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REVIEW 4 major objections 6 minor 40 references

Functional relevance based on the continuous Shapley value

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A continuous-game Shapley value turns every point of a functional input into a measurable player, giving any trained regression model a pointwise relevance curve.

desk verdict A well-built and useful functional-data interpretability heuristic whose central label—'continuous Shapley value'—is not backed by a verified convergence argument; the paper deserves serious refereeing, but the theory needs either proof or a more modest framing. read the letter →

arxiv 2411.18575 v2 pith:K35YLDL4 submitted 2024-11-27 stat.ML cs.AIcs.LGstat.AP

Pedro Delicado , Cristian Pachón-García This is my paper · ORCID
classification stat.MLcs.AIcs.LGstat.AP MSC 91A1262R10
keywords functionaldataanalysismodel-agnosticinterpretabilityShapleyvaluecontinuousgamesscalar-on-functionregressionghostvariablesrelevanceFDA
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

This paper proposes a way to answer, for a trained scalar-on-function prediction model, which points of the functional input actually matter for its predictions. The authors define a game whose players are the points $t$ in the interval where the predictor is observed, and whose payoff for a coalition $S$ is the coefficient of determination $R^2$ achieved when only the values of $X$ on $S$ are kept and the rest are replaced by their conditional expectation given $S$. The Shapley value of this continuous game yields a relevance function $R_f(t)$ that assigns an importance score to every point $t$, without retraining the model. If the construction is valid, any fitted functional predictor becomes auditable at pointwise resolution, and the method is presented as the first global, model-agnostic interpretability tool for functional data regression. The paper illustrates the method with simulated data and with the Tecator spectra.

What carries the argument

The load-bearing object is the Shapley value relevance function $R_f(t)$, built from the continuous game $\nu(S)=\tilde R^2(S)$, where $\tilde R^2(S)$ is the test coefficient of determination of the trained model $f$ evaluated on reconstructed curves $\tilde X_j^S(t)=X_j(t)\mathbf{1}_S(t)+\dot X_j(t)\mathbf{1}_{S^c}(t)$, with $\dot X_j(t)$ the conditional expectation of $X(t)$ given the observed values on $S$, estimated under a Gaussian-process assumption. The finite version of the game on a partition $\{I_1,\dots,I_n\}$ has Shapley value $\varphi_{\nu,i}=(1/n!)\sum_{\pi\in\Pi}[\tilde R^2(p_i^\pi\cup I_i)-\tilde R^2(p_i^\pi)]$, averaged over random permutations in practice, and the relevance function $R_f(t)=\sum_i \varphi_{\nu,i}(b_i-a_i)^{-1}\mathbf{1}_{I_i}(t)$ is the histogram density of these interval contributions. This machinery converts an infinite-player relevance problem into a sequence of finite Shapley-value computations, with the asymptotic theory for continuous games supplying the intended limit as the partition is refined.

What would settle it

Run the method on a deliberately discontinuous prediction rule, such as $f(X)=\max_t X(t)$, and refine the partition of $[0,1]$ while increasing the number of random permutations; if the resulting relevance function does not converge as the partition becomes finer, the assumed asymptotic value for $\nu(S)=\tilde R^2(S)$ does not exist for that model, and the finite approximation is not computing a well-defined continuous Shapley value.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Shapley values, originally defined for finitely many players, can be lifted to the infinite setting of functional regression by treating the continuum of points in the input interval as players and defining the payoff of each measurable subset $S$ as the test-set coefficient of determination of the already trained model when all points outside $S$ are replaced by conditional expectations. The resulting relevance function is computed through the asymptotic approach: for a partition of the interval into $n$ pieces, the finite Shapley values of the restricted game are averaged over random permutations, and the histogram of those interval contributions, renormalized by interval length, is the relevance curve $R_f(t)$. The paper also shows that this game is generally not a vector-measure game (a game whose payoff depends only on finitely many measures of the coalition), so no closed-form diagonal formula is available and the permutation-based finite approximation is not merely an algorithmic convenience but the operative definition.

Load-bearing premise

The construction depends on the way payoff is assigned to coalitions of points being one of the continuous games for which the Shapley value exists and is unique; the paper states that 'under certain conditions' the limit exists but does not verify those conditions for its specific payoff before using a finite approximation.

Editorial extensions

If this is right

  • Any trained functional predictor — linear, nearest-neighbour, or neural — can be audited with a single test set and no retraining, because the method needs only evaluations of the fitted model on reconstructed curves.
  • The relevance function inherits the Shapley efficiency property: integrated over the whole interval it recovers the model's total test $R^2$, so the curve is a genuine allocation of explained variability among regions of the input.
  • When the fitted model has no predictive skill on a target (a linear model facing a strongly non-linear map, for instance), the method returns a flat relevance curve, which serves as a diagnostic of model failure rather than a misleading ranking.
  • On the Tecator data, the relevance functions of all three algorithms agree that the interval $[1040,1050]$ dominates fat-content prediction and $[970,980]$ is negligible, matching the observed variability of the second-derivative curves.
  • The expected computational cost scales as $O(T^3|\Pi'|n)+O(mT^2|\Pi'|n)+O_f(m,T)|\Pi'|n$, driven by the number of random permutations and grid points; the accompanying ShapleyFDA package caches coalition values shared across permutations to reduce the work.

Reading between the lines

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

  • Beyond the paper, the same game construction should extend to functional classification by replacing $R^2$ with test accuracy or log-loss, and to multiple functional regressors by letting players carry a type label; the authors list these only as future directions.
  • Beyond the paper, the Gaussian-process assumption on the reconstruction step is the likely weak point: on strongly non-Gaussian functional data the conditional expectations become best linear predictions, so the method may rank points by their ability to predict missing segments rather than by their role in the response; a retraining-based LOCO comparison on simulated non-Gaussian data would test
  • Beyond the paper, the relevance curve offers a model-sanity check for functional linear models: discrepancies between $R_f(t)$ and the magnitude of the estimated coefficient $\beta(t)$ would flag regions where the linear approximation is distorted, a diagnostic that currently requires manual inspection of $\hat\beta$.
  • Beyond the paper, the difference between relevance curves computed at two partition resolutions could serve as a stability measure, revealing whether the model's signal is localized near specific points or spread over broad regions.
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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 / 6 minor

Summary. The paper proposes a global, model-agnostic interpretability method for scalar-on-function regression. For a trained predictor f and a test sample, it defines a set function ν(S) = ᷈R²(S), the coefficient of determination of f when the functional predictor is reconstructed on Sᶜ by its conditional expectation given the values on S (the ghost-variable idea). Using the asymptotic approach to Shapley values for games with a continuum of players, the authors define an interval Shapley value relevance function R_f(t) via finite interval partitions and random permutations, implement it in the Python package ShapleyFDA, and illustrate it on simulated scenarios and the Tecator data set. The central claim is that R_f(t) is a continuous Shapley value relevance function that identifies the points t most relevant for prediction, without retraining the model.

Significance. If the central claim is secured, the paper would be a useful contribution: it is, to my knowledge, one of the first systematic attempts at global, model-agnostic interpretability for functional predictors, and it offers pointwise relevance curves at no retraining cost. The ghost-variable imputation is a sensible way to avoid leave-one-point-out refitting, and the open-source package plus the reproducible simulation code are valuable assets. The simulations with known ground truth (linear, nonlinear, and discrete targets) give encouraging qualitative evidence that the method identifies relevant regions, and the Tecator example is a convincing illustration. However, the theoretical interpretation as a continuous Shapley value is not established in the manuscript, and several load-bearing assumptions of the continuous-game framework are left unverified.

major comments (4)
  1. [Section 2.3 and Section 3, Eq. (4)] The paper's central claim is that R_f(t) is a continuous Shapley value relevance function obtained by the asymptotic approach, but the required convergence is never established. Section 2.3 states only that 'under certain conditions' the limit of finite Shapley values over refining partitions exists and is unique, citing Neyman (1994); it does not state those conditions, and Section 3 does not verify them for the specific game ν(S)=᷈R²(S) in Eq. (3). The experiments in Section 5.1 fix n=20 intervals and |Π0|=1000 permutations, so the quantity computed in Eq. (4) is a finite-game Shapley value for one fixed partition. Without a proof that the finite Shapley values converge as n→∞, or at least a stability analysis in n, the pointwise curve R_f(t) is not shown to be a well-defined continuous object independent of the chosen partition and permutation sample.
  2. [Section 3.3] Section 3.3 shows that the game ν is not a vector measure game, so the diagonal formula cannot be used. This is a useful negative result, but it does not supply the positive membership needed for the asymptotic approach. The displayed expression for ν(S) depends on an S-dependent family of signed measures τ_k^{S+}, τ_k^{S-} and an S-dependent function ρ^S; no argument is given that the resulting game belongs to any class, such as pNA or another space on which the Neyman asymptotic value is known to exist and be unique. Consequently, the structural analysis in Section 3.3 leaves the convergence question open rather than resolving it.
  3. [Section 2.3 and Eq. (3)] The continuous-game framework in Section 2.3 defines games as nonnegative and monotone set functions ν: B → R+, but the proposed payoff ν(S)=᷈R²(S) is a coefficient of determination, which can be negative for poorly fitting models on subsets S. Indeed, Table 1 reports mean R²(I) values close to 0 for FLM under the nonlinear target, and subset R² values can fall below 0. The paper does not verify that ν is monotone or nonnegative, nor does it argue that the cited value theory extends to such set functions. This is a load-bearing assumption for invoking Aumann–Shapley theory.
  4. [Section 5.1 and Section 5.2] The experimental section provides no sensitivity analysis with respect to the two approximation choices on which the continuous interpretation depends: the number of intervals n and the number of random permutations |Π0|. All simulation scenarios use n=20 and |Π0|=1000 (5000 for Tecator), and no experiment varies the partition. Since the theoretical claim is about a limit over refining partitions, reporting results for, say, n ∈ {5,10,20,40} and a few permutation counts would be needed to support the claim that the observed curves are stable functional objects rather than artifacts of a particular finite partition.
minor comments (6)
  1. [Section 3.3] The notation ρ^S({τ_k^{S+}(S), τ_k^{S-}(S): k≥1}) is not precise: ρ^S is said to be defined on ℝ^N, but the argument is a countably infinite sequence of scalar values; the domain should be stated as a sequence space, or the displayed expression should be rewritten accordingly.
  2. [Section 3.3] The step replacing the finite-sample average by the population variance is written with '≈' and the reconstruction error is assumed to have zero mean. This is plausible given Kneip and Liebl (2020), but the approximation should be stated as a formal assumption or a limiting statement as m→∞, since the subsequent conclusion about the structure of ν rests on it.
  3. [Section 4] The complexity analysis is useful, but the statement that E(T_O)=E(T_M)=T/2 should be made conditional on the uniform random permutation model; otherwise the expectation over the random subset of intervals used in the Shapley sum is implicit.
  4. [Acknowledgments] The sentence 'founded by MICIU/AEI/...' should read 'funded by MICIU/AEI/....'
  5. [Section 5.1] The main-text figures show only the m=200, η=0.05 scenarios and the remaining cases are deferred to Appendix B; it would help the reader if the figure captions stated explicitly that the displayed curves are means over 100 replications and that the corresponding pointwise standard deviations are provided in the appendix.
  6. [Section 2.3] The definition of an S-admissible sequence uses subalgebras (P_j) and requires S ∈ P_1; when the finite approximations are based on interval partitions, the paper does not connect this definition to the specific choice of partitions in Eq. (4), which would clarify the notational link between ν_P and ν_{I_i}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the relevance function is a defined game-theoretic summary of a fixed prediction model, and the validation uses external ground truth.

full rationale

The paper's central construction is a definition, not a fitted prediction: Eq. (3) defines ν(S)=tilde R^2(S) from a pre-trained model f evaluated on ghost-variable reconstructions, and Eq. (4) computes finite Shapley values of the restricted game, aggregated into R_f(t). No parameter of R_f is tuned to reproduce the relevance curves shown in the simulations; those curves are compared with independently known target transformations (e.g., argmax of beta densities, discrete points), so the empirical validation is external to the method. The ghost-variable idea is taken from Delicado and Peña (2023), a self-citation, but it is not an unverified load-bearing premise: Section 3.1 derives the imputation from Gaussian conditional-expectation formulas and Section 3.2 shows it coincides with the Kneip-Liebl optimal reconstruction operator, an independent external result. The paper's own Section 3.3 shows the game is not a vector measure game, so the diagonal formula cannot be used; Section 2.3 cites Neyman (1994) for existence of an asymptotic value 'under certain conditions' without verifying those conditions for this ν. That is a correctness/completeness gap about convergence to a true continuous Shapley value, not a circular identification: the finite computation does not, by construction, equal the asymptotic value, and the claimed continuous interpretation is asserted rather than derived from the same input. No equation in the paper reduces R_f to its own inputs, and no fitted parameter is renamed as a prediction. Hence the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical or mathematical entities are postulated; the Shapley value relevance function is a derived summary statistic. The main hidden inputs are the Gaussian process assumption and the unverified convergence of the finite approximation to a continuous-game limit.

free parameters (3)
  • number of intervals n = 20
    Chosen by hand for all experiments; the relevance function is a histogram over these intervals and results may depend on the choice of n. No sensitivity analysis is provided.
  • number of random permutations |Π0| = 1000 (simulations), 5000 (Tecator)
    Used to approximate the Shapley sum; the convergence of the Monte Carlo approximation is not quantified.
  • observation grid size T = 101
    Functional data are discretized on a 101-point grid; all computations and the covariance matrix estimates depend on this grid.
assumptions (5)
  • standard math Shapley axioms and uniqueness for finite games, and existence and uniqueness of values for non-atomic games under the conditions in Aumann and Shapley (1974) and Neyman (1994).
    The paper builds its framework on these results without reproving them; this is accepted background.
  • domain assumption The functional predictor X follows a Gaussian process, so that E[X(t)|X(u), u in S] is linear in the observed values.
    Equation (5) uses the conditional expectation formula from multivariate normality; otherwise it is only a linear least squares predictor, not a true conditional expectation.
  • domain assumption The test data used to compute the payoff ν are independent of the training data, and the prediction model f is fixed before relevance is computed.
    Stated in Section 3 and needed so that R2(I) genuinely measures predictive performance on fresh data.
  • ad hoc to paper The payoff ν(S) = R2(f(\tilde X^S), y) with ghost-variable imputation is a valid measure of predictive relevance for the points in S.
    This is a modeling choice, not a consequence of an axiom system; the paper does not compare it against alternative payoff definitions.
  • ad hoc to paper The finite partition and finite permutation approximations converge to the continuous Shapley value for this specific game ν.
    The paper invokes existence and uniqueness conditions from the literature but does not verify them for ν(S) = R2(f(\tilde X^S), y).

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Cite this review

Pith. "Pith review of Functional relevance based on the continuous Shapley value." pith.science (2026). https://pith.science/paper/K35YLDL4

@misc{pith2026241118575,
  author       = {Pith},
  title        = {Pith review of: Functional relevance based on the continuous Shapley value},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K35YLDL4}},
  note         = {Machine review of arXiv:2411.18575}
}
read the original abstract

The presence of artificial intelligence (AI) in our society is increasing, which brings with it the need to understand the behavior of AI mechanisms, including machine learning predictive algorithms fed with tabular data, text or images, among others. This work focuses on interpretability of predictive models based on functional data. Designing interpretability methods for functional data models implies working with a set of features whose size is infinite. In the context of scalar on function regression, we propose an interpretability method based on the Shapley value for continuous games, a mathematical formulation that allows for the fair distribution of a global payoff among a continuous set of players. The method is illustrated through a set of experiments with simulated and real data sets. The open source Python package ShapleyFDA is also presented.

Figures

Figures reproduced from arXiv: 2411.18575 by the authors.

Figure 1
Figure 1. Example of a reconstructed function. (Left) Functional data set. (Middle) One functional data. Solid lines indicate intervals in S where the functional data is known, while dotted lines correspond to S c where the functional data is unknown. (Right) Reconstructed function. The values of the functional data for the intervals in S c (dashed lines) are computed as their conditional expectation given the values at the i… view at source ↗
Figure 2
Figure 2. Illustrative example of the Shapley value relevance function, which is able to detect that [0.4, 0.6) is the most relevant interval. when using a prediction model with a good predictive capacity, it is expected that the Shapley value relevance function assigns the majority of the relevance to player I3 = (0.4, 0.6]. Let us use the true regression function as predictive function: f(X ) = Υ(X ). The bottom left and ri… view at source ↗
Figure 3
Figure 3. Functional data sets. 5.1 Simulated data Regarding X , we consider three different ways of generation, namely Fourier expansion, sym￾metric Fourier expansion and Brownian trend (short name for Brownian motion with a trend). The first one consists of linear combinations of the Fourier basis, X F(t) = Pr s=0 Zsξs(t), where Zs are independent draws of a standard normal distribution, while {ξs(t)}s≥0 is the Fourier basi… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Beta density functions. software used to train those models, scikit-fda (Ramos-Carre˜no et al., 2024) contains an imple￾mentation of FLM and FKNN. In addition, Florian Heinrichs has an open source implementation of FNN available at GitHub (github.com/FlorianHeinrichs/f…
Figure 5
Figure 5. Figure 5: Mean Shapley value relevance functions for those scenarios with m = 200 and η = 0.05. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 5
Figure 5. Figure 5: It corresponds to using a discrete transformation when defining the target variable, [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: (Left) Boxplot of {Xj (0.15)} m j=1. (Middle) Boxplot of {|Xj (0.55)|}m j=1. (Right) Boxplot of {X 2 j (0.35)Xj (0.85)} m j=1 where, {Xj} m j=1 corresponds to the Brownian trend data set shown in the rightmost panel of [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Tecator data set. See also Boj et al. (2010). We proceed in the same way as in Section 5.1 to select the basis to represent the functional data. The R2 (I) are 0.962 for FLM, 0.9808 for FKNN and 0.8196 for FNN. Regarding FLM, the top panel of [PITH_FULL_IMAGE:figures/…
Figure 8
Figure 8. Figure 8: Shapley value analysis of Tecator data. (Top) Estimated beta function for the functional linear model using Tecator data set to predict the fat content. (Middle) Set of second derivatives. (Bottom) Shapley value relevance functions. 17 [PITH_FULL_IMAGE:figures/full_fi…
Figure 9
Figure 9. Figure 9: Mean Shapley value relevance functions for those scenarios with m = 200 and η = 0.25. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Mean Shapley value relevance functions for those scenarios with m = 500 and η = 0.05. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Mean Shapley value relevance functions for those scenarios with m = 500 and η = 0.25. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.