the most important feature is removed.
The generic classifier (x) = 1(x, X2 > 0) (X1 < 0) and the personalized classifier is hp [X,5 S) = S.S. S'. The most important feature for ho is X, and for hp is S.
Removing X₃ from hg: Without X2, the classifier reduces to the constant function
ho(X = 0. This leads incorrect prediction when = 1 and s' = 1.
Removing from hpp The personalized model becomes ht S S', which ignores S. This leads to an incorrect prediction when and = since the true label is y = but hpp =
All other combinations yield correct predictions even when the important feature is removed.
G-BoP incomp
8
8'
0
0
This yields the minimum group benefit of personalization is:
7
incomp
(ho, hp)) = min s,,1E(0,1)
C--ooPicommg (ho, Ap 8, 8') = -1
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Prediction Accuracy (0-1 Loss)
Explanation Quality- Incomp
Both generic and personalized models have perfect classification accuracy
Remove most important features
X
X 2
X
X
X
B=
B 0
==n
(X) = 1 + X2 > 0) 1(X, < 0)
hg X = 0
hp X = S'
S hp ( S-
Explanation:
X;: j: most important feature for ho
S most important feature for hp
model decision boundary
0 8 = = 0
1
0s=1
C(ho, 8)
Chh 8)
C(ho, 8)
C( h
0
0
0
0
0
0
8
G-BoP incomp
G-BoP
0
7x < 0
0
YP 0
0
Figure 8: Comparing generic model (ho) and a personalized model (hpp) on prediction and expla- nation (incomprehensiveness) Both achieve perfect accuracy, but removing each most important features yields different prediction performances. We find that 7/ while 7x <
C.4 PROOF FOR THEOREM 4.4
See Figure regressors.
9 for visualization of Theorem 4.4 for linear model with hg and hp Bayes optimal
Proof. A Bayes optimal regressor using subset of variables from indices in J C [l, t k] would given as:
(x, Σax Σ CaSS
jEJ.
jEJ.
3St
12++1
where h represents Bayes optimal regressor for the given subset and X, and S, are sub-vectors of X and S, using the indices in J.
In what follows, we denote \J as shorthand notation for [l,...t k] J.
From equation 9 and the definition of the true response Y = Ejse 0gXi + Σ Ej2441 0ySj-1, +e we obtain:
i+6 MSE(ho) Σ
Var Var(c),
(10)
j=t+1
MSE(k,) Var(c).
(11)
We define Jo and J, as a set of important features for hn and h Note that Jo and J, are the same across all samples for the additive model. Then, the sufficiency of the explanation for hg and h, is
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Explanation Quality Sufficiency
Prediction Accuracy (MSE Loss)
Explanation Quality Incomp.
Explanatio::
Only keep importatt Soatuse
Beeove nost inpprrinnt
mostimporoat feature
X
X
X
S
S
S
1x(X)) ,) = [ul(X) - ,)(
(x(X) =
x =X
(x(X) X
bo X
0
Kof X
(o( x|
x1p- [|A,()]
xl var(S)
(,(8)
b (8.))
var(8)
(4,(8) ,,(8,) var(X)
(x(K) - to (x,) var(X)
Figure 9: For linear model, absence of benefit in explanation quality means that there is also an absence of benefit in prediction accuracy, illustrated here (see Theorem We consider linear model Y = X + S - 6, with hg and hp Bayes optimal regressors. In this example, absence of benefit of personalization for the explanation quality, BoP-X = 0 evaluated terms of sufficiency (left column) means: [B[]/// - ho (oxX))ll [ = h (X) - h [ [?]] = var = 0. Then, absence of benefit of personalization for the explanation quality, BoP-
0 evaluated in terms of [[] (X) ho (,)()) + var(S) (X) var(S) = 0. This allows to conclude that, in terms of prediction accuracy (middle column): MSEa = MSEp and hence there is also no benefit of personalization in prediction :BoP-P 0.
comprehensiveness (right column) means: oo
ho (X [1
written as:
BI|l (X) ho (o(()0)11 Σ Var(
(12)
JEW
S5t
E|| Ellb,, Myp X Jn )11 = Σ
0z Var(X.) (X+)
Σ Var((S))-e)
(13)
JEW
JEW J2++1
jst
Similarly, the comprehensiveness of the explanation for hg and h2 written as:
E[]lho(X) - No (X, 10)))1111=1 Σ
ag Var(X,)
(14)
jEJo. JSt
E Billh, (X) - h (,,)1 1111
Σ
a Var(X.)
Σ Var((S))-o)
(15)
jeJ. i5t
JEJ, 324+1
Then, assumption of BoP-X for sufficiency becomes:
| ( - ho X,,,, Jo 0))13 B(]lA,(X) - (x,,,
(16)
Σ Var( = Σ Var( + Σ
a Var(S))-
(17)
Je
JEW JSt
JEWJ,
jst
#24+1
Similarly, our assumption of BoP-X = 0 for comprehensiveness becomes:
E ho x)
(X
Jo (.
(18)
Σ
a
Var(X.)
Σ
a²
Var (X.) + Σ
a Var(S
(19)
JoJo j5t
jeJ ist
jEJ j2t+i
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Summing both equations:
Σ aj (a) + Σ az Var(X,) Σ
az Var(X,) + Σ az Var(S)-1)
Jo JSt
JEJ 25t
WE
JEL
35t
324+1
Σ aj Var(X,)
Σ
aj (ar(())-)
jEJp
jEJ, 32++1
jst
Var(X) Var(X) + Var(S)
= Var(s) = O.
(20)
Since Var(S) = 0, we have that MSE(ha) MSE(hp) and thus: BoP-P = 0 which concludes the proof.
We can make the same claim with similar logic for classifier where Y is given as:
Y 1a a, + a44181 ................................ art + 0).
(21)
The derivations above are made at the population level, i.e., without distinguishing subgroups in the data. However, the reasoning also applies for subgroups, where we define subgroups to be defined by I(S 2 0) taking values in {0, In other words, if G-BoP surpsstrrooo.. hp = 0 and G-BoP incomp (bg. Ap.
0 then G-BoP p(ho- (h: 8) 0 for any {0, 1]. However, note that we can only make a statement on y(ho hp) (prediction accuracy) for the case where Ysutiiiiecyy (ho. hp) = 0 and iinnoompechnnsiveeess (hq. hp) if the following is true: the group realizing the minima in the three 7's is the same group.
D PROOF OF THEOREMS ON LOWER BOUNDS FOR THE PROBABILITY OF ERROR
As in Monteiro Paes et al. 2022), as:
will prove every theorem for the flipped hypothesis test defined
Ho: : T(h, hp: D)
Personalized hy performs worst: yields < 0 disadvantage
H1: Tho, hpi D) 2 0 + Personalized h, performs at least as good as generic ho-
where we emphasize that < O.
As shown in (Monteiro Paes et al. 2022). proving the bound for the original hypothesis test is equivalent proving the bound for the flipped hypothesis test, since estimating is hard as estimating -y. In every section that follows, Ho, H₃ refer to the flipped hypothesis test.
Here, we first prove a proposition that is valid for all of the cases that we consider in the next sections.
Proposition D.1. Consider Ho the set of distributions of data, for which the generic model ho performs better, i.e., the true q such that (hg. D) S C, and H₃ the set of distributions of data points for which the personalized model performs better, i.e., the true y is such that "(ho, (hh,,hh. D) 2 O. Denote Hy, Hi the set of distributions that are mixures of components in Ho and H, respectively. Consider a decision rule V that represents any hypothesis test. We have the following bound on the probability of error Pe:
min max P W POEH PiEH
2 (1-T(PP1Q).
for any Hy and any Q E Hi. Here TV refers to the total variation between probability distributions P and Q.
Proof. Consider by and hp fixed. Take one decision rule vi that represents any hypothesis test. Consider dataset such that Ho is true, i..,, D ~ P/j and dataset such that H, is true, i.e., D ~ P/-
might seem counterintuitive to use two datasets to compute the same quantity P.,. iee. one dataset to compute the first term in P.,, and dataset to compute the second term in P.. However, this
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is
just reflection of the fact that Pe is composite measure, with its two terms coming from two different settings: Hg true or Hy false, which are disjoint events. In the same way that Ho cannot be simultancously true and false, each term in P. considers one the other case, and we use one the other dataset.
We have:
Pe
Pr((eeecting Ho] Ho true) + Pr(Failing to reject Ho H, true))
(Pr(V(hn, App D, 1|D Po) Pr (N hp D,c) 0/D ~ Pt))
= (Pr(V(D) 1|D Po) + Pr ( 0 P.)) simplifying notations
2
= (1 - Pr(v(D) = 0/D ~ Po) + Pr ( 0 ~ Pi)) complementary event
2
= (1 Po(FF) + (F+)) writing F.g the event (DD) = 0
2
1 (Bo (E+))))
Now, we show that this equality holds when considering mixtures of components in Hg and mixtures of components in H1.
Consider a dataset distributed as Po E Hi that is mixture Po = Een w/ p(u) where w/ 2
and Σ w = 1, such that, for each component p(a) of the mixture, Hg is true.
Consider another dataset distributed as Pi E Hi that is a mixture P1 = Ene w p()), where w(() 2 0 and Σ/=1 w() = 1, such that, for each component p(5) of the mixture, H₃ is true.
For every j,J', since p(ù) Ho and p(5) E H1 we have:
Pe
1
1 (P (Fo 2
2
Since Σ/e wg = and J1
w -1
(1')
= 1 we have:
Jo
Pe =
W
(
Σ
w
6')
Pe
y'=1
Jo
w
Σ
wi
Pp
y'=1
( (5)pp
W/ w1 Pa
w w1 (0)) 1 (1 (1-(p())(Eg) (Ew) - p(')) (Ew)) (choosing to replace Pa its expression involving j.j')
2
Σ
(
[w] - (60')
p(5) p()pp())))) (Ew) + wft p(o'
Wi
ドー
J1
Σ
(5)
Σ
(o((")
(5) (5)
p(5) (1) Ev w0th p()))
J1
J1
J
w - Σ
(j')
(1') 11 (Ex) +
Σ
w (6))
(Ex))
=
wn
wi
2
5'=
='=1
3=1
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Ja
J1
J1 (using
(
(Ev) + Σ
(3) (l)p))
(i)
w
wi
1
=
1)
y²=1
da
0
())
(1
p( (EΦ) + Pi P((Ea))
(definition Pi)
2
-=1
(
Σ j=1
(5) p()) (Ev) Σ
1
Pi (B(W))
0
2
j=1
5=1
Jp
Jo
2
1
( p (Fv) + P((()))
0
(using
Σ
w0
())
1)
j=1
j=1
= - (1- Po(Es) + Pi (E+))) (definition of Po)
Hence, the equality holds for mixtures of components in Ho, H2 respectively. We denote Hy Hi the respective sets of such mixtures.
Now, we will bound this quantity:
min max Pe min max (1 (P((EN)-P((EW))
PoEH, V
W P.EH, 2
PiE
PeHli
max PoE
min W
(1 (Po(Ev)) - P1 (E+))) using minmax inequality
2
PIHH;
max P.E
(ax(()(Ee) - ((()))))
to minimize over V, we maximize (Po(Ev) Pi Pi(E))
2
Pieli
max
1
max (Po(A) (Po(A)) (P6(4)-)((4))) because the max is now over all possible events A
PoE
2
PiEHi
The maximization is broadened to consider all possible events A. This increases the set over which the maximum is taken. Because V/ is only subset of all possible events, maximizing over all events A (which includes v) will result in a value that is at least as large as the maximum over V. In other words, extending the set of possible events can only increase the maximum.
(1 (1-TV((P)) pm)]]
max P
by definition of the total variation (TV)
2
EHi
1 - min TV(Po Pi)
2
PoEHl
PiEH,
2 2 (1 - TV((VPP || Q)) for any P E Hy and Q E H².
This is true because the total variation distance TV(P || Q) for any particular pair P and Q cannot be smaller than the minimum total variation distance across all pairs. We recall that, by definition, the total variation of two probability distributions P, Q is the largest possible difference between the probabilities that the two probability distributions assign the same event A.
Since the total variation distance TV gives values in the interval [0, 1], the lower bound cannot exceed 1/2. This is expected. In binary hypothesis test with equal priors, decision rule corresponding to random guessing would yield P. = 1/2. Consequently, the optimal probability of error (the minimum all decision rules v/) must lie in the interval [0, 1/2].
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We note that, through this derivation, have recovered Le Cam's standard result (see (Le Cam 2012 Yu 1997) Lemma IJ).
Next, prove lemma that will be useful for the follow-up proofs.
Lemma D.2. Consider a random variable a such that E[a] = 1. Then:
B[(a
(22)
Proof. We have that:
BB[( 1) B[a² 2a + 1]
B[a"] 21E[a] + (linearity of the expectation)
R[aa]] + 1(E [a]
by assumption)
= B|a²] 1.
D.1 PROOF FOR THEOREM 5.1 ANY PROBABILITY DISTRIBUTION AND ANY NUMBER OF SAMPLES IN EACH GROUP
Below, we find the lower bound for the probability of error for any probability distribution of the BoP, and any number of samples per group.
Theorem (Theorem 5.1 restated). The lower bound writes:
1
1
(B) p(B)
min max P. 2 V POEH
1
(23)
2vd
Piel
1=1
where Po is mixture of distributions of data, for which the generic model ba performs better, i.e., the true 7 is such that (ho, hp. D) < O, and Pi is mixture of distribution of data points for which the personalized model performs better, i.e., the true is such that <(ho, hy: D) 2 Dataset 'D is drawn from an unknown distribution and has groups where d = 2*, with each group having m;j samples.
Proof. By Proposition D.I
have that:
1
min max P. 21 Q)) POE PIH
for any P E Hi and any Q Hi We will design two probability distributions P, Q defined on the N data points S1, Y, Y,),.., (NN, SN. Yn) of the dataset D compute interesting right hand side term.
An "interesting" right hand side term relies on distributions P, Q for which (V(P || Q) is small, i.e., probability distributions that are similar. To achieve this, we will first design the distribution Q E Hi, and then propose P a very small modification of Q, just enough to allow it to verify P E Hi-
that makes the lower bound tight possible, i.e.,
Mathematically, Q are distributions on the dataset D, i.e., of the random variables X,S.Y. Thus, we wish to design probability distributions on S1, YX). SN. YN).
i..d. realizations
However, we note that the dataset distribution is only meaningful in terms of how each triplet (X, Y,) impacts the value of the individual BOP B, Indeed, we recall that B, is function of the data point Z₁ = (X,, Sj. Yi), which we denote as f such that B; (Zz) cost(ho, X,, Y,) copt(hp. X,, Y.). Hence, any probability distribution on Z; will yield probability distribution on B, and any distribution on the dataset Zı, ..., ZN will yield distribution on B B..... BN
Conversely, let be given P(by, bx) 1I (.,b.) distribution on B BN defined by N independent distributions P; for 1, N, such that the support of each P; is restricted to the image
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of f. We propose to build probability distribution P(2 ... 2N) P((2)) on Zz,... ...... Zn that will ensure that f (Zz), [(ZN) is distributed P.
First, for each P. we restrict P. so that, for every value by that B; can take according to P.. there exists a unique z with positive density, concentrated as a Dirac at ži, such that we have f(;;) = b;- Existence is guaranteed since B, takes values in the image of f. Uniqueness is guaranteed because we can assign 0 mass to the potential non-unique values. Equivalently, f is a bijection from supp(P;) to the set of values taken by B, for each We note that the construction of P; (as a collection of Dirac measures centered at points 2; in the domain of f) makes supp( P.) a discrete set of points.
By this construction, we have restricted the support such that f acts as a bijection between supp C 2 and supp( P.) C for all i (and analogously for Qi and Qt). Let g = f denote the inverse mapping defined this restricted domain.
We define the distributions P and Q over the full dataset as the push-forward measures of P and Q under the component-wise mapping g. meaning P = ## P and Q 9* Q. By the Data Processing Inequality (DPI), applying any measurable function to a pair of distributions cannot increase their Total Variation distance. Applying the forward mapping f to the distributions P and Q yields: TV(P || Q) < TV(P || Q). Conversely, applying the inverse mapping g s to the distributions P and Q yields: (VP || Q) s TP Q). Thus, the Total Variation distance is preserved: (V(P Q) TV(P Q)
Thus, we design probability distributions P, on N i.i.d realizations of the "auxiliary random" variable B, with values in R, defined as:
B = ((ho(()) Y) (X, Y).
(24)
Intuitively, B, represents how much the triplet (X;, S;, Yi) contributes to the value of the BOP. b; >> O means that the personalized model provided a better prediction than the generic model on the triplet (I;, 8;, Mi) corresponding to the data point i. In what follows, we drop the "tilde" for notational convenience, such that we use P. Q to refer to P.Q.
Consider the event b (b1, ..., bN) E RN of N realizations of B. For simplicity in our computations, we divide this event into the groups, i.e. we write instead: b, = (b²) ..... b(m)), ) since each group j has my samples. Thus, we have: b = {6, [b(²) b==1..d, "i=1..d,kkI... k...... indexed by j. k where j = 1...d is the group in which this element is, and 1. is the index of the element in that group.
Design Q. Next, continue designing distribution Q (since we have justified that we can define them on B) on this set of events that will (barely) verify H1, i.e,, such that the expectation of B according to Q will give y O. We recall that y means that the minimum benefit across groups is O, implying that there might be some groups that have a > 0 benefit.
Given p as distribution with mean μ
0, , we propose the following distribution for Q
2,(b,) = II
p(b) (b)) ), for every group j = 1....d
k=1
Q(b) II Q,(b,)) j=1
We verify that we have designed Q correctly, i.e., we verify that Q E Hj When the dataset is distributed according to Q, we have:
7 min C.(ho, 8) Ch 8) .ES
min Eq (0 X), Y) S = s] (X), Y) S = s] (by definition of group cost)
#ES
min Eql (0 (), Y) ((h, Y) S = s] (by linearity of expectation)
*E8
BolB | S (by definition of random variable B)
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min 0 (by definition of the probability distribution B)
Thus, find that y 0, which means that 7 2 0, i... Q H1, and hence Q HHi.
Design P. Next, we design P as a small modification of the distribution Q. We recall that P E Ho means that 7 I where 0 in the flipped hypothesis test. This means that, under Hn, there is one group that suffers a decrease in performance of |e] because of the personalized model.
Given p distribution with Jr 0, and pt distribution with mean μ
0, we have:
Py(b,)
II p(b (k)
for every group L....d,
Pb(b,) II p (0) (0) for every group j 1...dd
k=1
d
P(b) d
(b) where: Pj(b= P,(b)) Σ j=1
II Py((b)) 8
Intuitively, the distribution P represents the fact that the personalized model worsen performances by |c| for one group. We construct the mixture by assuming that this group can be either group 1, or group 2, etc, or group d, and consider these to be disjoint, equiprobable events.
We verify that we have designed P correctly, i.c., we check that each of its component is in Ho- When the dataset is distributed according one component Pj, we have:
min C, s) Chp. s)
#ES
min E [B S = s] (same computations as for Q E H1)
SES
min(e, ..... 0) (since exactly group has mean c)
= (since e < 0).
Thus, we find that y which means that y S c, i.e. Pj E Hy for every j. i.e., Ho-
Compute total variation (V || Q). We have verified that Q E Hi and that P E Hy. We these probability distributions to compute the lower bound for P.. First, we compute their total variation:
1 TV(P
Q)
P(b by) Q(bi...... .... ...,bb))) db))dd,...db (TV for probability density functions)
2 bs....,,
1 d
1
7 Σ
(b)b)
II Pb II
(b,b
dbp...db; (definition of P, Q)
2 by
1=1
j'4
j=1
1 d (b,)
1
à Σ
II Pb(by)
II j=1
Q (b,)
dbi dbi...db (adding missing j' j)
2
by....bb
P(,)))
j==1
1 2 (,))
1
Σ
(ggb,
II (b,b,) j=1
db (P3 =Q by construction)
2 bs.....s
7
P,,b,)
1=1
==1
d
(b,b,)
II (b(b,)
db....bb (extracting the product)
2
d
P3(b,)
1=1
1
(5 P3(b,)
EQ
Σ
(recognizing an expectation with respect to Q)
2
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p (6,²))
(definition P3 and p)
p(b) (
1/2
p (6ft)
1 BQ 2
Gauchy-Schwaalte
p()))))
Auxiliary computation to apply Lemma D.2 to prove that the expectation of the first term is 1. We have:
Next, we will apply Lemma D.2 For this, need
P (
Eq
d
((b(1))
j=
Thy p² (6,5))
IQQQ
(linearity of expectation)
p(()
123
(b(())
Σ
IQQQ
(rearranging the product)
p(O(t))
my
p' (b(b))
Σ II
EQQ
(product of independent variables)
p(
ps )
E
(definition of Q)
p(b("))
p' (b) p(b) pbb)db (definition of expectation in
pt (b)db (simplify)
myy
(probability density function integrates to 1)
(term independent k)
(term independent of j)
Continue by applying Lemma D.2
This auxiliary computation shows that we meet the assumption
of Lemma D.2 Thus, we continue the computation of the lower bound of the TV by applying
Lemma D.2
pe (6(())
1
TV(P QQ) <
BQ 2
-1
Lemma D.2 D.2
p(b)))) (8)
1 EQ
IT T 1f (0()))))
my
((((()))
2
2j
1
defining 2j
Im Im p(6)5))
k= p(
1=1
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Q
Σ
Z
-1
1
expanding the square of the sum
2
d²
3='==
d
1
BQ
Σ
23.28
j=1
where we split the double sum to get independent variables in the second term.
We get by linearity of the expectation, ElaXX + +bY = [F[x] bE[;]
1
(V(P)
Q)
EQ
Zyy 1
L
1.j' "1
1
EQ
Σ
E
-1
2
13==
mh
(b
p (6,5))
pr 1r
)])
1
2
EQ
II
Σ
EQ
II km
II
1
2
d
P(B(R))
p(o
A=
(b(
1
Σ
IEQ
2
d
P(b
(K)
172
o' (6
+
Σ
IQQ
EQ
-1
(product of independent variables)
Lk=1 p(()t)) (K)
p(b jr
(K)
5-'=1
(proge) p(of()))
27
my Σ
1
Ep
2
d2
1=
my
)
ps (b D(6(5))
Σ
II
Ep
Epp
(product of independent variables and def. of Q)
)(
5.3
(
27
my Rem
II Ep
Σ
1
(auxiliary computation below)
2
d2
k=
5. 1.5'=1. 1J
Amm
27
1
my
II 1=1
Ep
1
(term independent of k)
2
d2
p(b(M))
j. 1
1
Σ
my
Ep
(=))(k)) )(6
27
+
(dl d)
(term independent of j)
2
12
1=1 A=1
1
(F(B) p(B)
m,
Ep
(d d)
(term independent of k)
2
d2
(D((B) 27 my
1
2
d2
Ep
p(BB)
+1 d
-
(distribute 1/d?)
1=1
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(D'(B) ((B)
27 my
Bp
(simplify)
2
d2 j=1
d
[B() P(B)
27 mn
Σ
Ep
(extract 1/Vd)
2vd
+
Σ
(pf
2
2Vd
p(b)
p(b)db
-
(definition of expectation)
1
Σ
pf (b)2 db p(b)
my
(simplify p(b))
2Vd
1
(B)
m
Epp
-1
(def of expectation)
2Vd
Auxiliary computation in
We show that:
p' (
E,
pf (b)
p(b) pbb)db
p" (b)db simplify p(b)
probability density function p integrates to 1.
Final result: This gives the final result:
min max Pe 2 (1 - (VTV(PP
Poe PIE
1
1
(B)
mj
-1 1 -1)
$
min max P >
1
Epp
W Potf
2 d
p(B)
j=1
PiEH
D.2 PROOF FOR COROLLARY
5.2 ANY DISTRIBUTION AN EXPONENTIAL FAMILY
We consider fixed exponential family in it natural parameterization. i.c., probability distributions of the form:
fx 0) h(x)) exp(@ T(x) A())),
(25)
where / the only parameter varying between two distributions from that family, i.e., the functions T and A are fixed. We recall few properties of any exponential family (EF) that will be useful in our computations.
First, the moment generating function (MGF) for the natural sufficient statistic T(x) is equal to:
M (t) exp (A(9 t) A(0)).
Then, the moments for T(x), when 0 is scalar parameter, are given by:
F(T] (0)
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V[T) (0).
Since the variance is non-negative V[T) O, this means that have A² 0 and thus A' is monotonic and bijective. We will use that fact in the later computations.
In the following, we recall that the categorical distribution and the Gaussian distribution with fixed variance o² are members of the exponential family.
Example: Categorical distributions as a EF The categorical variable has probability density function:
K
p( π) exp
log TT
exp
2i log T
1
Ik
log
k
K
exp
log
T
K
2k + log
1
Σ
T
T
where we have used the fact that
1
K-1 ===
Tk.
We note that we need to use the PDF of the categorical that uses a minimal (i.e., 1) set of parameters. We define h(x), T(x), 0 E R- and A(0) as:
h(x) T(x) = I,
TR
Tk
0% = log
= log
for I,...
1
>k=1
K K-1
TTk
K-1
K Σ
k= The
A(O) = - log
1
Tk
log
=
log
log
A=
1
k=1
Tk
k=1 Tk
which shows that the categorical distribution is within the EF. For convenience have defined 0 setting it to as per the Equation above.
Now, adapt these expressions for the case of Categorical variable with only K 3 values I = -11, x2 = 1 and
such that π& = O, i.c., there is no mass on the 23 = 0, and we denote
T = P1 and æT2 = P2 and πΣ P1 P2 O. We get:
h(x)== =
T(x) z,
0 log
Pi and 02 = 0 by convention, above, 0g llog
T3
p2
D22
A(02) log (c t ex)) log (e + 1 0) log
log
21
+1
log
P1 P2
+
1
Example:
Gaussian distribution with fixed variance as EF fixed variance has probability density function:
The Gaussian distribution with
(x
2))
p (a 1)
exp
2
20
2
2xp
112
exp
V 2
20
2
2.r.. - 112
exp
exp
V
20
202
2/11
112
exp
exp
2
2
20
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We define T(x), 0 R and A(0) as:
x²
h(x)
exp
V2T22
20
T(x) I,
0 1
0²
n1² 026²
A(O)
20² = 2
which shows that the Gaussian distribution with fixed variance o² is within the EF.
Corollary (Corollary 5.2 (restated)). The lower bound for the exponential family with any number of samples in each group writes:
)
M,(2500 Mp(A0)3
min max P.
1
-1
PoE R REH
2vd
j=l
Proof. By Theorem 5.1 we have:
m, pf (B)
1
min max Pe POE PIEH!
Ep
2
2
(B))
where p. pf are distributions with means and respectively.
Plug in the exponential family Under the assumption of an exponential family distribution for the random variable B, we have:
min max Pe W PotH
Pie
7
d
M(B) exp 0 (TB) A(09)
my
1
E
-
1
2
2vd
d
h(B) exp (0. T(B) A(00))
3
d
EXp(O..((B)) A(85))
m;
Epr
simplifying h
2
2d
d j=1
exp (T))).((B).... A(80))
my 7
d
E
exp(O'.((B) - exp(() exp ((10)) -1
properties of exp
2
2
d j=1
Σ
Epc
exp(A(8") - -(()))
2
2vd
d
3m1
m3
- 7
eexp
(00- - 80) -)(B))
(properties of exp and rearranging terms)
d
m,
1
exp(A((0)) - A Epr
exp((( - +8)T(BB))
-1
2
2
d j=1
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d
exp(A((0)) - MB)) -
2
2vd
1=1
(def. of MGF of Mp.(t) = Ey [exp(t T())) with A0 = - 00)
We define AO = - On- Here, we will apply the properties EF regarding moment generating functions, i.e., for the p" with natural parameter 0g:
Mp (t) exp (A(0. + t) - A(0.)) = Mp (-10) = exp (A(6)) - A(0.))
= Mp (A0) = exp (A(20, 06) A(fc))
And, for p associated with natural parameter 0::
Mt(() exp (A(6o A(0b)) Mp( --) exp (A(20) 0.) A(0o))
Mx(20) = exp (A(O.) - A(0)))
Mp(A0)2 = exp (2A(0.) - 2A(0)))
M,(2A0) exp (A(20, -00 A(0o))
So, that we have on the hand:
M- (A0) exp (A(0o) A(O2)) exx (A(20. - 0o) - A(O.))
and on the other hand:
(2A0) exp (A(20, 06) A(Oo))
M (10)
exp (2A(0.)) 2A(()))
exp (A(20, 06))
1
cxp (2A(0t) 2A(0a)) exp (A(0o))
exp (A(20. 0o))
exp (2A(0.) A(O)))
exp (A(20. 0o) + A(0)) - A(O)) - A(O))
exp (A(0)) A(O.) + A(20, 0o) A(O.))
exp (A(0o) -((0)) exp (A(20. - 000 - A(0.)
Consequently, we get two equivalent expressions for our final result:
$
d
1
exp(A(09) A(89)) exp (A(20. 0p) A(00)))
2
2vd
1=1
(Mpr (-A)M -1 1)
(first expression)
2
2v dd
j=1
7
M,((20) M₂(A0)²
)
-
-1
(second expression)
2
2vd
j=1
We will use the second expression.
D.3 PROOF FOR CATEGORICAL BoP
Here, we apply the exponential family result found in D.2 to find the lower bound for a categorical distribution.
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Corollary D.3. [Lower bound for categorical individual BoP for any number of samples in each group (Monteiro Paes et al. [ The lower bound writes:
7 {+)
min max Pe
Σ
(1 + 4E ms
-1
PoE Pif
2
2vd
where Px,S.Y is distribution of data, for which the generic model ho performs better, i.e., the true is such that (ha, hp, D < O, and Qx.s y is distribution of data points for which the personalized model performs better, Le., the true is such that ~(ho, hy, D) 2
Proof. By Corollary 3.2 we have:
M2(200) m M2(A0)3
-1
min max Pe 2 PEE PIEH
2
Plug in Categorical assumption
We find the bound for the categorical case.
We need choose or such that Ep [
We have:
Epp [ = pi + 1-2-
We choose i I-c and ps Itc
For the categorical, we have 0 and:
0g log
P1
log
1/2
P2
1/2
Pi
-
log
log
1
A(0p) = log Ho + 1) = log(2)
A(0₂) = log (c" 1) = log
1-c +1 ++1=
= =log
(1-c+1+
log
2
+E
1 l+E
A(20.) = log (0 +1)
log
((e2))2 21)
(1-
2
log
+1 +1
1
- 2e + 2
log
1 + 2: + +1)
2e + 2 1 2E E2
log
1+2e Ee2
2 2c²
log
+ 2E + c
We also have: A0 Oz. Accordingly, we have:
M, Mv() exp (A(0) 20) A(Oo))
= =exp (A(O.) - A(0o))
2
=exp (log
- log(22)
1+c
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1
2
= exp log
2
l+c)
1
1+6 E
M₂(2A0) = exp (A(28, 00)) A(Bo))
= exp (A(20.) - A(0)))
=ex log
+
llog(2)
+ 2e + +
1
2 + 26
eep
21 + 62
c+
1 +
And the lower bound becomes:
min max Pe 2 - TV(PII
W PoEH. REH
M,(220)
min max P. 2
1
1
PoEH, v
2v d
(A0)
Pi PEH
my
1+-22 I+2c+e7
1
1
2
2
1
1+c2
m
1
1+2c++2
2
1+26++2
1
1 Σ
-1)
1
(1 + 2 my
2vd
1=1
D.4 MAXIMUM ATTRIBUTES (CATEGORICAL BoP) FOR ALL PEOPLE
In the case where dataset is drawn from an unknown distribution and has d groups where d d= 2 with each group having m [N/d] samples, Corollary D.3 becomes:
2vd (1+ 7
1 1
min max Pe 2 2
PoEHL
PyEH
Corollary D.4 (Maximum attributes (categorical) for all people). Consider auditing personalized classifier hy to verify whether it provides a gain of = 0.01 to each group on an auditing dataset D. Consider an auditing dataset with N x 109 samples, or one sample for each person on earth. If hp uses more than k > 17 binary group attributes, then for any hypothesis test, there will exist pair of probability distributions Px,s, E Ho, Qx,G. E H₂ for which the test results in a probability of error that exceeds 25%
P. 1
4210 17
min max Px
(26)
4
Qx.yeEH
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D.5 PROOF FOR GAUSSIAN BoP
Here, we do the proof assuming that the BoP is a normal variable with a second moment bounded by o².
Corollary D.5. [Lower bound for Gaussian BoP for any number of samples in each group] The
lower bound writes:
min max Pe 2 1 1 2vd
m o²
exp
1
V POEHo
PIEH;
5=1
where Px.s.Y is a distribution of data, for which the generic model ho performs better, i.e,, the true such that (ho, hpg D) < O, and Qx.s.y is distribution of data points for which the personalized model performs better, i.., the true is such that 7(ho, kt, D) > O.
Proof. By Corollary 5.2 we have:
M2(2A0) m, Mp(AA)3
min max P. PoE PIEHÇ
2
1
2
Plug in Gaussian assumption
We find the bound for the Gaussian case. For the Gaussian, we
have:
14
0o
0
0. Pe
o 20
A(O))
a g2 e
A(O.)
2
20
82 240
26
A(20.) =
2
o²
because 1i0 = 0 and μr. e by construction. Thus, we also have: A0 0.-
Accordingly, we have:
2²
Mp(20) exp (A(Oo + A0) A(Oo)) exp (A(O.) - A(Oo)) = exp
20
262
My(2A0) exp (A(6, + 226) - A(O))) = exp (A(20. 06)) = exp (A(28.)) = exp
o²
And the lower bound becomes:
1
min max P. (1-117 TV Q))
PoEH
2
RElli
Mp(2A0)
min max Pe
1-
P.E
2
2vd
P.EHi
2
122
exp
2
2/d
d
1
exp
20
2 exp
--
-
2
2vd
d
exp
(200)
39

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Published as a conference paper at ICLR 2026
22 exp
m
2
2vd
exp
(02)
j=1
1
1
2
my
exp
1
2
2v d
d
0²
1
1
mye
cxp
2
2vd
In the case where each group has a different standard deviation of their BoP distribution, this becomes:
1
myc²
1
exp
2
2vd
j=1
D.6 MAXIMUM ATTRIBUTES (GAUSSIAN BoP) FOR ALL PEOPLE
In the case where dataset is drawn from an unknown distribution and has d groups where d 2², with each group having [N/d] samples, Corollary D.5 becomes:
me²
min max Pe 2 2 P.E
exp
2vd
02
P.EHi
Corollary D.6 (Maximum attributes (Gaussian BoP) for all people). Consider auditing a personalized classifier hg to verify if it provides a gain of e = 0.01 to each group on an auditing dataset D.. Consider an auditing dataset with a = 0.1 and N on earth. If he uses more than k 2 23 binary group attributes, then for any hypothesis test there will exist a pair of probability distributions Px,s,y E Ho, Qx,s,y H2 for which the test results in probability of error that exceeds 25%.
10 samples, one sample for each person
2 23
min W
max Px.
Pe 2
(27)
4
Qx,cyEE
D.7 PROOF FOR THE SYMMETRIC GENERALIZED NORMAL DISTRIBUTION
We solve for the bound assuming the BoP is a symmetric generalized Gaussian distribution.
Symmetric Generalized Gaussian The symmetric generalized Gaussian distribution, also known as the exponential power distribution, is generalization of the Gaussian distributions that includes the Laplace distribution. A probability distribution this family has probability density function:
3
exp (-(1=-m1))
(28)
2aT(1/3)
with mean and variance:
V[x] a²I(3/3)
E[ = μ,
(29)
T(1/3)
We can write the standard deviation = av
P(18)) T(3/3)
= an(3) where we introduce the notation
(33) =
r(z3)
This notation will become convenient in our computations.
r(33))
40

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Published as a conference paper at ICLR 2026
Example: Laplace
The Laplace probability density function is given by:
Jr m
[(x Jr, b)
exp
(30)
2b
which is the family for and 8 since the gamma function verifies T(1) = 1)!
0! 1.
Proposition D.7. [Lower bound for symmetric generalized Gaussian BoP for any number of samples in each group] The lower bound writes:
1
(- B
-e|
|B 1
)
my
min P.
1
Eye
exp
POE 2
2vd
a
PEE
where Px.S.Y is a distribution of data, for which the generic model ho performs better, i.e,, the true ? is such that (ho, hp, D) < O, and Qx,s is distribution of data points for which the personalized
model performs better, ie., the true is such that ((h, htp D) >
Proof. By Theorem 5.1 have:
my (B) p(B)
1
min max Pe POEH
1
B
-
2
P.H
j=1
Plug the symmetric generalized Gaussian distribution variable B follows exponential power distribution, we continue the computations as:
Under assumption that the random
B-c a
d
rexp
min max Pe V P
1
2
2vd
[3
P,EHi
1=1
exp
9
(1 -1
( my
1
Σ
B
2
1
2vd
By
cxp
a
exp
j=l
(property of exp)
1
1
B
E
B
9
m -1
]
1
Bys
exp
-
(property of exp)
2
2vd
7 1
1
1
B
|B|
my
1
Σ
Ryp
exp
(property of exp)
2
2dd
D.8 PROOF FOR LAPLACE BoP
Here, we do the proof assuming that the BoP a Laplace distribution (for more peaked than the normal variable).
Corollary D.8. [Lower bound for a Laplace BoP for any number of samples in each group] The lower bound writes:
$
d
myje
min max Pe > P P.EH
2
1
2
d
exp
j=1
41

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Published as a conference paper at ICLR 2026
where Px.S,Y is distribution of data, for which the generic model bu performs better. i.e., the true such that (hy, hp. D) < O, and Qx.s.Y Y is a distribution of data points for which the personalized model performs better, i... the true is such that 7(ho, hp, D) > 0.
Proof. By Proposition D.7
we have:
(_IB
c/
|B
my
min max Pe y PoEH. PIEH
2
2
Epe
exp
a
j=1
Plugging in our values of and 3 shown satisfy the Laplace probability density function we get:
B - B } m,
1
1
1
Bpr
exp
1
2vd
Using bounds Since finding the worst case lower bound, will find functions that upper and lower bound BB e| |. This function is lower bounded by and upper bounded by -e since < 0. Indeed, since e 0, there are three cases:
0< < B e: this gives B - el - |B] = B - 6 - B = -e
< 0 < B e: this gives B
B B - B = =2 - <> 2E = since
0< B
B
< this gives B c|
[ B B=c.
Thus, we have:
- B c - B < - and:
E < B E B < -
B
[B]
b
B
-
|B]
- 2
Z
b
(-9) 2 exp
B
|B]
2 exp (
exp
b
Thus, applying the expectation gives:
(-r)
c
|B]
(
Eys
exp
E
exp
>
Ey
exp
b
(-r) >
B
cl B
exp
Eyp
exp
exp
b
because the lower and upper bounds do not depend on B.
All the terms in these inequalities are positive, and the power function is increasing on positive numbers. Thus, we get:
my
B - cl |B|
my
CC) my
exp
Eye
exp
exp
6
d
exp
(-))
5
E)
m,
d
Epp
exp
B cl
|B
d
1
Σ
exp
E
m;
1=1
1=1
1=1
Σ
exp
mce
1
Σ
Eps
exp
B
cl
|B]
my
exp
mye
b
42

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Published as a conference paper at ICLR 2026
d
exp
m3e
2
d ==1
d
E
p
exp
c
my
12
Σ
d
exp
m,e
b
-
1
1=1
==
$
m,e
exp
1=1
[ cl
-
|B|
my
Eyp
exp
7
j=1
mge
exp
2vd
1
d 1=1
exp
mye
-1)
B
B
)
my
Σ
Epp
exp
2vd
b
Σ
m,e
exp
2dd
b
Back to Probability of error To maximize Pa. we take the function that gives us the lower bound. Plugging this upper bound back into equation for P.
my,
min max Pe Pot
2
2vd
exp
1
P.H
In the case where each group has a different scale parameter of their BoP distribution, this becomes:
1
min max Pe W POE
Σ
exp
mye bj
2
2vd
P.EHi
j=l
such that for the unflipped hypothesis testing with 0 get:
$ 1
1
1
mye
min max Pe
1
exp
W PoE PiH
2
2vd
bj
1=1
D.9 MAXIMUM ATTRIBUTES (LAPLACE BOP) FOR ALL PEOPLE
In the case where dataset is drawn from an unknown distribution and has d groups where d 2 with each group having m [N/d] samples, Corollary D.8 becomes:
(1
me
- 1 11t)
min max Pe > 2 2vd
exp
P.E PEH
Corollary D.9 Maximum attributes (Laplace) for all people). Consider auditing a personalized classifier hy to verify if it provides gain of 0.01 to each group auditing dataset D.
43

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Consider auditing dataset with = 0.1 and N = =8 10 samples, one sample for each person on earth. If hy uses more than 2 26 binary group attributes. then for any hypothesis test there will exist pair of probability distributions Px,s, Ho, Qx,Gyy E H1 for which the test results in
probability of error that exceeds 25%.
1
2 26 = min max
Pe 2
(31)
E Qx,CCEHi
4
E LIMITS ON ATTRIBUTES AND SAMPLE SIZE
This section derives theoretical limits on the number of personal attributes and the sample size required per group ensure that the probability of error remains below practitioner-speeffied threshold.
Corollary E.1. Let N be the number of participants, and assume that each group ...... has my = m = [ samples. To ensure that the probability of error verifies min max Pe < 4. the number of binary attributes must be chosen such that le S kmaxs where:
(N log(e² + +1) (Categorical BoP)
In
kmar
W
NN
(Gaussian BoP, variance o²)
In
WWW
(Laplace BoP. scale b).
1
where W is the Lambert W function.
F MIMIC-III EXPERIMENT RESULTS
Below is all supplementary material for the MIMIC-III experiment. This includes G-Bol distribution plots and plots showing how incomprehensivenes. and sufficiency change over the number of features removed.
F.1 EXPERIMENT PLOTS
Experiment Setup. We assume that the practitioner uses 70/30 train-test split for both tasks and compare two neural network models: personalized model with one-hot encoded group attributes (hp) and a generic model without them (hg). Regression outputs are normalized to zero mean and unit variance.
Explanation Method and Explanation Evaluation metric. We assume that the practitioner gen- erates the most important features of models using Integrated Gradients from Captum explanation method (Sundararajan et al.) 2017). We assume that they use sufficiency and incom- prehensiveness as our explanation evaluation metries, where 50% of features are either kept or removed.
Integrated Gradients extracts the most important features of each model by computing input-feature attributions by integrating gradients along path from baseline to the input. To evaluate BoP using sufficiency and incomprehensiveness, we set such that 50% of features are kept or removed. Plots below depict how sufficiency and incomprehensiveness change for different values of as well as show the individual BoP distributions. We use Integrated Gradients for its efficiency, interpretability, and broad adoption, though our framework supports any attribution method.
In the following section, we show supplementary plots for the regression task on the auditing dataset. We show the distribution of the BoP across participants for all three metrics we evaluate. We overlay Laplace and Gaussian distributions to see which fit the individual BoP distribution best, illustrating that prediction and incomprchensiveness
best fit by Laplace distributions and sufficiency by Gaussian distribution. Additionally, we show how incomprehensiveness and sufficiency change for the number of important attributes that are kept removed.
44

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Squared Error Prodiction Difference per Subgroup
Whote
Taplaoe bmo 1968 Fouredion pion 0001 amn 5825
Taplare: 603'1 heo 1974
Couretin: pan 0023 omn 3283
Ukle laphas
20la
Twrosit
Jagasce
1.0
1.5
Dirferenca NouWilat, Lot
Nourlati, 18:45
Eaploon p=c.0000 000.0100, mmn..577
Tapince:
y=n 0012. ozn 050.3525 3525
K31779
Tuba laghen 11 Counraa
Staae: B.322
20 Lagase 11
1.5
Diffrrnnca 7.00
Differenon Loas
Figure 10: Individual prediction cost for all groups using the square error loss function.
Squared Error Incompprehensiveness Difference per Subgroup
Laplace: 5-00 00.000000 2-0.5854
Laplane: 0,0882, 3m0.5825 Conseeaam p- 0.1211, n-B1-000
Canrecan an 2-110000f m-11.526
8.720
Dota
Teta
Domaity
1.0
Laploe
0.5
0.0
0.0
NentWlibe,
NonWhlll,
Lapisce:
Sidder: 1-0,26011 m-:0.5572
Cimmnotion (==000061, m=0.5737
Tots
Tota
Tainsity
Laploe 31
1.0
Lepioe
Gazara
0.3
0.5
0.0
Difforemce T.oss
Figure 11: Individual incomprehensivvnness cost for all groups using the square error loss function.
G ADDITIONAL DATASET RESULTS
The following are the experimental results for G-BoP and G-BoP x on the UCI Heart (Janosi et al. 1989) and MIMIC-III Kidney injury dataset (Johnson et 2016) utilizing three explainer methods through Captum: Integrated Gradients Sundararajan al. (0017). Shapley Value Sampling (Strumbelj & Kononenko 2010), and DeepLIFT (Shrikumar et al. 2017). Interestingly, we see a large amount of agreement across these explainer methods: in nearly all cases, groups that benefited were harmed remain consistent across methods, although the amount by which this occurs varies. We compute Ejim- the value of e for which the lower bound of P. surpasses 25% for the Shapley Value Sampling Method on the UCI Heart dataset to illustrate the full pipeline.
45

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Squared Error Sufficiency Difference per Subgroup
White, 45
White 18-45
Luphauee 0.1015, li 0.7265
Luplatac: 0.1576, 0.6777
Caussiam pmo 1816 cm3.9e00
Gesnerumm pmm) 2t22), umm.
0.6
Sildews 0.9922
Stader: 0.9446
Date
Darsity
Laplase
Decsity
0.6
Taplace
0.4
0.1
0.2
0.0
-3
Difforanice In pas
Difference II Lass
Laslace: 1--0.3305, b-0.7421
Laplace: p=0022 2-00.2336, D=0.716c
Consesian: p-0.10MM. w...M,,--J..0 -1.96W
Gessolamn pr 1280, 2-0.96887
0.6
Iato
n6-
Data
Density
0.-
Derssiy
0.4
Tapdere
-2
Differennee in Loss
Difference in. Loss
Figure 12: Individual sufficiency cost for all groups using the square error loss function.
Table 4: Experimental results the UCI Heart test with columns for DeepLift (D.L.,, Integrated Gradients (L.G.), and Shapley Value Sampling (S. (SS....). The classification task is predicting heart disease presence and the regression task is predicting ST depression induced by exercise. All available features are used, and negative entries appear in red. Using our framework, we computed Qim (for the S.V.S. explainer method) where the lower bound on Pa surpasses 25%. In classification, élim 0.1156 for all metrics; in regression, élim 0.0163 for prediction (Laplace), 0.02 for incomprehensiivenes (Laplace), and 0.153 for sufficiency (Gaussian). Given an 0.002, none of these tests are reliable.
Classification Results
Group
Prediction
Incomp. D.L.
Suff. D.L.
Incomp. LG.
Suff. LG.
Incomp. S.V.S.
Suff. S.V.S.
Female,
0.0000
0.0000
-0.0435
0.0000
-0.0435
0.0000
-0.0870
Female, 18-45
0.0000
-0.1429
0.0000
-0.1429
0.0000
0.0000
0.0000
Male, 45+
0.0588
-0.0588
-0.0784
-0.0588
-0.1373
-0.0588
-0.1176
Male, 18-45
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
All Pop.
0.0440
-0.0330
-0.0440
-0.0330
-0.0750
-0.0220
-0.0769
Minimal BoP
0.0000
-0.1429
-0.0784
-0.1429
-0.1373
-0.0588
-0.1176
Regression Results
Group
Prediction
Incomp.
Suff.
Incomp. LGG
Suff. LG
Incomp. S.V.S.
Suff. S.V.S.
Female, 45+
-0.3077
0.3528
0.1385
0.0980
0.2040
0.1747
0.3332
Female, 18-45
0.0521
-0.0004
0.1067
-0.0438
0.1774
-0.0207
0,0222
Male, 45+
0.0914
0,0286
0,0531
0,0173
0.1381
0,0315
0.1617
Male, 18-45
-0.1410
0.1239
0.4293
0.1384
0.4365
0.1360
0.3592
All Pop.
-0.0333
0.0791
0.1833
0.0523
0,2035
0.0779
0.2258
Minimal BoP
00.007
-0.0004
0.0531
-0.0438
0.1381
-0.0007
0.0222
46

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Published as a conference paper at ICLR 2026
Incomprehensiveness per Group for Varying Values
White, 45+
White, 18-45
squered eivor Loss
squaree error Loss
1.2
1.0
1.0
0.8
White, 45+ hg
White, 18-15
20
30
50
80
20
50
80
Top Features Removed
Top Features Removed
NonWhite,
NonWhite, 18-45
squared ecrox Loos
1.1
squared, crror Loos
1.2
1.0
0.3
0.8
NouWhile, 45e Rg
0.8
NoaWhile, 18-45 Ap
NonWhile,
50 NonWhite,
20
50
00
20
Features Removed
Sufficiency per Group for Varying Values
White,
squared esror Loos
squared error Locs
2.0
2.0
1.0-
2.0
65- fly White, 45 fg
0.5
0.5
White, 18-45
20
40 Top Features Removed
50
80
10
20
40
50
70
80
Top Peatures Removed
NonWhite, 45-
NonWhite, 18-45
Squared atvor Loss
senarre arvor Loss
2.5
1.5-
1.5
1.0-
NonWhite, 45+ Rg
1.0
NonWhile, 16-45
NonWhite,
0.
NonWhite
20
50
20
50 Top Features Removed
80
Removed
Figure 13: Values of Sufficiency and Incomprchensiveness across varying r top features selected using the square error loss function. Values are found for ho and hp.
47

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Table 5: Experimental results on the MIMIC-III Kidney test set, with columns for DeepLift (D.L.), Integrated Gradients (I.G.), and Shapley Value Sampling (S. V.S.); negative values appear in red. The regression task predicts hours to the next continuous renal replacement therapy (CRRT). For classification, the target is patient mortality during the same hospital admission. Features include recent lab measurements (e.g., sodium, potassium, creatinine) prior to CRRT, along with patient age, hours in the ICU at CRRT administration, and the Sequential Organ Failure Assessment (SOFA) score at admission.
Classification Results
Group
Prediction
Incomp. D.L
Suff. D.L.
Incomp. LG
Suff. LG
Incomp. S.V.S.
Suff. S.V.S.
Female, 45+
0.0392
0.0392
-0.0784
0.0392
-0.0784
0.0392
-0.0196
Female, 18-45
0.0000
0.0000
0.3636
0.0000
0.3636
0.0000
0.3636
Male, 45+
0.0164
-0.0164
0.0820
-0.0164
0.0984
-0.0164
0.0000
Male, 18-45
0.0000
0.0000
-0.0833
0.0000
-0.0833
0.0000
0.1667
All Pop.
0.0224
0.0074
0.0296
0.0074
0.0370
0.0074
0.0370
Minimal BoP
0.0000
-0.0164
-0.0833
-0,0164
-0.0833
-0.0164
-0.0196
Regression Results
Group
Prediction
Incomp.
Suff.
Incomp. LG
Suff. LG
Incomp. S.V.S.
Suff. V.S.S.
Female, 45+
0.7582
0.1440
-0.5722
0.1322
-0.6185
0.1380
-0,5414
Female, 18-45
0.5639
0.0177
-0,3255
0.0404
-0,2543
0.0649
00,1077
Male, 45+
0.3449
0.0258
-0.1180
0.0299
-0.1368
0.0310
-0.1518
Male, 18-45
0.4869
-0.1016
-0.1639
-0,0997
-0,1571
-0.0892
40,2124
Pop.
-0.0093
0.0595
-0.3097
0.0584
-0,3311
0.0635
-0.3167
Minimal BoP
-0.0933
-0.1016
-0.5722
-0.0997
-0.6185
-0.8922
-0.5414
48


Reviews:
[{"review_id": "cQdKwtSLWI", "reviewer": "Reviewer_bsq1", "summary": "The paper discusses personalization and explainability of machine learning models and prove the relationship between these two. The paper then discusses hypothesis testing, lower bound, practical aspects of the theory and concludes with experiments.", "questions": "None", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I am not really an expert in this field so just offering my best evaluation here. \n\nThe paper is well written and provides important insights on the personalization and explainability. The sections on their relationship are well-thought and the theoretical aspects are sound.", "weaknesses": "I am not familiar with the relevant literature but the paper is quite original and contains insightful theories.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "ufpLYFeaUm", "reviewer": "Reviewer_dhvG", "summary": "The paper aims to measure how personalisation in machine learning impacts model performance and explainability. The authors show that, for the Bayes optimal classifier, accuracy and explainability are not necessarily aligned. However, when the data distribution follows an additive model, they show that there is a strong relationship between accuracy and explainability. The paper presents a statistical test to determine whether personlisation leads to improvements in accuracy and/or explainability.", "questions": "- Is it easy to construct distributions that show Theorem 4.1 and 4.2? I would be interested in how often this arises in practice.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Paper is well written and clear for the most part\n- Figure 3 and example scenario in Section 6 may be helpful for practitioners who might use the framework", "weaknesses": "**Weak Problem Statement**\n\nThere are two facets here:\n1. Does this actually occur in practice?\n2. If yes, what are the ramifications?\n\nThe authors don't really address these questions. I'd imagine the most salient case here is when we see a benefit in predictive accuracy but not in explainability (which kind of seems to be the case in Table 2?). For example, Monteiro Paes et al. (2022), demonstrate that personalisation may lead to harm on the second page (Table 1). Even still, we need to **infer** why a decrease in incomprehensiveness (or sufficiency) is undesirable. The authors should explicitly discuss the harm here.\n\nI know that the authors state that the BoP framework is metric agnostic, but they should note that they have chosen these metrics to use throughout the paper, including in theoretical results.\n\nThe main motivation in this paper is:\n> A common intuition is that if personalization improves prediction, it must also improve explanations (Del Giudice, 2024) (lines 189-190).\n\nHowever, this doesn't seem to be the point of Del Giudice (2024). \"Explanation quality\" in this paper is different. Del Giudice is talking about the **model's** ability to represent real-world phenomena:\n> The cost is that maximizing the predictive accuracy of a model tends to sacrifice its ability to represent the underlying phenomenon in an accurate and interpretable fashion (page 24).\n\nThis differs from the explanation metrics one might use in XAI, which aims to measure whether explanations faithfully explain **the model** (not nature).\n\n**Results**\n> Insight: The choice of improvement threshold $\\varepsilon$ is key.\n\nThe paragraph here is just repeating the theoretical result in words; its quite trivial. The authors need to elaborate on this further.\n\nMoreover, I can't seem to find the results that accompany the first and last insight.\n\n**Overall Takeaway**\n\nThe authors conclude that there are limits to testing whether there is a benefit to personlisation. However, Monteiro Paes et al. have already demonstrated this in their work. Instead, if the authors have shown a tighter bound, they should frame the takeaway in light of those results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "yZH7H5HYJ3", "reviewer": "Reviewer_BgiC", "summary": "The paper assesses the impact of personal information on explanation fidelity, fairness and model performance claiming that impact on predictions and explanations can be different.", "questions": "see above.\n\n* Why do you need your framework and cannot just assess fairness using one of the many existing ways, ie.:\n1) Add personalization features -> compute fairness metrics\n2) Remove personalization features -> compute fairness metrics\n3) Compare", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "* XAI is a hot topic with many open questions and investigating personalization and faithfulness is an interesting angle\n* The paper aims at a strong mathematical underpinning paired with real-world examples", "weaknesses": "* The paper should early on state that they rely on importance scores, which covers only a subset of XAI methods though arguably the most important one (as the authors claim). Also the emphasis is on tabular data, which is a further narrowing down. Both should be clear from the abstract.\n\n* The setting is not fully clear. On the one hand, the title should contain fairness rather than prediction (pointing towards overall prediction accuracy) as fairness seems to be the focus and not prediction accuary as also stated \"We propose a framework to evaluate whether this weaker fairness criterion is satisfied, both theoretically and empirically, rather than proposing corrective algorithms.\"   . In turn, also some of the abstract and introduction might require rephrasing. On the other hand, H1 points towards an improvement of all groups. It is better to make one clear case on either of the two.\n\n* There is no passage in the appendix (or main part) for theorm 5.1. called like \"proof of theorem 5.1\" ( it might have been called proof for theorem D.3 / 4.3 (which exist) or so), but I could not find the proof. While this might be a minor issue, it essentially makes the work non-verifyable.\n\n* Given the inaccessibility of the proof (prior comment), I focused on the intution provided, which could be expanded. Essentially, the paper appears to say that if there is colinearity among features (i.e., the personalization ones and others) , faithfulness can be jeopardized.  While this sounds reasonable, it is not clear, why that must be really the case. It seems not really specific to personalization but a rather general issue with colinearity. Do you agree?\n\n* The case-study style evaluation is not appreciated. It seems to jeopardize generalization and also makes an assessment clear given by all the practioners advice - a reference to an Appendix is of little help here, but some kind of convincing overview would be better. The promise of providing clear datasets characteristics (mentioned in abstract) to see when personalization hurts seems to be missing - or where is it? \n\nDetail: use faithfulness/fidelity in the main part of the paper, not just in the Appendix, as it is a well-known term and clarifies some of our complex arguign, ie.  the sentence \"this is by design:....\" is obsolete if you clearly speak of fidelity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "3N0ulBIQsg", "reviewer": "Reviewer_k2Uy", "summary": "The paper introduces a unified framework to evaluate how *personalization* in machine learning affects both **prediction accuracy** and **explanation quality** across groups.  \nIt extends the *Benefit of Personalization (BoP)* from binary classification to regression and explanation metrics (sufficiency, incomprehensiveness), proving that personalization can improve or harm interpretability even when accuracy is unchanged.  \nThe authors derive finite-sample limits on the reliability of hypothesis tests for personalization and apply their framework to healthcare datasets (MIMIC-III, UCI Heart), showing that personalization effects are often untestable given dataset size and attribute structure.", "questions": "1. Does the framework apply beyond feature-attribution explainers (e.g., to counterfactual or surrogate explanations)?  \n2. Could the additive-model alignment in Thm. 4.3 generalize to linear or kernel settings?  \n3. How robust are the lower-bound results to distributional misspecification (e.g., Laplace vs Gaussian)?  \n4. Can the framework test *absence of harm* rather than *presence of benefit*?\n\n\n### Comment on Table 3 (Appendix B):\nTable 3 could be made more consistent with Table 1, particularly regarding the inclusion of s∖Js_{\\setminus J}s∖J​ in the definition of incomprehensiveness. Clarifying this alignment would improve coherence across the theoretical and empirical sections.\nIt would also help readers if the caption or surrounding text explicitly stated that higher BoP values in each row indicate greater benefit for subgroup sss, reinforcing the interpretability of the metrics.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- Extends BoP theory to continuous and explanation-based settings, bridging personalization, fairness, and explainability.  \n- Offers clear theoretical results (Thms. 4.1–4.3) showing prediction and explanation gains can diverge; provides a rare link between them under additive models.  \n- Derives practical finite-sample error bounds that guide when personalization tests are statistically valid.  \n- Empirical study (MIMIC-III) effectively illustrates the theoretical limits, highlighting cases where high empirical gains are statistically meaningless.  \n- Well-motivated and relevant to fairness, interpretability, and trustworthy AI.", "weaknesses": "- The explanation method is left vague in the main text; it becomes clear only in Appendix A. Stating early that the framework is demonstrated with Integrated Gradients, DeepLIFT, and SHAP would help.  \n- Heavy notation and scattered definitions (Tables 1 & 3) slow reading; concise examples or visual summaries could clarify.  \n- Some theorems (e.g., 4.3) claim necessity (“always”) rather than possibility (“can”)—the text should match this (compare lines 232 vs 242)\n- Too much of the intuition (e.g., Fig. 6) is tucked into appendices, reducing accessibility. You have some nice plots!\n- Limited discussion of connections to fairness of recourse (implying/not implying fairness of prediction) literature, despite thematic overlap.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)