Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

iLOCO: Distribution-Free Inference for Feature Interactions

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper introduces iLOCO, a model-agnostic metric for pairwise and higher-order feature interactions, along with distribution-free confidence intervals for it, computed efficiently via minipatch ensembles.

desk verdict A clean new interaction importance metric with a real gap in the minipatch coverage theory for null pairs, and no code or data to back the empirical claims. read the letter →

arxiv 2502.06661 v2 pith:AIXTAT5H submitted 2025-02-10 stat.ML cs.LG

Camille Little , Lili Zheng , Genevera Allen This is my paper · ORCID
classification stat.MLcs.LG
keywords featureinteractionsiLOCOinferenceminipatchensemblesdistribution-freefunctionalANOVAconfidenceintervalsinterpretablemachinelearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that feature interactions can be quantified and given confidence intervals without distributional assumptions. It defines iLOCO for a feature pair as the difference between removing each feature alone and removing both together, and shows under a functional ANOVA structure that this score equals the variance contributed by the joint interaction terms containing both features. It then proves that both a data-splitting version and a minipatch ensemble version yield asymptotically valid confidence intervals under mild moment and stability conditions. If correct, this gives practitioners a model-agnostic way to test whether a detected interaction is real or just noise, at scales where existing interaction metrics become computationally infeasible.

What carries the argument

The argument runs on the functional ANOVA decomposition f*(X) = Σ_u g_u(X_u) with E[g_u] = 0 and E[g_u g_v] = 0 for u ≠ v. This orthogonality turns the error-increase terms Δ_T into sums of squared norms, and the alternating sum over subsets T of S collapses, via the inclusion-exclusion identity of Lemma 1, to the single sum over terms u that contain S. For inference, the minipatch ensemble constructs leave-one-observation-out predictions by averaging only over minipatches that exclude the observation and the relevant features; the proof then decomposes the estimation error into a CLT term, a minipatch-sampling error, a leave-one-out bias, and a stability term, each controlled by Assumptions 3-7.

What would settle it

Take a population model with correlated features, for example correlated Gaussians with f*(X) = X1 X2 + X1, where the functional ANOVA components are not orthogonal; compute the population iLOCO* directly from the definition and check whether Proposition 1's equality iLOCO* = E[g_{12}(X_{12})^2] holds, with any mismatch showing the variance interpretation fails outside the orthogonality assumption. For the inference theorem, simulate with a base learner whose predictions across minipatches are unbounded or with the minipatch size n too large relative to N, and check whether 90% intervals cover the true iLOCO-MP target substantially below 0.9.

Watch

Extended reading notes

Core claim

The central claim is that the interaction Leave-One-Covariate-Out score, defined as iLOCO_{j,k} = Δ_j + Δ_k − Δ_{j,k}, isolates the signal that features j and k carry only through joint and higher-order terms. Under a functional ANOVA assumption with orthogonal components, Proposition 1 shows iLOCO*_{j,k} = Σ_{u ⊇ {j,k}} E[g_u(X_u)^2] in squared-error regression, and twice that in Bernoulli classification with absolute error; hence positive scores indicate genuine interaction variance. The paper further claims valid asymptotic coverage for confidence intervals built from the data-splitting estimator (Theorem 1) and from the minipatch ensemble estimator (Theorem 2), the latter avoiding data splitting and requiring only a bounded third moment, Lipschitz error functions, bounded prediction differences across minipatches, and suitable minipatch sizes and counts.

Load-bearing premise

The claim that iLOCO measures pure interaction variance rests on the assumption that the regression function can be decomposed into additive components that are uncorrelated with each other; with correlated features, that orthogonality generally fails and the score no longer separates interaction effects from main effects.

Editorial extensions

If this is right

  • Users can rank all feature pairs by iLOCO and read off which interactions are statistically significant, without assuming a data-generating distribution.
  • A negative iLOCO value signals a correlated, individually important pair that LOCO-style measures miss, giving a practical detector for that hard case.
  • The S-way definition extends the same logic to triple and higher-order interactions, with a generalized version of Proposition 1 in the supplement.
  • Minipatch ensembles make the metric and intervals computable at scales where the H-statistic and FaithSHAP become infeasible, per the paper's timing table.
  • The inference target for iLOCO-MP conditions on all the data, so users get sharper estimates than data-splitting while still having asymptotic coverage guarantees.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • In real data with correlated features, the orthogonal-decomposition identity in Proposition 1 may fail; the score's behavior as a signed detector (negative for correlated main effects, positive for true interactions) may survive even when its variance interpretation does not, but that survival is not proved in the paper.
  • The coverage theorem for iLOCO-MP depends on the minipatch-size condition n = o(σ_{j,k}^{MP}/(LD√N)); in practice σ is unknown, so users may want a diagnostic or a data-splitting cross-check before trusting intervals in small samples.
  • The same alternating-sum construction could be applied to other importance notions, such as Shapley-based or partial-dependence-based measures, to produce interaction analogues, since the inclusion-exclusion argument is generic.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes iLOCO, an interaction importance score defined as Δ_j + Δ_k − Δ_{j,k} for a feature pair (j,k), with extensions to higher-order interactions. Under a functional ANOVA assumption, Proposition 1 shows that the population iLOCO equals the sum of squared norms of all ANOVA components containing both features, giving the score an interpretation as interaction variance. The paper then develops distribution-free confidence intervals for iLOCO via data splitting (iLOCO-Split, Theorem 1) and via minipatch ensembles (iLOCO-MP, Theorem 2), and validates the method on synthetic simulations and two real datasets. The iLOCO-Split inference is a standard CLT argument conditional on the training split. The iLOCO-MP theory is imported from prior work on LOCO inference for individual features and adapted to the interaction score.

Significance. If the results were fully established, the paper would provide the first model-agnostic, distribution-free inferential procedure for feature interactions, and the minipatch formulation would give a substantial computational advantage over existing interaction metrics (Table 1). The Definition 1 inclusion-exclusion identity is clean and parameter-free, and Proposition 1 is proved carefully under the stated orthogonality assumption; these are genuine strengths. The iLOCO-Split coverage theorem is a standard but useful application of the classical CLT. However, the headline iLOCO-MP coverage theorem is vacuous as stated because its minipatch-size assumption cannot be satisfied by any integer sequence, and the proof does not address degenerate null pairs. The empirical coverage experiment does not resolve this gap because the null pair is not specified and the simulation parameters violate the theorem's assumptions. The computational and metric contributions are valuable, but the central inferential claim for iLOCO-MP is not currently supported.

major comments (4)
  1. [Section 3.2, Assumption 6; Appendix C, Assumption 12] Assumption 6 requires n = o(σ^MP_{j,k}/(L D √N)). For any fixed nonzero σ^MP_{j,k} and fixed L,D, the right-hand side tends to zero as N→∞, while n is a positive integer, so no sequence of minipatch sizes satisfies the condition. For σ^MP_{j,k}=0 the assumption is undefined, and Assumptions 3 and 7 also fail. Thus Theorem 2 (Theorem 4 in the appendix) has no non-vacuous regime as stated. The proof's ε^(2) bound is 4 L D n / (σ^MP_{j,k} √N), which would require n = o(σ^MP_{j,k} √N / (L D)), not n = o(σ^MP_{j,k} / (L D √N)); the condition likely should be imposed on n/N rather than on n. The paper must correct this rate and verify the theorem under the corrected assumptions, or state precisely the asymptotic regime in which the theorem applies. This is load-bearing because the iLOCO-MP coverage guarantee is a headline contribution.
  2. [Section 3.2, Theorem 2; Section 4, Figure 3] The theorem assumes σ^MP_{j,k}>0 through the normalized third-moment condition and the rates in Assumptions 6-7. For a true null interaction with no main effects involving either feature, h_{j,k}(X,Y) is identically zero, so σ^MP_{j,k}=0 and the theorem is inapplicable. This is precisely the regime relevant to the paper's stated goal of testing whether an interaction is significant. The Figure 3 coverage experiment on a 'null feature pair' does not resolve this: the pair is not identified, and if it is a zero-variance pair the coverage is trivially 1, while if it has main effects then σ>0 but the experiment still violates Assumption 6 because n=20%N. The authors should specify the null pair, report the estimated σ^MP for that pair, and either extend the theory to the degenerate case or state the nondegeneracy condition explicitly.
  3. [Appendix C, proof of Theorem 4] The proof of convergence of ε^(3) and of the variance estimator is not self-contained. After bounding γ_loss(h_{j,k}) ≤ 16 L^2 D^2 n^2/(N−1)^2, the paper states that 'the rest of the arguments in [2] follow directly,' but the argument connecting this stability bound to ε^(3) → 0 is omitted, and the same is true for the consistency of σ̂^2_{j,k}. Since Theorem 4 is explicitly imported from the authors' previous work on LOCO inference, the adaptation to the interaction score should be shown in enough detail that a reader can verify that no term involving the interaction structure changes the analysis. Relatedly, Assumptions 3 and 9 should state E|·|^3 ≤ C, not E(·)^3 ≤ C, since the proof uses absolute moments.
  4. [Section 2.2, Assumption 1 and Proposition 1; Section 2.5] Proposition 1 relies on the zero-mean and zero-correlation conditions of Assumption 1, which Proposition 3 establishes only when features are independent. For correlated features, which the paper explicitly addresses in Section 2.5, the functional ANOVA components are generally not orthogonal, and iLOCO no longer equals the sum of interaction variances. The claim that negative iLOCO identifies 'individually important but correlated feature pairs' is therefore not supported by Proposition 1. The paper should make clear that the 'pure interaction importance' interpretation holds only under Assumption 1, and should either provide a separate theoretical statement for the correlated-feature regime or present the correlated-feature application as heuristic.
minor comments (5)
  1. [Abstract and Section 1.1] The statement that 'no methods exist to conduct statistical inference for feature interactions' is too strong; it should be qualified as 'to the best of our knowledge,' especially given the rapidly growing literature on interaction inference.
  2. [Section 4, Figure 3 caption] The caption says 'coverage ... for a null feature pair' but does not identify which pair is used. The paper should state the pair and the corresponding SNR and main-effect configuration so the reader can determine whether σ^MP is zero or positive.
  3. [Section 4.2, Car Evaluation case study] The text says 'a minipatch size of m = 20%M and n = 20%M'; the second quantity should presumably be 20%N, matching the notation used elsewhere.
  4. [Algorithm 2] The leave-one-out predictions use denominators such as Σ_b I(i∉I_b)I(j,k∉F_b); for finite B these sums can be zero, leaving the predictor undefined. The algorithm should specify how to handle minipatches where the required exclusion pattern never occurs, for example by discarding such samples or adding a small floor.
  5. [Section 3.2] The text contains a typo: 'asymptomatically valid' should be 'asymptotically valid.'

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the main self-citation supplies proof technique rather than the target conclusion.

full rationale

The central derivation chain is self-contained. Definition 1 fixes iLOCO = Δj + Δk − Δj,k, and Proposition 1 evaluates this quantity under the functional-ANOVA Assumption 1. The appendix proof expands Δ*_T = Σ_{u∩T≠∅} E[g_u²] and applies the inclusion-exclusion identity (7) to obtain iLOCO*_S = Σ_{u⊇S} E[g_u²]. Nothing is fitted in this step, and the conclusion is not used to define the metric, so there is no self-definitional or fitted-input circularity. The iLOCO-Split coverage theorem is a conditional CLT argument relying on the stated third-moment assumption; the proof cites Lei et al. for a variance-consistency step, not the authors' own work. The iLOCO-MP coverage theorem (Theorem 2 in the main text, Theorem 4 in Appendix C) is the only place where the authors' prior minipatch LOCO paper [14] is used; the main text says 'the proof follows closely from that of [14]' and the appendix repeatedly invokes 'the same arguments as' [2]. However, the appendix supplies the actual error decomposition into h − E[h] + ε^(1) + ε^(2) + ε^(3), bounds the ε terms, and gives the stability bound γ_loss(h_j,k) ≤ 16L²D²n²/(N−1)²; the citation is therefore a reference to shared proof machinery rather than a black-box import of the coverage conclusion. This is a minor self-citation, not constructional circularity. Empirically, iLOCO is benchmarked against external methods such as H-statistic, FaithSHAP, and Iterative Forests, and the coverage simulation evaluates intervals against a Monte Carlo estimate of the stated conditional target using 10,000 new test points, so 'prediction' is not a renamed fit. A separate correctness concern, not a circularity, is that Assumptions 6/12–13 divide by σ^MP_{j,k}, so the theorem's rate conditions are vacuous for a null pair where σ^MP = 0; this limits the stated inference guarantee but does not make the argument circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The metric definition has no fitted constants, so there are no free parameters. The conceptual content rests on the functional ANOVA orthogonality assumption and technical moment/boundedness conditions inherited from prior LOCO inference work. No new entities are posited.

assumptions (4)
  • domain assumption Assumption 1: f*(X) admits a functional ANOVA decomposition with zero mean and zero correlation between different component functions.
    Invoked in Section 2.2 and Appendix D to prove Proposition 1. Without orthogonality, iLOCO* does not collapse to the sum of interaction variances.
  • domain assumption Assumption 2/8: bounded third moment of the per-sample iLOCO score for iLOCO-Split.
    Used to satisfy the Lyapunov condition in the proof of Theorem 1/3. Reasonable but not verified on real data.
  • domain assumption Assumptions 3-7/9-13: bounded third moment, Lipschitz error, bounded minipatch prediction difference D, minipatch size n=o(σ/(LD√N)), and B large enough.
    Technical conditions for Theorem 2/4. The condition n=o(σ/(LD√N)) implicitly requires the population variance σ_{j,k}^{MP} to be positive, which fails for null interactions.
  • domain assumption Data are i.i.d.
    Assumed in both theorems; typical for this literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of iLOCO: Distribution-Free Inference for Feature Interactions." pith.science (2026). https://pith.science/paper/AIXTAT5H

@misc{pith2026250206661,
  author       = {Pith},
  title        = {Pith review of: iLOCO: Distribution-Free Inference for Feature Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIXTAT5H}},
  note         = {Machine review of arXiv:2502.06661}
}
read the original abstract

Feature importance measures are widely studied and are essential for understanding model behavior, guiding feature selection, and enhancing interpretability. However, many machine learning fitted models involve complex interactions between features. Existing feature importance metrics fail to capture these pairwise or higher-order effects, while existing interaction metrics often suffer from limited applicability or excessive computation; no methods exist to conduct statistical inference for feature interactions. To bridge this gap, we first propose a new model-agnostic metric, interaction Leave-One-Covariate-Out (iLOCO), for measuring the importance of pairwise feature interactions, with extensions to higher-order interactions. Next, we leverage recent advances in LOCO inference to develop distribution-free and assumption-light confidence intervals for our iLOCO metric. To address computational challenges, we also introduce an ensemble learning method for calculating the iLOCO metric and confidence intervals that we show is both computationally and statistically efficient. We validate our iLOCO metric and our confidence intervals on both synthetic and real data sets, showing that our approach outperforms existing methods and provides the first inferential approach to detecting feature interactions.

Figures

Figures reproduced from arXiv: 2502.06661 by the authors.

Figure 1
Figure 1. Validation of iLOCO Metric. Part A shows the success probability of identifying an interaction pair in nonlinear classification scenarios (i) and (iii) using an MLP classifier. Part B presents the success probability of detecting an important, correlated feature pair across varying correlation strengths. 3.2 iLOCO Inference via Minipatches Recall that after fitting minipatch ensembles, estimating the iLOCO metric is… view at source ↗
Figure 2
Figure 2. Comparative Evaluations. Success probability of detecting feature pair (1, 2) across SNR levels for KRBF, RF, and MLP classifiers on nonlinear classification simulations (i) and (ii). Assumption 5. The prediction difference between the predictors trained on different minipatches are bounded by D at any input value X. Assumption 6. The minipatch sizes (m, n) satisfy m M , n N ≤ γ for some constant 0 < γ < 1, and n = … view at source ↗
Figure 3
Figure 3. Theory Validation. Coverage of 90% confidence intervals for a null feature pair in synthetic regression simulation (i) using KRBF, MLP, and RF as the base estimators. expected to rise with increasing SNR. Across all model classes and scenarios, iLOCO-Split (blue) and iLOCO-MP (red) consistently outperform baseline methods, especially at higher SNRs. Notably, FaithSHAP (black) exhibits inflated success probabilities … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: iLOCO (computed via iLOCO-MP) marginal confidence intervals ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Are machine learning interpretations reliable? A stability study on global interpretations

    stat.ML 2025-05 conditional novelty 6.0 of 10

    Popular machine learning interpretation methods are frequently unstable under small data perturbations, and interpretation stability does not track prediction accuracy.

Reference graph

Works this paper leans on

70 extracted references · 63 canonical work pages · cited by 1 Pith paper

  1. [2]

    Permutation importance: a corrected feature importance measure

    André Altmann, Laura Toloşi, Oliver Sander, and Thomas Lengauer. “Permutation importance: a corrected feature importance measure”.Bioinformatics 26.10 (2010), pp. 1340–1347

    work page 2010

  2. [1]

    A review of uncertainty quantification in deep learning: Techniques, applications and challenges

    Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U Rajendra Acharya, et al. “A review of uncertainty quantification in deep learning: Techniques, applications and challenges”. Information fusion 76 (2021), pp. 243–297

    work page 2021

  3. [3]

    Exploring feature interactions in the wild: the new feature-interaction challenge

    Sven Apel, Sergiy Kolesnikov, Norbert Siegmund, Christian Kästner, and Brady Garvin. “Exploring feature interactions in the wild: the new feature-interaction challenge”.Proceedings of the 5th international workshop on feature-oriented software development. 2013, pp. 1–8

    work page 2013

  4. [4]

    Predictive inference with the jackknife+

    RinaFoygelBarber,EmmanuelJ.Candès,AadityaRamdas,andRyanJ.Tibshirani.“Predictive inference with the jackknife+”.The Annals of Statistics49.1 (2021), pp. 486–507.doi: 10. 1214/20-AOS1965. url: https://doi.org/10.1214/20-AOS1965

    doi:10.1214/20-aos1965 2021

  5. [5]

    Iterative random forests to discover predictive and stable high-order interactions

    Sumanta Basu, Karl Kumbier, James B. Brown, and Bin Yu. “Iterative random forests to discover predictive and stable high-order interactions”.Proceedings of the National Academy of Sciences 115.8 (2018), pp. 1943–1948. doi: 10.1073/pnas.1711236115 . eprint: https: //www.pnas.org/doi/pdf/10.1073/pnas.1711236115 . url: https://www.pnas.org/doi/ abs/10.1073/p...

    doi:10.1073/pnas.1711236115 2018

  6. [6]

    Car Evaluation.UCIMachineLearningRepository.DOI:https://doi.org/10.24432/C5JP48

    MarkoBohanec. Car Evaluation.UCIMachineLearningRepository.DOI:https://doi.org/10.24432/C5JP48. 1988

    doi:10.24432/c5jp48 1988

  7. [7]

    Distributional conformal prediction

    Victor Chernozhukov, Kaspar Wüthrich, and Yinchu Zhu. “Distributional conformal prediction”. Proceedings of the National Academy of Sciences118.48 (2021), e2107794118

    work page 2021

  8. [8]

    Detecting gene–gene interactions that underlie human diseases

    Heather J Cordell. “Detecting gene–gene interactions that underlie human diseases”.Nature Reviews Genetics10.6 (2009), pp. 392–404

    work page 2009

Show all 70 references
  1. [9]

    Epistasis: what it means, what it doesn’t mean, and statistical methods to detect it in humans

    Heather J Cordell. “Epistasis: what it means, what it doesn’t mean, and statistical methods to detect it in humans”.Human molecular genetics11.20 (2002), pp. 2463–2468

    2002

  2. [10]

    Significance tests of feature relevance for a black-box learner

    Ben Dai, Xiaotong Shen, and Wei Pan. “Significance tests of feature relevance for a black-box learner”. IEEE transactions on neural networks and learning systems35.2 (2022), pp. 1898– 1911

    2022

  3. [11]

    Difford’s Guide for Discerning Drinkers

    Diffords. Difford’s Guide for Discerning Drinkers. 2025. url: https://www.diffordsguide. com/cocktails (visited on 01/30/2025)

    2025

  4. [12]

    Techniques for interpretable machine learning

    Mengnan Du, Ninghao Liu, and Xia Hu. “Techniques for interpretable machine learning”. Communications of the ACM63.1 (2019), pp. 68–77

    2019

  5. [13]

    Predictive learning via rule ensembles

    Jerome H. Friedman and Bogdan E. Popescu. “Predictive learning via rule ensembles”.The Annals of Applied Statistics 2.3 (Sept. 2008). doi: 10 . 1214 / 07 - aoas148. url: https : //doi.org/10.1214%2F07-aoas148

    2008

  6. [14]

    Inference for Interpretable Machine Learning: Fast, Model-Agnostic Confidence Intervals for Feature Importance

    Luqin Gan, Lili Zheng, and Genevera I. Allen. “Inference for Interpretable Machine Learning: Fast, Model-Agnostic Confidence Intervals for Feature Importance” (2022).doi: 10.48550/ ARXIV.2206.02088. url: https://arxiv.org/abs/2206.02088. 14

    2022 doi

  7. [15]

    Feature Importance for Uncertainty Quantification In Agent-Based Modeling

    Gayane Grigoryan and Andrew J Collins. “Feature Importance for Uncertainty Quantification In Agent-Based Modeling”.2023 Winter Simulation Conference (WSC). IEEE. 2023, pp. 233– 242

    2023

  8. [16]

    Variable importance assessment in regression: linear regression versus random forest

    Ulrike Grömping. “Variable importance assessment in regression: linear regression versus random forest”.The American Statistician63.4 (2009), pp. 308–319

    2009

  9. [17]

    The impact of non-additive genetic associations on age-related complex diseases

    Marta Guindo-Martínez, Ramon Amela, Silvia Bonàs-Guarch, Montserrat Puiggròs, Cecilia Salvoro, Irene Miguel-Escalada, Caitlin E Carey, Joanne B Cole, Sina Rüeger, Elizabeth Atkinson, et al. “The impact of non-additive genetic associations on age-related complex diseases”. Natu...

    2021

  10. [18]

    A Class of Statistics with Asymptotically Normal Distribution

    Wassily Hoeffding. “A Class of Statistics with Asymptotically Normal Distribution”.The Annals of Mathematical Statistics(1948), pp. 293–325

    1948

  11. [19]

    Discovering Additive Structure in Black Box Functions

    Giles Hooker. “Discovering Additive Structure in Black Box Functions”.Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’04. Seattle, WA, USA: Association for Computing Machinery, 2004, pp. 575–580.isbn: 1581138881. doi: 1...

    2004

  12. [20]

    Generalized functional anova diagnostics for high-dimensional functions of dependent variables

    Giles Hooker. “Generalized functional anova diagnostics for high-dimensional functions of dependent variables”.Journal of computational and graphical statistics16.3 (2007), pp. 709– 732

    2007

  13. [21]

    Explaining explanations: Axiomatic feature interactions for deep networks

    Joseph D Janizek, Pascal Sturmfels, and Su-In Lee. “Explaining explanations: Axiomatic feature interactions for deep networks”.Journal of Machine Learning Research22.104 (2021), pp. 1–54

    2021

  14. [22]

    SPEX: Scaling Feature Interaction Explanations for LLMs

    Justin Singh Kang, Landon Butler, Abhineet Agarwal, Yigit Efe Erginbas, Ramtin Pedarsani, Kannan Ramchandran, and Bin Yu. “SPEX: Scaling Feature Interaction Explanations for LLMs”. arXiv preprint arXiv:2502.13870(2025)

    2025 arXiv

  15. [23]

    Black-box tests for algorithmic stability

    Byol Kim and Rina Foygel Barber. “Black-box tests for algorithmic stability”.Information and Inference: A Journal of the IMA12.4 (2023), pp. 2690–2719

    2023

  16. [24]

    Predictive inference is free with the jackknife+- after-bootstrap

    Byol Kim, Chen Xu, and Rina Barber. “Predictive inference is free with the jackknife+- after-bootstrap”. Advances in Neural Information Processing Systems33 (2020), pp. 4138– 4149

    2020

  17. [25]

    Distribution- free predictive inference for regression

    JingLei,MaxG’Sell,AlessandroRinaldo,RyanJTibshirani,andLarryWasserman.“Distribution- free predictive inference for regression”.Journal of the American Statistical Association113.523 (2018), pp. 1094–1111

    2018

  18. [26]

    High-dimensional interaction detection with false sign rate control

    Daoji Li, Yinfei Kong, Yingying Fan, and Jinchi Lv. “High-dimensional interaction detection with false sign rate control”.Journal of Business & Economic Statistics40.3 (2022), pp. 1234– 1245

    2022

  19. [27]

    Statistical and functional studies identify epistasis of cardiovascular risk genomic variants from genome-wide association studies

    Yabo Li, Hyosuk Cho, Fan Wang, Oriol Canela-Xandri, Chunyan Luo, Konrad Rawlik, Stephen Archacki, Chengqi Xu, Albert Tenesa, Qiuyun Chen, et al. “Statistical and functional studies identify epistasis of cardiovascular risk genomic variants from genome-wide association studies”...

    2020

  20. [28]

    Analysis of regression in game theory approach

    Stan Lipovetsky and Michael Conklin. “Analysis of regression in game theory approach”.Applied stochastic models in business and industry17.4 (2001), pp. 319–330

    2001

  21. [29]

    Explicit feature interaction- aware uplift network for online marketing

    Dugang Liu, Xing Tang, Han Gao, Fuyuan Lyu, and Xiuqiang He. “Explicit feature interaction- aware uplift network for online marketing”.Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2023, pp. 4507–4515

    2023

  22. [30]

    Feature Importance and Uncertainty Quantifi- cation of Machine Learning Model in Materials Science

    Zhichen Liu, Akash Singh, and Yumeng Li. “Feature Importance and Uncertainty Quantifi- cation of Machine Learning Model in Materials Science”.ASME International Mechanical Engineering Congress and Exposition. Vol. 87684. American Society of Mechanical Engineers. 2023, V011T12A007

    2023

  23. [31]

    Why epistasis is important for tackling complex human disease genetics

    Trudy FC Mackay and Jason H Moore. “Why epistasis is important for tackling complex human disease genetics”.Genome medicine6.6 (2014), p. 42

    2014

  24. [32]

    Definitions, methods, and applications in interpretable machine learning

    W James Murdoch, Chandan Singh, Karl Kumbier, Reza Abbasi-Asl, and Bin Yu. “Definitions, methods, and applications in interpretable machine learning”.Proceedings of the National Academy of Sciences116.44 (2019), pp. 22071–22080

    2019

  25. [33]

    Epistasis—the essential role of gene interactions in the structure and evolution of genetic systems

    Patrick C Phillips. “Epistasis—the essential role of gene interactions in the structure and evolution of genetic systems”.Nature Reviews Genetics9.11 (2008), pp. 855–867

    2008

  26. [34]

    PLINK: a tool set for whole-genome association and population-based linkage analyses

    Shaun Purcell, Benjamin Neale, Kathe Todd-Brown, Lori Thomas, Manuel AR Ferreira, David Bender, Julian Maller, Pamela Sklar, Paul IW De Bakker, Mark J Daly, et al. “PLINK: a tool set for whole-genome association and population-based linkage analyses”.The American journal of hu...

    2007

  27. [35]

    A Shapley–Owen Index for Interaction Quan- tification

    Giovanni Rabitti and Emanuele Borgonovo. “A Shapley–Owen Index for Interaction Quan- tification”. SIAM/ASA Journal on Uncertainty Quantification7.3 (2019), pp. 1060–1075. doi: 10.1137/18M1221801 . eprint: https://doi.org/10.1137/18M1221801 . url: https: //doi.org/10.1137/18M1221801

    2019 doi

  28. [36]

    Bootstrapping and sample splitting for high-dimensional, assumption-lean inference

    Alessandro Rinaldo, Larry Wasserman, and Max G’Sell. “Bootstrapping and sample splitting for high-dimensional, assumption-lean inference”.The Annals of Statistics47.6 (2019), pp. 3438–

    2019

  29. [37]

    Conformalized quantile regression

    Yaniv Romano, Evan Patterson, and Emmanuel Candes. “Conformalized quantile regression”. Advances in neural information processing systems32 (2019)

    2019

  30. [38]

    A comprehensive review of feature based methods for drug target interaction prediction

    Kanica Sachdev and Manoj Kumar Gupta. “A comprehensive review of feature based methods for drug target interaction prediction”.Journal of biomedical informatics93 (2019), p. 103159

    2019

  31. [39]

    Explaining deep neural networks and beyond: A review of methods and applications

    Wojciech Samek, Grégoire Montavon, Sebastian Lapuschkin, Christopher J Anders, and Klaus- Robert Müller. “Explaining deep neural networks and beyond: A review of methods and applications”. Proceedings of the IEEE109.3 (2021), pp. 247–278

    2021

  32. [40]

    The hardness of conditional independence testing and the generalised covariance measure

    Rajen D Shah and Jonas Peters. “The hardness of conditional independence testing and the generalised covariance measure”.The Annals of Statistics48.3 (2020), pp. 1514–1538

    2020

  33. [41]

    Gene interactions in human disease studies—evidence is mounting

    Pankhuri Singhal, Shefali Setia Verma, and Marylyn D Ritchie. “Gene interactions in human disease studies—evidence is mounting”.Annual Review of Biomedical Data Science6.1 (2023), pp. 377–395. 16

    2023

  34. [42]

    The use of polynomial splines and their tensor products in multivariate function estimation

    Charles J Stone. “The use of polynomial splines and their tensor products in multivariate function estimation”.The annals of statistics(1994), pp. 118–171

    1994

  35. [43]

    The Shapley Taylor Interac- tion Index

    Mukund Sundararajan, Kedar Dhamdhere, and Ashish Agarwal. “The Shapley Taylor Interac- tion Index”.Proceedings of the 37th International Conference on Machine Learning. Ed. by Hal Daumé III and Aarti Singh. Vol. 119. Proceedings of Machine Learning Research. PMLR, 13–18 Jul 20...

    2020

  36. [44]

    Conformal prediction under covariate shift

    Ryan J Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. “Conformal prediction under covariate shift”.Advances in neural information processing systems32 (2019)

    2019

  37. [45]

    MP-Boost: Minipatch boosting via adaptive feature and observation sampling

    Mohammad Taha Toghani and Genevera I Allen. “MP-Boost: Minipatch boosting via adaptive feature and observation sampling”.2021 IEEE International Conference on Big Data and Smart Computing (BigComp). IEEE. 2021, pp. 75–78

    2021

  38. [46]

    Faith-Shap: The Faithful Shapley Interaction Index

    Che-Ping Tsai, Chih-Kuan Yeh, and Pradeep Ravikumar. “Faith-Shap: The Faithful Shapley Interaction Index”.Journal of Machine Learning Research24.94 (2023), pp. 1–42.url: http: //jmlr.org/papers/v24/22-0202.html

    2023

  39. [47]

    How does this interaction affect me? interpretable attribution for feature interactions

    Michael Tsang, Sirisha Rambhatla, and Yan Liu. “How does this interaction affect me? interpretable attribution for feature interactions”.Advances in neural information processing systems 33 (2020), pp. 6147–6159

    2020

  40. [48]

    Can I trust you more? Model-agnostic hierarchical explanations

    Michael Tsang, Youbang Sun, Dongxu Ren, and Yan Liu. “Can I trust you more? Model-agnostic hierarchical explanations”.arXiv preprint arXiv:1812.04801(2018)

    2018 arXiv

  41. [49]

    Decorrelated variable importance

    Isabella Verdinelli and Larry Wasserman. “Decorrelated variable importance”.Journal of Machine Learning Research25.7 (2024), pp. 1–27

    2024

  42. [50]

    Epistasis regulates genetic control of cardiac hypertrophy

    Qianru Wang, Tiffany M Tang, Nathan Youlton, Chad S Weldy, Ana M Kenney, Omer Ronen, J Weston Hughes, Elizabeth T Chin, Shirley C Sutton, Abhineet Agarwal, et al. “Epistasis regulates genetic control of cardiac hypertrophy”.Research square(2023), rs–3

    2023

  43. [51]

    Testing conditional independence in supervised learning algorithms

    David S Watson and Marvin N Wright. “Testing conditional independence in supervised learning algorithms”.Machine Learning110.8 (2021), pp. 2107–2129

    2021

  44. [52]

    Efficient nonparametric statistical inference on population feature importance using Shapley values

    Brian Williamson and Jean Feng. “Efficient nonparametric statistical inference on population feature importance using Shapley values”.International conference on machine learning. PMLR. 2020, pp. 10282–10291

    2020

  45. [53]

    A general framework for inference on algorithm-agnostic variable importance

    Brian D Williamson, Peter B Gilbert, Noah R Simon, and Marco Carone. “A general framework for inference on algorithm-agnostic variable importance”.Journal of the American Statistical Association 118.543 (2023), pp. 1645–1658

    2023

  46. [54]

    Feature selection for huge data via minipatch learning

    Tianyi Yao and Genevera I Allen. “Feature selection for huge data via minipatch learning”. arXiv preprint arXiv:2010.08529(2020)

    2020 arXiv

  47. [55]

    Minipatch learning as implicit ridge-like regularization

    Tianyi Yao, Daniel LeJeune, Hamid Javadi, Richard G Baraniuk, and Genevera I Allen. “Minipatch learning as implicit ridge-like regularization”.2021 IEEE International Conference on Big Data and Smart Computing (BigComp). IEEE. 2021, pp. 65–68. 17

    2021

  48. [56]

    Cis-epistasis at the LPA locus and risk of cardiovascular diseases

    Lingyao Zeng, Sylvain Moser, Nazanin Mirza-Schreiber, Claudia Lamina, Stefan Coassin, Christopher P Nelson, Tarmo Annilo, Oscar Franzén, Marcus E Kleber, Salome Mack, et al. “Cis-epistasis at the LPA locus and risk of cardiovascular diseases”.Cardiovascular Research 118.4 (202...

    2022

  49. [57]

    Floodgate: inference for model-free variable importance

    Lu Zhang and Lucas Janson. “Floodgate: inference for model-free variable importance”.arXiv preprint arXiv:2007.01283 (2020)

    2020 arXiv

  50. [58]

    Explainable machine learning in image classification models: An uncertainty quantification perspective

    Xiaoge Zhang, Felix TS Chan, and Sankaran Mahadevan. “Explainable machine learning in image classification models: An uncertainty quantification perspective”.Knowledge-Based Systems 243 (2022), p. 108418. 18 A Inference Algorithms Algorithm 1iLOCO-Split Estimation and Inferenc...

    2022

  51. [60]

    Split the data into disjoint training and test sets:(X(1), Y(1)), (X(2), Y(2))

  52. [61]

    Train prediction models ˆf, ˆf −j, ˆf −k, ˆf −(j,k) on (X(1), Y(1)): ˆf (X) = H(X(1), Y(1))(X), ˆf −j(X) = H(X(1),−j, Y(1))(X −j), ˆf −k(X) = H(X(1),−k, Y(1))(X −k), ˆf −(j,k)(X) = H(X(1),−(j,k), Y(1))(X −(j,k))

  53. [62]

    For theith sample in the test set, compute the feature importance and interaction scores: ˆ∆j(X (2) i , Y(2) i ) = Error(Y (2) i , ˆf −j(X (2),−j i )) −Error(Y (2) i , ˆf (X (2) i )), ˆ∆k(X (2) i , Y(2) i ) = Error(Y (2) i , ˆf −k(X (2),−k i )) −Error(Y (2) i , ˆf (X (2) i )),...

  54. [63]

    Calculate iLOCO metric for each test samplei: \iLOCOj,k(X (2) i , Y(2) i ) = ˆ∆j(X (2) i , Y(2) i ) + ˆ∆k(X (2) i , Y(2) i ) − ˆ∆j,k(X (2) i , Y(2) i )

  55. [64]

    Let N test be the sample size of the test data. Obtain a1 − α confidence interval foriLOCOsplit j,k : Csplit j,k = iLOCOj,k − zα/2ˆσj,k√ N test , iLOCOj,k + zα/2ˆσj,k√ N test , where iLOCOj,k = 1 N test N test X i=1 \iLOCOj,k(X (2) i , Y(2) i ) ˆσj,k = vuut 1 N test − 1 N test...

  56. [65]

    , B • Randomly subsample n observations Ib ⊂ [N ] and m features Fb ⊂ [M ]

    Perform Minipatch Learning: forb = 1, . . . , B • Randomly subsample n observations Ib ⊂ [N ] and m features Fb ⊂ [M ]. • Train prediction model ˆfb on (XIb,Fb, YIb): ˆfb(X) = H(XIb,Fb, YIb)(X Fb)

  57. [66]

    Obtain predictions: LOO prediction: ˆf−i(Xi) = 1PB b=1 I(i /∈ Ib) BX b=1 I(i /∈ Ib) ˆfb(Xi) LOO + LOCO (featurej): ˆf −j −i (X −j i ) = 1PB b=1 I(i /∈ Ib)I(j /∈ Fb) BX b=1 I(i /∈ Ib)I(j /∈ Fb) ˆfb(X −j i ) LOO + LOCO (featuresj, k): ˆf −(j,k) −i (X −(j,k) i ) = 1PB b=1 I(i /∈ ...

  58. [67]

    Calculate LOO Feature Occlusion: ˆ∆j(Xi, Yi) = Error(Yi, ˆf −j −i (X −j i )) −Error(Yi, ˆf−i(Xi)) ˆ∆k(Xi, Yi) = Error(Yi, ˆf −k −i (X −k i )) −Error(Yi, ˆf−i(Xi)) ˆ∆j,k(Xi, Yi) = Error(Yi, ˆf −(j,k) −i (X −(j,k) i )) −Error(Yi, ˆf−i(Xi))

  59. [68]

    Calculate iLOCO Metric for each samplei: \iLOCOj,k(Xi, Yi) = ˆ∆j(Xi, Yi) + ˆ∆k(Xi, Yi) − ˆ∆j,k(Xi, Yi)

  60. [69]

    Obtain a 1 − α confidence interval foriLOCOMP j,k : CMP j,k = iLOCOj,k − zα/2ˆσj,k√ N , iLOCOj,k + zα/2ˆσj,k√ N where iLOCOj,k = 1 N NX i=1 \iLOCOj,k(Xi, Yi) ˆσj,k = vuut 1 N − 1 NX i=1 \iLOCOj,k(Xi, Yi) − iLOCOj,k 2 Output: CMP j,k 20 B Inference Theory for iLOCO-Split We res...

  61. [70]

    for our functional ANOVA decomposition. Now we recall the definition ofiLOCO∗ S and plug in∆∗ T above into iLOCO∗ S: iLOCO∗ S = X T ⊆S,T ̸=∅ (−1)|T |+1∆∗ T = X T ⊆S,T ̸=∅ (−1)|T |+1 X u⊆[M ]:u∩T ̸=∅ E[g2 u(Xu)] = X u⊆[M ] E[g2 u(Xu)] X T ⊆S,T ̸=∅ (−1)|T |+1I(u ∩ T ̸= ∅). In th...

    2000

  62. [3469]

    url: https://doi.org/10.1214/18-AOS1784

    doi: 10.1214/18-AOS1784. url: https://doi.org/10.1214/18-AOS1784

    doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.