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arxiv: 2605.10533 · v1 · submitted 2026-05-11 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

ConfoundingSHAP: Quantifying confounding strength in causal inference

Dennis Frauen, Eyke H\"ullermeier, Maresa Schr\"oder, Marie Brockschmidt, Maximilian Muschalik, Santo M.A.R. Thies, Stefan Feuerriegel, Valentyn Melnychuk

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Pith reviewed 2026-05-12 03:19 UTC · model grok-4.3

classification 💻 cs.LG
keywords causal inferenceconfoundingShapley valuesadjustment setsobservational dataexplainable AI
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The pith

ConfoundingSHAP attributes confounding strength to individual covariates through a targeted Shapley game over adjustment sets.

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

The paper develops a method to identify which observed covariates act as confounders when treatment assignment is unknown in observational data. It defines a custom Shapley game whose value function measures how much an adjustment set reduces bias from confounding, then attributes that value back to each covariate. This produces rankings of confounding strength that differ from standard SHAP uses focused on treatment-effect heterogeneity. A TabPFN-based estimator evaluates the many required adjustment sets without repeated model refits. The resulting attributions are shown to highlight driving covariates across several datasets.

Core claim

ConfoundingSHAP is a Shapley-based method for attributing confounding strength to individual covariates. We propose a Shapley game targeted to infer the confounding strength of the covariates. Our resulting Shapley values differ from the standard applications of SHAP explanations on causal targets, such as understanding treatment effect heterogeneity, which are ill-suited for our task. Second, as our task requires evaluating the value function over many adjustment sets, we provide a scalable TabPFN-based estimation that avoids exhaustive refitting. We demonstrate the practical value across various datasets, where ConfoundingSHAP provides informative explanations of which observed covariates

What carries the argument

The custom Shapley value game whose value function is defined on adjustment sets to isolate the confounding bias removed by each covariate.

If this is right

  • Analysts gain an explicit ranking of which covariates most influence the treatment-outcome dependence.
  • Adjustment decisions can be prioritized by the magnitude of attributed confounding strength.
  • The method supplies an explanation of the observed treatment assignment mechanism without assuming it is known.
  • Scalable evaluation makes the attributions feasible even when the number of covariates is moderately large.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same game structure could be adapted to quantify the strength of other bias sources such as selection bias.
  • Rankings from ConfoundingSHAP might serve as input features for automated confounder-selection procedures in high-dimensional settings.
  • In longitudinal data the method could be extended to time-varying confounders by defining value functions over time-indexed adjustment sets.

Load-bearing premise

The value function over adjustment sets isolates confounding strength rather than other forms of statistical dependence or model misspecification.

What would settle it

A controlled simulation with known true confounders and non-confounders in which the method assigns low strength to the true confounders or high strength to non-confounders would falsify the approach.

Figures

Figures reproduced from arXiv: 2605.10533 by Dennis Frauen, Eyke H\"ullermeier, Maresa Schr\"oder, Marie Brockschmidt, Maximilian Muschalik, Santo M.A.R. Thies, Stefan Feuerriegel, Valentyn Melnychuk.

Figure 1
Figure 1. Figure 1: Shown is the difference be￾tween variable importance obtained by applying SHAP to a CATE estimator [38] vs. of confounding strength (ours). In causal inference, treatment assignment is often not ran￾dom, so treated and untreated groups can differ system￾atically in their characteristics [37, 58, 64]. This gives rise to observed confounding, where variables influence both treatment assignment and the outcom… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of our ConfoundingSHAP method. Therein, residual confounding bias is defined as the gap between observational and causal contrasts and is attributed to individual variates locally and globally. Several explanation methods have been developed for causal machine learning [56, 59], but with different target objects. Examples are methods for variable importance of propensity scores (to explain the tre… view at source ↗
Figure 3
Figure 3. Figure 3: Overview of ConfoundingSHAP. Our method defines confounding bias under restricted adjustment ( A ), introduces a Shapley game over this bias functional ( B ) and then estimates coalition values with tabular foundation models ( C ). ν(S) := −E[bS(XS)] . (7) Why the negative sign? We define ν(S) = −E[bS(XS)] with a negative sign, so that, when comparing a smaller subset S to the larger subset S ∪ {j}, a posi… view at source ↗
Figure 5
Figure 5. Figure 5: Medium-dimensional covariate set￾ting. Synthetic 17-covariate dataset with five confounders. ⇒ The confounders have the largest Shapley values and are recovered in the top five ranks across seeds. not require retraining. TabPFN is used only to obtain the coalition-value estimates νˆ(S) efficiently ( 3 ). The confounding-bias game itself remains unchanged and is the one defined in Section 4. Estimating Shap… view at source ↗
Figure 6
Figure 6. Figure 6: ⇒ Global confounding strength dis￾appears under randomly assigned treatment [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: SUPPORT right heart catheterization study. Confounding strength aver￾aged over 10 runs. ⇒ Several covariates are clinically plau￾sible confounders. Aim. Finally, we demonstrate the clinical value of Confounding￾SHAP by using the SUPPORT right heart catheterization study [10], where the treatment is right-heart catheterization (RHC) within the first 24 hours in the intensive care unit (ICU). The outcome is … view at source ↗
Figure 9
Figure 9. Figure 9: Approximation quality at fixed budget B = 1024 across increasing numbers of covariates. We next vary the coalition budget for synthetic problems across 25, 50, 75, and 100 covariates, each containing 40% true confounders. Figures 10 and 11 show that increasing the budget improves both attribution mass and confounder recovery. This confirms the expected accuracy-compute tradeoff: larger budgets provide more… view at source ↗
Figure 10
Figure 10. Figure 10: Absolute Shapley mass assigned to true confounders as a function of coalition budget. [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Fraction of true confounders recovered among the top 40% of covariates as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Approximation quality for 200 covariates across increasing budgets. [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: CATE explanation versus confounding bias attribution on a synthetic dataset with 4 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: PEHE after removing top-ranked, random, or lowest-ranked covariates on the ACIC 2016 [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Global Shapley values for synthetic datasets with 11 covariates: exact versus approximate [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Local Shapley values for the 11-covariate synthetic dataset with exact computation. [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Global Shapley values for the 17-covariate synthetic dataset with RegressionMSR ap [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Stability of approximate ConfoundingSHAP values over 10 runs in the 11-covariate synthetic setting. F.5 Stability over repeated synthetic datasets We additionally test stability across independently generated synthetic datasets. This evaluates whether the method recovers the intended covariate roles across different samples from the same data-generating process, rather than relying on a favorable random d… view at source ↗
Figure 19
Figure 19. Figure 19: Dataset-level stability for the four-covariate synthetic setting with exact computation. [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Dataset-level stability for the 11-covariate synthetic setting with exact computation. [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Dataset-level stability for the 11-covariate synthetic setting with RegressionMSR approx [PITH_FULL_IMAGE:figures/full_fig_p028_21.png] view at source ↗
read the original abstract

In causal inference, confounders are variables that influence both treatment decisions and outcomes. However, unlike as in randomized clinical trials, the treatment assignment mechanism in observational studies is not known, and it is thus unclear which covariates act as confounders. Here, we aim to generate insight for causal inference and answer: which of the observed covariates act as confounders? We introduce ConfoundingSHAP, a Shapley-based method for attributing confounding strength to individual covariates. Our contributions are twofold. First, we propose a Shapley game targeted to infer the confounding strength of the covariates. Our resulting Shapley values differ from the standard applications of SHAP explanations on causal targets, such as understanding treatment effect heterogeneity, which are ill-suited for our task. Second, as our task requires evaluating the value function over many adjustment sets, we provide a scalable TabPFN-based estimation that avoids exhaustive refitting. We demonstrate the practical value across various datasets, where ConfoundingSHAP provides informative explanations of which observed covariates drive confounding and thereby helps to provide more insight for causal inference in practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces ConfoundingSHAP, a Shapley-value method that attributes confounding strength to individual covariates in observational causal inference. It defines a game whose value function is evaluated over adjustment sets (distinct from standard SHAP applications to treatment-effect heterogeneity), and substitutes TabPFN for exhaustive refitting to achieve scalability. The authors claim that the resulting attributions identify which observed covariates drive confounding and thereby supply practical insight for causal analysis, with demonstrations on various datasets.

Significance. If the value function successfully isolates confounding strength rather than general dependence, the approach would supply a useful diagnostic for high-dimensional observational studies, helping analysts decide which covariates to adjust for and interpret the sources of bias. The TabPFN-based estimator is a pragmatic engineering contribution that removes the computational barrier to evaluating the exponential number of adjustment sets.

major comments (3)
  1. [demonstration / experiments] The empirical evaluation (demonstration section) reports no quantitative metrics, error bars, or comparisons against ground-truth confounding strengths or alternative methods. Without these, the claim that ConfoundingSHAP 'provides informative explanations' cannot be assessed and the central practical contribution remains unsupported.
  2. [§3 (Shapley game definition)] The value function of the proposed Shapley game is defined over adjustment sets but contains no explicit mechanism (contrast, bias term, or causal contrast) that subtracts direct predictive effects, mediation, or collider bias. Consequently, the attributions may reflect any form of statistical dependence rather than confounding strength alone; this is load-bearing for the method's validity.
  3. [§2 / §3] The manuscript asserts that the resulting Shapley values 'differ from the standard applications of SHAP explanations on causal targets' yet provides no formal comparison or counter-example showing that standard SHAP on the outcome or CATE would fail to recover the same ranking of confounders.
minor comments (2)
  1. [§3] Notation for the value function v(S) and the adjustment-set game should be introduced with a single, self-contained definition rather than scattered across paragraphs.
  2. [experiments] The abstract states that the method was 'demonstrated on various datasets' but the main text should include a table summarizing dataset characteristics, number of covariates, and sample sizes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of ConfoundingSHAP's potential utility. We address each major comment below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: The empirical evaluation (demonstration section) reports no quantitative metrics, error bars, or comparisons against ground-truth confounding strengths or alternative methods. Without these, the claim that ConfoundingSHAP 'provides informative explanations' cannot be assessed and the central practical contribution remains unsupported.

    Authors: We agree that the demonstration section would be strengthened by quantitative support. The current version uses qualitative illustrations on real datasets to highlight interpretability where ground truth is unavailable. In revision we will add a new subsection with synthetic experiments that have known confounding structures, reporting metrics such as ranking correlation with true confounders and precision at identifying them. We will also compare against baselines (e.g., outcome-model feature importance) and include error bars from repeated simulations. revision: yes

  2. Referee: The value function of the proposed Shapley game is defined over adjustment sets but contains no explicit mechanism (contrast, bias term, or causal contrast) that subtracts direct predictive effects, mediation, or collider bias. Consequently, the attributions may reflect any form of statistical dependence rather than confounding strength alone; this is load-bearing for the method's validity.

    Authors: The value function is explicitly the absolute change in the estimated average treatment effect when the covariate is added to the adjustment set: v(S) = |τ̂ − τ̂_S|, where τ̂ is the unadjusted estimate and τ̂_S is the estimate after adjusting for S. This focuses on impact to the causal quantity rather than predictive power. We will revise §3 to state this definition more prominently and add a limitations paragraph discussing mediation and collider effects under the maintained no-unmeasured-confounding assumption for the observed covariates. revision: yes

  3. Referee: The manuscript asserts that the resulting Shapley values 'differ from the standard applications of SHAP explanations on causal targets' yet provides no formal comparison or counter-example showing that standard SHAP on the outcome or CATE would fail to recover the same ranking of confounders.

    Authors: We will add a new comparison subsection (and appendix counter-example) showing that standard SHAP applied to an outcome model or CATE model ranks variables by predictive strength, including non-confounders such as pure outcome predictors. The counter-example uses a simple DAG in which a non-confounder receives high standard SHAP but low ConfoundingSHAP; we will also report empirical ranking differences on one of the real datasets. revision: yes

Circularity Check

0 steps flagged

No significant circularity in ConfoundingSHAP derivation

full rationale

The paper defines a Shapley game over adjustment sets whose value function measures changes in causal effect estimates, then computes attributions via the standard Shapley formula and approximates the many evaluations with TabPFN. This construction does not reduce any claimed prediction to a fitted parameter by definition, nor does it rely on a self-citation chain or imported uniqueness theorem to justify the central result. The value function is an explicit modeling choice whose correctness is an external assumption rather than a tautology, and the method is presented as a novel targeting of Shapley values rather than a renaming of a known pattern. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method assumes that a well-defined value function over adjustment sets can isolate confounding strength and that TabPFN provides an accurate enough surrogate for the many required evaluations.

axioms (2)
  • domain assumption A value function over adjustment sets exists that isolates confounding strength
    Stated in the abstract as the target of the Shapley game.
  • domain assumption TabPFN can approximate the required value-function evaluations without refitting
    Presented as the scalability solution.

pith-pipeline@v0.9.0 · 5519 in / 1145 out tokens · 35245 ms · 2026-05-12T03:19:48.711583+00:00 · methodology

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