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Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory

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Pith's one-line read Feature-based explanations reduce to two choices: the feature distribution used for imputation and the aggregation of higher-order interactions.

desk verdict A genuinely useful two-axis framework for perturbation-based explanations, but the exact placement of Integrated Gradients and other gradient methods under 'partial b-fANOVA' is wrong and needs a caveat. read the letter →

arxiv 2412.17152 v2 pith:6AOXCV5V submitted 2024-12-22 cs.LG stat.ML

Fabian Fumagalli , Maximilian Muschalik , Eyke Hüllermeier , Barbara Hammer , Julia Herbinger This is my paper · ORCID
classification cs.LGstat.ML
keywords feature-basedexplanationsfunctionalANOVAShapleyvalueinteractionsMöbiustransformsensitivityanalysisexplainableAIunifiedframework
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 aims to show that most feature-based explanations—local and global, perturbation- and gradient-based—are not separate algorithms but instances of a single construction. That construction pairs a functional ANOVA (fANOVA) decomposition, which decides how strongly the feature distribution enters through baseline, marginal, or conditional imputation, with a game-theoretic aggregation, which decides how strongly higher-order interactions enter through pure, partial, or full effects. The paper proves identities that place SHAP variants, SAGE, PFI, PDP, integrated gradients, DeepSHAP, and others on one grid, and it demonstrates the grid on synthetic and real-world models. If the framework is right, apparent disagreements between explanation methods trace back to these two choices rather than to arbitrary implementation details, giving practitioners a principled way to choose, compare, and interpret methods.

What carries the argument

The load-bearing object is the fANOVA decomposition generated by a value function F_S(x)=\int F(x)\,dP(x_{-S}), together with the Möbius transform of the resulting explanation game. The fANOVA component recursively defines effects f_S(x)=\int F(x)\,dP(x_{-S})-\sum_{T\subset S}f_T(x), so the choice of P completely determines one dimension of the framework; the Möbius transform m(S)=\sum_{T\subseteq S}(-1)^{s-t}\nu(T) is the pure additive contribution of a set in cooperative game theory, and Theorem 4 shows that pure, partial, and full influence measures are precisely different summaries of this transform. Theorem 1 supplies the bridge to gradient methods by proving that b-fANOVA effects coincide with Taylor interactions when F is represented by its Taylor series around the baseline.

What would settle it

Take a one-dimensional ReLU network F(x)=\max(0,x-c) with a baseline b<c and an instance x_0>c. The exact b-fANOVA effect is $f^{{(b)}}$_1(x_0)=F(x_0)-F(b)=x_0-c, but the Taylor series of F around b is identically zero in a neighborhood of b, so Theorem 1's equality fails. This is a concrete model where the framework's gradient-method categorization is not exact.

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Extended reading notes

Core claim

The central claim is that every feature-based explanation in the framework's catalog is determined by two components. First, the choice of distribution P in the fANOVA value function F_S(x) = \int F(x)\,dP(x_{-S}) specifies how the feature distribution enters the explanation: baseline (b-fANOVA) sets P to Dirac masses at a point b, marginal (m-fANOVA) integrates over the joint marginal distribution, and conditional (c-fANOVA) integrates over p(x_{-S} \mid x_S). Second, the game-theoretic aggregation of the Möbius transform m(S)=\sum_{T\subseteq S}(-1)^{s-t}\nu(T) yields three regimes: pure effects keep only the isolated effect m(S), partial effects (Shapley value, generalized values, Shapley interactions) spread higher-order terms with index-specific weights, and full effects include all higher-order terms involving the set. Theorem 1 identifies b-fANOVA effects with Taylor interactions, which is what places gradient-based methods in the framework; Theorem 2 states when the three decompositions coincide; Theorem 3 identifies the Möbius transform of the sensitivity game with the variance of c-fANOVA effects for independent features; and Theorem 4 gives the summary rules that make pure, partial, and full effects increasingly sensitive to higher-order interactions.

Load-bearing premise

The identification of b-fANOVA effects with Taylor interactions in Theorem 1 assumes that the model F is represented by its infinite Taylor series around the baseline b; for ReLU networks and tree ensembles this analyticity fails, so the framework's placement of gradient-based methods is only approximate for those model classes.

Editorial extensions

If this is right

  • Baseline, interventional, and observational SHAP are all partial individual effects that differ only in whether the imputation is b-, m-, or c-fANOVA.
  • PDP and M-plots compute pure joint effects of m- and c-fANOVA, so their centering choices align directly with fANOVA components.
  • SAGE and PFI/CFI are the partial and full individual effects of the risk game, meaning they answer different questions about performance loss rather than competing versions of the same question.
  • For independent features, the Möbius transform of the sensitivity game is the variance of the corresponding c-fANOVA effect, linking global sensitivity indices for dependent data to the same grid.
  • Gradient-based methods such as Integrated Gradients, DeepSHAP, and Integrated Hessians summarize b-fANOVA effects, subject to the Taylor-representability condition in Theorem 1.

Reading between the lines

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

  • The paper does not make this explicit, but the framework suggests that comparing explanation methods is a matter of comparing two dials; a practitioner could use it to pre-register exactly which distributional and interaction assumptions a method encodes.
  • An implication beyond the paper is that benchmark disagreements between attribution methods can be diagnosed by holding one dial fixed and varying the other, rather than by treating methods as incommensurable.
  • One testable extension is to quantify how far gradient-based attributions deviate from exact b-fANOVA effects on non-analytic models such as ReLU networks; Theorem 1 guarantees equality only when the Taylor representation exists.
  • The appendix already points toward a broader grid: replacing the three imputations with retraining-based value functions would add model fitting as a third dimension, a direction the paper identifies as future work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper proposes a unified framework for feature-based explanations by combining functional ANOVA (fANOVA) with cooperative game theory. Three fANOVA decompositions (baseline, marginal, conditional) capture the influence of the feature distribution, while three game-theoretic summary types (pure, partial, full) capture increasing influence of higher-order interactions. The framework is instantiated on local, sensitivity, and risk explanation games, and a catalog of existing XAI methods (SHAP variants, IG, DeepSHAP, SAGE, PFI, Sobol indices, etc.) is mapped to pairs (fANOVA type, interaction summary). The paper includes proofs of the main theorems in the appendix, a practitioner guide, and experiments on synthetic and real-world data.

Significance. If the categorization is accurate, the paper provides a valuable two-dimensional taxonomy that clarifies why explanation methods disagree: the choice of imputation distribution and the choice of higher-order interaction summary. The core mathematical machinery (Möbius inversion relating local games to fANOVA effects, Theorem 4's summary of the Möbius transform) is correct and well-presented. Concrete strengths are the machine-checkable-style appendix proofs, the explicit treatment of pure/partial/full effects in Tables 2 and 3, the reproducible code link, and the clear, falsifiable claim that each method is characterized by an imputation family and an interaction family. The empirical sections usefully illustrate the framework, especially the synthetic experiments varying feature correlation. The main weakness is that the catalog places Integrated Gradients and Integrated Hessians as exact instances of the partial b-fANOVA effect, but their weights depend on Taylor monomial degree rather than subset size, so that placement is not exact.

major comments (2)
  1. [Section 3.1, Theorem 1 and Appendix A.1] Integrated Gradients is catalogued as the partial individual b-fANOVA effect, but Eq. (5) shows that its weight on a b-fANOVA interaction I(κ) is kappa_i / sum(kappa), which depends on the degree vector, not the uniform Shapley weight 1/s. A concrete counterexample makes the gap concrete: for F(x1,x2)=x1^2*x2 with baseline b=(0,0) and instance x0=(1,1), the local b-fANOVA game has ν(empty)=0, ν({1})=0, ν({2})=0, ν({1,2})=1, so the Shapley partial effect is (0.5, 0.5). Integrated Gradients, however, assigns weights 2/3 and 1/3 because the only Taylor interaction has κ=(2,1). Thus IG is not the partial b-fANOVA effect, and Table 4's placement is inaccurate unless it is explicitly qualified as a degree-weighted generalization. The same issue affects Integrated Hessians, whose weights are product-weighted by degrees. The Limitations section (§7) states that some gradient methods summarize derivatives on a more fine-grained level, but the main text and Figure 1 present the placements without this caveat. I request that the authors either correct the classification (e.g., mark IG/IH as 'degree-weighted partial b-fANOVA') or add a prominent qualification in Section 4.1 and Figure 1.
  2. [Section 3.1, Theorem 1 and Appendix A.1] Theorem 1 identifies b-fANOVA effects with Taylor interactions only when F is represented by its Taylor series expanded around the baseline b. This analyticity assumption is load-bearing: deep networks with ReLU activations and tree ensembles are not globally Taylor-representable, so for such models the mappings of DeepSHAP, Integrated Gradients, and Integrated Hessians to b-fANOVA effects are at best approximate. The paper should state this assumption explicitly wherever the gradient-method taxonomy is used (Section 4.1 and Appendix E.1.2), rather than only in the Limitations section. The authors should also indicate whether the degree-weight issue in the previous comment is independent of this analyticity concern; the counterexample above uses a polynomial, so it is not an artifact of non-analyticity.
minor comments (4)
  1. [Section 1, Contributions] In the bullet 'Interpretations for three types of explanations ...', the word 'undermined' appears where 'underlined' or 'supported' is presumably intended; please correct the typo.
  2. [Title and abstract] The title and abstract contain 'ANOV A' with an extra space; this appears to be a formatting artifact and should be fixed to 'ANOVA'.
  3. [Section 3.2, Corollary 1 and Remark 1] Corollary 1 is essentially the Möbius inversion theorem applied to the inclusion ordering, as the authors themselves note in Remark 1. This is fine, but the framing as a 'corollary' of the framework should be softened if the reader might otherwise over-interpret it as a new prediction; the current text already handles this, so this is only a presentation suggestion.
  4. [Figure 1 and Table 4] Since Integrated Gradients and Integrated Hessians are not exact Shapley partial effects, Figure 1 and Table 4 would benefit from a footnote or asterisk indicating the degree-weighted nature of these methods, to avoid misleading practitioners who read the table as an exact correspondence.

Circularity Check

1 steps flagged · score 2.0 of 10

One acknowledged definitional identity (Möbius inversion) is non-load-bearing; the framework's categorizations and experiments are not circular.

  1. self definitional [Section 3.2, Corollary 1 and Remark 1]
    "Corollary 1. The MT m(loc) x0 of the local explanation game ν(loc) x0 is the fANOVA effect fS evaluated at x0, m(loc) x0(S) = Σ T ⊆S (−1)s−tFT (x0) = fS(x0), i.e., the pure additive contribution of the features in S in the fANOVA decomposition. Remark 1. Corollary 1 follows directly from the definitions of the MT and the fANOVA components. In fact, both are special cases of the Möbius inversion theorem (Rota, 1964, Proposition 2) with the inclusion ordering."

    The local explanation game is defined as ν(loc)(S) = F_S(x0), and the fANOVA effect f_S(x0) is defined from the same value functions F_S by inclusion-exclusion (Eq. 1). The Möbius transform of ν(loc) is exactly that same inclusion-exclusion, so the equality m(loc)(S) = f_S(x0) holds by construction rather than as an independently discovered prediction. The paper transparently labels it as 'follows directly from the definitions', and the central contribution—categorizing existing explanation methods along imputation and interaction dimensions—does not depend on this identity being an empirical output. Hence the circularity is real but minor and non-load-bearing.

full rationale

Apart from the acknowledged definitional identity in Corollary 1, the paper's derivation chain is self-contained. Theorem 1 is proved from the Taylor expansion and the inclusion-exclusion form of b-fANOVA, Theorems 2 and 3 are proved from the definitions of the value functions and variance, and Theorem 4 follows from standard game-theoretic summary formulas. The categorization of methods such as SHAP, SAGE, PFI, Sobol' indices, PDP, and M-plots is checked against their known definitions rather than against quantities fitted in this paper. The experiments use correctly specified synthetic models and real-world benchmarks; no fitted parameter is relabeled as a prediction. The self-citations (Muschalik et al. 2024a,b; Fumagalli et al. 2023, 2024) concern implementation details and approximation algorithms, not load-bearing theoretical premises. The placement of Integrated Gradients and Integrated Hessians under 'partial b-fANOVA' is approximate because their Taylor-interaction weights differ from Shapley weights; that is a correctness/accuracy issue, not circularity. Overall, the central claim has independent content and the only definitional reduction is openly acknowledged, so a low score is appropriate.

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

The central framework does not introduce fitted constants or new physical entities. It rests on standard Möbius inversion and variance decomposition, plus two domain assumptions: Taylor representability for the gradient-method branch, and the removal-by-imputation semantics for feature influence. No free parameters enter the theorems; empirical illustrations use hyperparameters such as 512 background samples and 30 repetitions, which do not affect the theoretical claims.

assumptions (4)
  • domain assumption F is representable by its Taylor series around baseline b for the explained instance x0 (Theorem 1).
    Used to equate b-fANOVA effects with Taylor interactions, which underpins the placement of gradient-based attribution methods such as DeepSHAP and Integrated Gradients in the framework.
  • standard math The Möbius inversion theorem and inclusion-exclusion principle hold for the set function lattice (Lemma 1, Corollary 1).
    Standard combinatorial identities used to show that the Möbius transform of the local explanation game equals the fANOVA effect and to derive the sensitivity game's Möbius transform.
  • standard math For independent features, fANOVA effects on disjoint feature sets are independent and the variance decomposition holds (Section 2.1, Theorem 3).
    Underlies the sensitivity game result that the Möbius transform equals the variance of the c-fANOVA effect for independent features.
  • domain assumption Feature influence is meaningfully represented by the value function F_S(x) = E[F(x_S, X_-S)] under baseline, marginal, or conditional distributions (Definitions 1 to 3).
    The entire framework adopts the removal-by-imputation semantics common to SHAP, SAGE, and PFI; the paper does not justify this as the correct causal semantics, only as the shared structure of existing methods.

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Cite this review

Pith. "Pith review of Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory." pith.science (2026). https://pith.science/paper/6AOXCV5V

@misc{pith2026241217152,
  author       = {Pith},
  title        = {Pith review of: Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AOXCV5V}},
  note         = {Machine review of arXiv:2412.17152}
}
read the original abstract

Feature-based explanations, using perturbations or gradients, are a prevalent tool to understand decisions of black box machine learning models. Yet, differences between these methods still remain mostly unknown, which limits their applicability for practitioners. In this work, we introduce a unified framework for local and global feature-based explanations using two well-established concepts: functional ANOVA (fANOVA) from statistics, and the notion of value and interaction from cooperative game theory. We introduce three fANOVA decompositions that determine the influence of feature distributions, and use game-theoretic measures, such as the Shapley value and interactions, to specify the influence of higher-order interactions. Our framework combines these two dimensions to uncover similarities and differences between a wide range of explanation techniques for features and groups of features. We then empirically showcase the usefulness of our framework on synthetic and real-world datasets.

Figures

Figures reproduced from arXiv: 2412.17152 by the authors.

Figure 1
Figure 1. Categorization of selected feature-based explanations with our framework: [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Pure (a) and full (b) individual effects and two￾way interactions for the global sensitivity game of an XG￾Boost model trained on California housing. Blue and red colors denote a reduction and increase of variance. fects. For b- and m-fANOVA, the comparison between pure, partial, and full clearly shows the presence of fea￾ture interactions between x1, x2, and x3 with stronger effects between the first two features. … view at source ↗
Figure 2
Figure 2. Local explanations for the instance x = (1, 1, 1, 1) averaged over 30 repetitions of varying random seeds (fluc￾tuation is shown by error bars). Note that the model is cor￾rectly specified as F(x) = 2x1 +2x2 +2x3 +x1x2 +x1x2x3. explanation games, types, and effects. 5 EXPERIMENTS We now showcase our framework on synthetic and real￾world datasets. For m-fANOVA, we use a background dataset of 512 samples, and for c-fA… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Individual (left) and second order partial in￾teraction (2-SVs, right) effects for the local explanation game of a sentiment analysis language model. While “but” has a negative pure effect, the partial and full effects are positive, indicating positive higher-order int…
Figure 5
Figure 5. Figure 5: Individual local explanations for the instance [PITH_FULL_IMAGE:figures/full_fig_p036_5.png]
Figure 6
Figure 6. Figure 6: Individual local explanations for the instance [PITH_FULL_IMAGE:figures/full_fig_p037_6.png]
Figure 7
Figure 7. Figure 7: Local Explanation Game: Pure (a), partial (b), and full (c) feature influences for an XGBoost model fitted on Titanic. The first row depicts the individual explanations where (b) corresponds to interventional SHAP. The second row shows the pure, partial (2-SVs, 2-SII) …
Figure 8
Figure 8. Figure 8: Global Sensitivity Game: Pure (a), partial (b), and full (c) feature influences for an XGBoost model fitted on Titanic. The first row depicts the individual explanations, the second row interactions, and the third row the joint effects up to order 2. The column (c) cor…
Figure 9
Figure 9. Figure 9: Global Risk Game: Pure (a), partial (b), and full (c) feature influences for a neural network fitted on California housing. The first row depicts the individual explanations where (b) corresponds to SAGE and (c) corresponds to PFI. The second row shows the pure, partia…
Figure 10
Figure 10. Figure 10: Global Risk Game: Pure (a), partial (b), and full (c) feature influences for a neural network fitted on Bike sharing. The first row depicts the individual explanations where (b) corresponds to SAGE and (c) corresponds to PFI. The second row shows the pure, partial (2-…
Figure 11
Figure 11. Figure 11: Local Explanation Game: Pure (a), partial (b), and full (c) feature influences for a sentiment analysis language model fine-tuned on the IMDB dataset (Tsai et al., 2023). The first row depicts the individual explanations where (b) corresponds to baseline SHAP using th…

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Reviewed August 11, 2026 · model on record in the stance chip above.