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REVIEW 4 major objections 5 minor 14 references

On Model Extrapolation in Marginal Shapley Values

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that marginal Shapley values, which average over features in ways that force the model into sparse or impossible regions, can be made extrapolation-free by computing attributions inside strata of a causal feature and then…

desk verdict The two-feature stratified-vs-causal equivalence is real and the math checks out; the paper is a solid, honest contribution, but the method is not yet defined beyond the toy, so treat the broader claim as a hypothesis. read the letter →

arxiv 2412.13158 v1 pith:4NXVNEXK submitted 2024-12-17 stat.ML cs.LG

Ilya Rozenfeld This is my paper · ORCID
classification stat.MLcs.LG
keywords Shapleyvaluesmodelextrapolationmarginalaveragingconditionalcausalstratifiedfeatureattributioncorrelatedfeatures
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 claims that marginal Shapley values, which average over features in ways that force the model into sparse or impossible regions, can be made extrapolation-free by computing attributions inside strata of a causal feature and then redistributing the stratum-dependent baseline according to the causal direction. For the model $f(X)=\beta_0+\beta_1 X_1+\beta_{12} X_1 X_2$ with $X_2=I(X_1>0)$, this stratified method reproduces exactly the causal Shapley values that use conditional averaging, for both causal directions $X_1\to X_2$ and $X_2\to X_1$. The result matters because it suggests causal attributions can be obtained from marginal averaging alone, avoiding the cost and complexity of conditional expectations. On a real motor-claims example, the stratified approach changes the attribution of Bonus Malus and exposes an age-related interaction that standard marginal Shapley values do not show.

What carries the argument

The central object is the stratified Shapley value: compute the usual marginal-average Shapley values separately on each stratum of a causal feature, obtaining a stratum-specific reference $\phi_0$, then use the assumed causal direction to decide which feature receives the non-constant part of $\phi_0$, leaving a common reference $\xi$. The toy model carrying the argument is the linear spline $f(X)=\beta_0+\beta_1 X_1+\beta_{12} X_1 X_2$ with $X_2=I(X_1>0)$, and the identity that makes the claim concrete is the match between the stratified formulas in Table 4 and the causal Shapley formulas in Table 6 when $\xi=\beta_0+\beta_{12}\gamma$ with $\gamma=E[X_1X_2]=1/\sqrt{2\pi}$. This identity does the work of showing that conditional averaging can be replaced by within-stratum marginal averaging for this model class.

What would settle it

Compute stratified marginal and causal conditional Shapley values for the linear spline model with an additional within-stratum correlation, for instance $X_3$ correlated with $X_1$ inside each sign region; if the two sets of attributions diverge, the claimed equivalence between stratified marginal averaging and causal conditional averaging is refuted.

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Extended reading notes

Core claim

On its own terms, the paper establishes that stratifying by a causally upstream feature, computing Shapley values separately within each stratum using marginal averaging, and then transferring the non-constant portion of the stratum reference value to the causal feature produces attributions identical to causal Shapley values for the linear spline model $f(X)=\beta_0+\beta_1 X_1+\beta_{12} X_1 X_2$ with $X_2=I(X_1>0)$. The equality holds with shared baseline $\xi=\beta_0+\beta_{12}\gamma$, where $\gamma=E[X_1X_2]=1/\sqrt{2\pi}$, and holds in both causal directions: under $X_1\to X_2$ the attribution $\phi_2=0$ with all baseline transfer going to $X_1$, and under $X_2\to X_1$ the transfer goes to $X_2$. The paper presents this equivalence as evidence that causal Shapley values, normally defined through conditional averaging and requiring conditional expectation estimates, can be recovered from extrapolation-free marginal averaging whenever the correct stratification is known.

Load-bearing premise

The method presupposes that the analyst knows the true causal direction and the correct stratification, and that stratifying on that feature eliminates all problematic extrapolation; if either assumption fails, the stratified values will not equal a meaningful causal attribution.

Editorial extensions

If this is right

  • If the stratified method is correct, causal Shapley values for this model class can be computed without conditional averaging, which is usually more complicated and computationally expensive.
  • The method intentionally violates the Symmetry axiom of the original Shapley framework, so attributions are causal rather than purely game-theoretic.
  • The freedom to choose $\xi$ lets an analyst answer different questions, such as comparing an observation to a region average, a representative observation, or the overall model mean, though extreme $\xi$ choices can distort attributions.
  • On the French motor-claims data, stratifying by driver age yields Bonus Malus attributions with much stronger age dependence than standard marginal Shapley values, including a low-score signal for very young drivers that standard values hide.
  • The resulting attributions depend on the analyst's causal assumptions and stratification choice, so they are only as meaningful as those inputs.

Reading between the lines

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

  • Editorial inference: the equivalence suggests a broader recipe—whenever a causal variable partitions the feature space so that no residual within-stratum correlation forces off-manifold evaluations, marginal averaging inside each stratum may approximate causal Shapley values; this can be tested on any model with a known causal graph.
  • Editorial inference: the method would likely weaken if the stratifying variable is continuous and strata are chosen by bins, because extrapolation reappears at bin boundaries; the paper's real-data demonstration sidesteps this by treating age as discrete.
  • Editorial inference: the comparison between stratified and conditional causal Shapley values could be used as a diagnostic—disagreement would signal a misspecified causal direction or residual within-stratum dependence.
  • Editorial inference: the caution about choosing $\xi$ points to a practical heuristic not proposed in the paper: anchor $\xi$ to a quantity that minimizes the distortion of feature attributions, such as the observed average prediction, or select $\xi$ to preserve efficiency.
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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

4 major / 5 minor

Summary. This paper studies the extrapolation problem in marginal (interventional) Shapley values for correlated features. Using a two-feature linear spline model f(X) = β0 + β1X1 + β12X1X2 with X2 = I(X1 > 0), it shows that linear and constant extrapolation conventions lead to different and unintuitive Shapley decompositions. It then proposes a 'stratified' approach: compute Shapley values within strata of the causal parent feature using within-stratum background data, then redistribute stratum-specific baselines based on causal direction. For the toy model, the resulting Table 4 with ξ = β0 + β12γ exactly reproduces the causal Shapley values in Table 6 for both directions X1 → X2 and X2 → X1. The method is illustrated on French motor third-party liability claims data, stratifying on driver age.

Significance. The paper's closed-form computations are internally consistent; I verified Tables 1, 2, 4, and 6 and the Appendix B derivation. The two-feature equivalence between stratified marginal averaging and causal Shapley values is a genuinely interesting observation and is stated with an explicit caveat ('at least for a simple problem considered here'). If the construction could be generalized, it would offer a computationally cheaper route to causal Shapley values. However, the significance is currently limited to the toy setting: no general definition of strata, no multi-feature baseline redistribution rule, and no validation against causal Shapley values in the real-data example are provided.

major comments (4)
  1. [Section 3, Table 4] The method is not defined in general. No rule is given for selecting the stratifying variable(s), for handling more than two features, for continuous stratifiers with many levels, or for distributing the stratum baseline among multiple features in a DAG. The real-data application in Section 4 uses a single ad hoc transfer of the entire non-constant baseline to DrivAge. Because the abstract claims a general 'approach,' this under-specification is load-bearing; the paper should either provide a general construction or explicitly limit the claim to the two-feature example.
  2. [Appendix B, Table 6, Conclusion] The equivalence with causal Shapley values is demonstrated only for a two-feature model with deterministic dependence and linearity within strata. No argument is given that within-stratum marginal averaging equals causal conditional averaging for general DAGs or for M > 2 features. The conclusion is appropriately cautious, but the abstract and Section 3's statement that 'causal Shapley values can be computed without conditional averaging' are stronger than what is proven.
  3. [Section 4, Figures 4-7] Stratifying on DrivAge does not by itself avoid extrapolation for the remaining features: they are still marginalized jointly within each age stratum. The manuscript notes the high Area/Density correlation and drops Area, but that is a preprocessing choice, not a property of the method; other correlations among the remaining features can still force off-support evaluations. Moreover, no causal Shapley values are computed on the real data, so the real-data section provides no evidence for the replication claim.
  4. [Table 4, Section 3] The reference constant ξ is free in the stratified approach, and the paper discusses several possible choices. The equality with causal Shapley values holds only for ξ = β0 + β12γ = E[f], which is indeed the natural causal φ0, but the paper should state this dependence explicitly and explain how ξ would be chosen in a multi-feature problem. As written, the replication of causal Shapley values appears to require knowing the causal baseline in advance.
minor comments (5)
  1. [Section 2.2] The phrase 'liner extrapolation' should be 'linear extrapolation'.
  2. [References [12] and [13]] The URLs for references [12] and [13] appear to be swapped: [12] points to the SHAP 'Explaining a simple OR function' page while [13] points to the scikit-learn user guide. The in-text citation for Histogram-Based Gradient Boosting should point to the scikit-learn documentation.
  3. [Appendix B, Eq. (B.2)] At x1* = 0, the expression (1 + sign(x1*))/2 gives 0.5, but X2 is defined as I(X1 > 0), so the conditional expectation is discontinuous at 0; the statement should be qualified as holding almost everywhere.
  4. [Section 4, Figure 4] The caption of Figure 4b should state explicitly that DrivAge is excluded from the 'rest of the features' panel.
  5. [Section 4] The description of how the age Shapley value is computed ('a non-constant portion of reference model should be transferred to the driver's age feature') is informal; please specify the formula, e.g., φ_age = f(x) - ξ - Σ_{j≠age} φ_j.

Circularity Check

1 steps flagged · score 3.0 of 10

The toy derivation is self-contained, but the stratified baseline ξ is a free constant and the claimed match to causal Shapley values is obtained by setting ξ to the causal φ0, making the constant part match by construction.

  1. self definitional [Section 3, paragraph after Table 4; Appendix B, Table 6]
    "The remaining constant portion of φ0 is denoted by ξ, and there is a freedom in choosing its value. ... As shown in Appendix B, this approach leads to the same result as in Table 4 with ξ = β0 + β12γ."

    In Table 4, the stratified φ0 is the free constant ξ and φ1 is β0 + β1x1* − ξ. In Table 6, the causal φ0 is β0 + β12γ and φ1 is β1x1* − β12γ. The paper's statement that the stratified approach 'leads to the same result' as causal Shapley values is made true by setting the free stratified baseline exactly equal to the causal baseline. Thus the constant part of the equivalence is satisfied by construction rather than derived. The x1-dependent terms still match without any parameter choice, so the circularity is limited to the baseline offset.

full rationale

The central toy-model derivation is otherwise self-contained and is benchmarked against the external causal Shapley formulas of Heskes et al. and Frye et al., which are not derived from the present paper. The only true circular element is the free reference constant ξ: Table 4 leaves it unspecified, and the equivalence with Table 6 is announced for ξ = β0 + β12γ, which is exactly the causal φ0. This makes the constant portions of φ1 and φ2 match by construction, although the slope terms in x1 match without fitting. There is no load-bearing self-citation chain: reference [6] is cited for the conditional-vs-marginal background, but the extrapolation analysis and stratified construction are independent. The real-data section is a demonstration rather than a prediction, and its limitations (e.g., strata chosen only on driver age) are scope issues rather than circularity. Overall, the partial baseline-matching circularity warrants a score of 3, not higher, because the non-constant parts of the comparison are independent and the external benchmarks are genuine.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The ledger contains one free constant ξ, several causal-direction assumptions, and standard Shapley/causal-Shapley background. No new particles, forces, mediators, or conserved quantities are introduced; the stratified approach is a method, not an entity.

free parameters (1)
  • ξ (reference constant) = User-selected; causal-match value β0+β12γ
    Table 4 leaves ξ free: "there is a freedom in choosing its value." The equality with causal Shapley values is obtained by setting ξ=β0+β12γ, so the reported replication relies on this chosen value.
assumptions (6)
  • standard math Shapley axioms and the unique solution formula (efficiency, dummy, symmetry, additivity)
    Section 1 uses these without proof as the foundation of the attribution framework.
  • domain assumption Causal Shapley value definitions from Heskes et al. and Frye et al., retaining only causally valid terms with conditional averaging
    Appendix B adopts these formulas as the reference truth that the stratified approach is claimed to replicate.
  • domain assumption The causal direction X1→X2 or X2→X1 is known and supplied by the analyst
    Section 3 introduces this causal information by assumption to redistribute the stratum-specific baseline φ0.
  • domain assumption X1~N(0,1), X2=I(X1>0), and γ=E[X1X2]=1/sqrt(2π)
    This is the model in Eq. (3.1), used for all closed-form value functions and for the claimed extrapolation effects.
  • domain assumption In the real data, driver age causes bonus-malus, and the Area/Density correlation can be handled by dropping Density
    Section 4 states "it is obvious that the age is the cause of the Bonus Malus score" and drops Density due to high correlation with Area.
  • domain assumption Stratum-specific background data removes extrapolation for the stratified feature
    Section 4 uses same-age background subsamples, but no diagnostic is provided to show that all remaining off-manifold evaluations are gone.

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Cite this review

Pith. "Pith review of On Model Extrapolation in Marginal Shapley Values." pith.science (2026). https://pith.science/paper/4NXVNEXK

@misc{pith2026241213158,
  author       = {Pith},
  title        = {Pith review of: On Model Extrapolation in Marginal Shapley Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NXVNEXK}},
  note         = {Machine review of arXiv:2412.13158}
}
read the original abstract

As the use of complex machine learning models continues to grow, so does the need for reliable explainability methods. One of the most popular methods for model explainability is based on Shapley values. There are two most commonly used approaches to calculating Shapley values which produce different results when features are correlated, conditional and marginal. In our previous work, it was demonstrated that the conditional approach is fundamentally flawed due to implicit assumptions of causality. However, it is a well-known fact that marginal approach to calculating Shapley values leads to model extrapolation where it might not be well defined. In this paper we explore the impacts of model extrapolation on Shapley values in the case of a simple linear spline model. Furthermore, we propose an approach which while using marginal averaging avoids model extrapolation and with addition of causal information replicates causal Shapley values. Finally, we demonstrate our method on the real data example.

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Reference graph

Works this paper leans on

14 extracted references · 11 canonical work pages

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