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On Measuring Intrinsic Causal Attributions in Deep Neural Networks

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Pith's one-line read This paper claims that the intrinsic causal contribution of each input feature to a trained neural network's output can be computed uniquely from observational data, without interventions, given a known acyclic causal graph over the…

desk verdict A legitimate, mostly sound adaptation of intrinsic causal contributions to neural-network attribution, with a clean identifiability argument, but the real-data results rest on assumed causal graphs and the experiments are thinner than the theory. read the letter →

arxiv 2505.09660 v1 pith:2VURI2EK submitted 2025-05-14 stat.ML cs.LG

Saptarshi Saha , Dhruv Vansraj Rathore , Soumadeep Saha , Utpal Garain , David Doermann This is my paper · ORCID
classification stat.MLcs.LG
keywords intrinsiccausalcontributionnormalizingflowSobol'indicesneuralnetworkattributionidentifiabilityglobalexplanationsstructuralmodelShapleyvalue
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 a trained neural network's input features can be assigned intrinsic causal contributions (ICC): the part of a feature's effect on the network's output that is not merely inherited from its upstream causes. The central claim is that these contributions are identifiable from observational data alone, given a known acyclic causal graph over the inputs, and can be estimated post-hoc by fitting a causal normalizing flow to the input distribution. If this is right, global explanations of model behavior can separate a feature's own causal role from effects it merely passes along, something ordinary Shapley or permutation importance cannot do. The authors also connect variance-based ICC to Sobol' sensitivity indices, and report that ICC-based attributions outperform several existing global explanation methods on a stability measure called PGU.

What carries the argument

The central object is the causal normalizing flow: a parametric triangular monotone increasing (TMI) map Fθ together with a fully factorized latent noise distribution Pθ, trained to match the observational input distribution. Its role is to recover the exogenous noise variables U behind each input feature, because ICC is defined as the uncertainty reduction in Ŷ from conditioning on those noise variables. Known identifiability results for triangular maps, namely that two matching flows differ only by independent component-wise invertible transformations, make the ICC value invariant across equally good flow fits. A second piece of machinery is Lemma 1, which identifies conditioning on noise with conditioning on observed ancestors and with the corresponding do-intervention, turning a rung-three causal quantity into an observational one.

What would settle it

Fit two differently initialized causal normalizing flows to the same synthetic dataset generated by a known chain SCM A→B→C→Ŷ, compute the ICC of B with each flow, and check that the values agree to Monte Carlo tolerance; then re-run after adding a spurious edge A→C to the supplied graph and measure how much B's attribution moves.

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Extended reading notes

Core claim

The paper treats a neural network as a structural causal model over the input features X and the prediction Ŷ, and defines the intrinsic causal contribution of feature Xj as the change in conditional uncertainty of Ŷ caused by perturbing its exogenous noise Uj, after conditioning on a set I of other noise variables. Symmetrizing over all topological orderings gives ICC_To, and over all permutations gives ICC_Sh; with variance as the uncertainty measure, both decompose the total variance of Ŷ. The key identifiability result (Theorem 7) states that if two causal normalizing flows both match the same observational distribution P(X, Ŷ), their ICC values coincide, because the flows can differ only by component-wise invertible transformations of the exogenous variables, which leave the relevant conditional expectations unchanged. Lemma 1 supplies the bridge from interventions to observation: for a topological ordering, φ(Ŷ|U_T) = φ(Ŷ|X_T) = φ(Ŷ|do(X_T)), so the contributions are computable from observational conditionals. A Monte Carlo algorithm based on the Jansen estimator with randomized quasi-Monte Carlo sampling estimates the required conditional variances.

Load-bearing premise

The load-bearing premise is that the true cause-effect relationships among the input features are known in advance and involve no hidden common causes; if that graph is misspecified, the attributions silently measure the wrong causal quantity.

Editorial extensions

If this is right

  • Post-hoc global attributions can be computed for any trained network from its input data and a known causal graph, with no intervention data or retraining.
  • Attributions respect causal order: a feature's score is its own added contribution, separate from the effect it inherits from upstream features.
  • Variance-based ICC gives an exact additive decomposition of the prediction's variance across input features, so attribution scores are comparable and sum in a meaningful way.
  • When input features are independent, ICC reduces to a Sobol'-index-style variance decomposition, linking explainability to classical global sensitivity analysis.
  • On the three datasets tested, ICC-based explanations produced smaller prediction-gap-on-perturbation (PGU) values than GAM, SP-LIME, and permutation feature importance, indicating more stable global explanations.

Reading between the lines

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

  • If the identifiability result transfers from variance to other uncertainty measures satisfying the stated axioms, the same flow-based pipeline could produce entropy-based or divergence-based causal attributions without new theory.
  • Because the graph is assumed rather than learned in the real-world demonstrations, a practical extension is to run the pipeline over a small set of plausible graphs and report the spread in attributions, turning graph uncertainty into an explicit error bar.
  • The framework could be pointed at fairness questions: the intrinsic contribution of a protected attribute, stripped of effects inherited through mediators, is a concrete, computable target for fairness audits.
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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

3 major / 7 minor

Summary. This paper proposes a post-hoc framework for quantifying intrinsic causal contributions (ICC) of input features to the output of a trained neural network. The authors treat the input generation process as a structural causal model (SCM), define ICC through exogenous noise variables, and introduce two aggregation schemes: Shapley-based ICC and a topological-order average ICC_To. They prove efficiency, nullity, symmetry, and sensitivity properties; with a variance-based uncertainty measure they obtain a causal variance decomposition and relate ICC to Sobol indices under independent inputs. To make ICC identifiable from observational data, they fit causal normalizing flows (CNFs) to the input distribution and invoke component-wise identifiability of triangular maps. Experiments on synthetic data, AutoMPG, and COMPAS compare the resulting attributions with GAM, SP-LIME, and permutation feature importance using the PGU metric.

Significance. If the theoretical claims hold, the paper makes a useful contribution: it supplies an identifiability argument (Theorem 7) via existing CNF results, proposes a topological-order averaging scheme that reduces permutation ambiguity, and connects causal ICC with Sobol indices (Theorem 4). The theoretical chain in Sections 3-7 is coherent and, conditional on Assumptions 1 and 2, the central claim is defensible. The paper also creditably avoids overclaiming rung-3 causal semantics by relying on Lemma 1 to read ICC_To from observational conditionals. However, the practical significance depends on the input causal graph being correct and causally sufficient, and the real-data graph assumptions are not verified; this limits the support for the empirical claims in Section 8.

major comments (3)
  1. [Section 7 / Assumption 1 / Appendix F] The central identifiability result is conditional on a correct, confounder-free causal graph, but the real-data experiments do not establish that this assumption holds. Appendix F states that the AutoMPG ground-truth graph is unknown, is adopted from Reddy et al. (2023a), and is 'validated through consultations with GPT-3.5'; the COMPAS graph is only 'inspired by' Plecko and Bareinboim (2024). If the topological order or a parent set is misspecified, the fitted CNF can still match P(X), but the recovered noise variables U will not be the true causal noises, so Lemma 1's equality phi(Yhat|U_T) = phi(Yhat|do(X_T)) fails and the reported ICC values estimate a different quantity. No amount of observational data can detect this misspecification under the stated assumptions. The authors should either use datasets with verified causal graphs, perform a sensitivity analysis over plausible graphs, or explicitly reframe the real-data results as illustrative and conditional on assumed graphs; the abstract and Section 8 currently overstate the empirical support.
  2. [Section 7, Algorithm 1] Algorithm 1 is not executable as written. Step 3 uses undefined variables epsilon_V and epsilon_W, and the expressions N(F_theta(epsilon_V)) and N(F_theta(epsilon_W)) do not match the quantities u_M, u_N, and u_Q defined in steps 1-2. Since this algorithm is the core estimator for ICC_Sh, the pseudocode must be corrected (likely epsilon_V = u_N and epsilon_W = u_Q) before the experiments can be reproduced.
  3. [Appendix D, Theorem 2] The proof of the variance monotonicity inequality is garbled and does not establish the claim. The text invokes the law of total variance and concludes 'E(V(X)) >= E_Y(V(X|Y))', which is not a meaningful comparison, and 'interchanging Y and Z' does not yield Var_Y(E(X|Y)) <= Var_{Y,Z}(E(X|Y,Z)). The inequality itself is true and follows directly from applying the law of total variance to E[X|Y,Z], but the proof as written is invalid. This theorem underpins the positivity/monotonicity of the proposed variance-based ICC and the decomposition in Corollary 3, so the proof needs to be rewritten.
minor comments (7)
  1. [Section 4.1] The definition of the coalition value phi(Yhat|do(X_T)) := sum_{x_T} phi(Yhat|do(X_T=x_T)) ptilde(x_T) is confusing: the left-hand side notation suggests a single interventional quantity, and the averaging over x_T with ptilde(x_T) should be defined precisely and kept consistent with Lemma 1.
  2. [Section 5, Corollary 3] The transition from unnormalized to normalized phi should be flagged. As written, the decomposition V(Yhat) = sum_j ICC_To is correct only for the unnormalized convention phi(Yhat|U_I) = V_{U_I}(E(Yhat|U_I)); under the normalized convention introduced immediately afterward, the identity becomes 1 = sum_j ICC_To.
  3. [Appendix D, Property 1 proof] The overbrace in the telescoping sum reads phi(Yhat|U_V), which should be phi(Yhat|U) or phi(Yhat|U_[p]); this is a typo in a proof of a central property.
  4. [Theorem 4, Eq. (7)] The second display in Theorem 4 writes ICC_phi(X_j -> Yhat) without a superscript; since ICC_To and ICC_Sh coincide under independent inputs, the equality should say so explicitly.
  5. [Appendix D, Property 3 proof] The binomial coefficients in the Property 3 proof use n where the paper elsewhere uses p; please make the notation uniform.
  6. [Section 8, COMPAS analysis] The claim that single-feature classifiers resolve the discrepancy between ICC_To and ICC_Sh in COMPAS is not supported: the reported F1 scores are 0.8972, 0.8964, and 0.8912, which are nearly equal, so this check does not show that recidivism is the most critical feature. Please either remove this sanity check or state explicitly what conclusion it supports.
  7. [Appendix E, Algorithm 2] Algorithm 2's notation x_Q = (x_M-I, x_NI) is ambiguous; please define it elementwise, e.g., x_{Q,i} = x_{M,i} for i not in I and x_{Q,i} = x_{N,i} for i in I.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: ICC identifiability follows from cited external identifiability results plus explicit graph assumptions, not from the paper's own fitted quantities.

full rationale

The paper's derivation chain is not circular. Intrinsic causal contribution is defined in Eq. (2) in terms of exogenous noises U, and the central identifiability claim (Theorem 7) rests on Lemma 1, which is cited from Janzing et al. (2024), and on the componentwise-invertibility theorem of Javaloy et al. (2023) and Xi and Bloem-Reddy (2023), both external to this author group. Theorem 7 establishes invariance of ICC under componentwise reparameterization of U; that invariance is a nontrivial consequence of measurability of the conditional expectation, not an assumption built into the definition of ICC. The Sobol-index relation in Theorem 4 is proved from the variance decomposition and Owen's Shapley-Sobol theorem, again an external result, so it is a genuine equivalence rather than a renaming. The real-data causal graphs are assumed from prior work and GPT-3.5 consultation, and the COMPAS graph is inspired by an external source; those are external modeling assumptions. If the graph is misspecified or hidden confounders exist, the recovered U variables are not the true causal noises and the estimated ICC targets a different quantity, but that is a validity/identifiability concern under the stated assumptions, not a circular reduction. The only self-citation in the paper, Saha and Garain (2022), appears in a general related-work list and is not load-bearing for any theorem or experiment. No fitted parameter is relabeled as a prediction, and no result is forced by a self-citation chain. Therefore the appropriate finding is no significant circularity.

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

No numeric free parameters are fitted to make the central derivation work. The CNF parameters are estimated from data as part of the method, and the causal graphs are assumed, not fitted. The choice of variance as phi is a modeling choice, not a fitted constant. No new physical or latent entities are postulated; intrinsic causal contribution is a quantity from prior work (Janzing et al., 2024).

assumptions (3)
  • domain assumption The causal graph G among input features is acyclic and contains no latent confounders (Assumption 1).
    Section 3, Assumption 1. Required for Lemma 1, which equates conditioning on U_T with do(X_T) and underpins identifiability of ICC_To.
  • domain assumption The input SCM is diffeomorphic and causally sufficient: F invertible with differentiable inverse, and U components independent (Assumption 2).
    Section 7, Assumption 2. Needed for the CNF identifiability theorems (Theorems 6 and 7) that make ICC_Sh invariant across fitted flows.
  • domain assumption For real datasets, the true causal graph is replaced by an assumed graph: AutoMPG graph from Reddy et al. validated via GPT-3.5, COMPAS graph from Plecko and Bareinboim.
    Appendix F. The attributions are only as good as this graph, and this is the weakest point for the real-world experiments.

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Pith. "Pith review of On Measuring Intrinsic Causal Attributions in Deep Neural Networks." pith.science (2026). https://pith.science/paper/2VURI2EK

@misc{pith2026250509660,
  author       = {Pith},
  title        = {Pith review of: On Measuring Intrinsic Causal Attributions in Deep Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VURI2EK}},
  note         = {Machine review of arXiv:2505.09660}
}
read the original abstract

Quantifying the causal influence of input features within neural networks has become a topic of increasing interest. Existing approaches typically assess direct, indirect, and total causal effects. This work treats NNs as structural causal models (SCMs) and extends our focus to include intrinsic causal contributions (ICC). We propose an identifiable generative post-hoc framework for quantifying ICC. We also draw a relationship between ICC and Sobol' indices. Our experiments on synthetic and real-world datasets demonstrate that ICC generates more intuitive and reliable explanations compared to existing global explanation techniques.

Figures

Figures reproduced from arXiv: 2505.09660 by the authors.

Figure 1
Figure 1. An example of a causal view of a NN with three input features. White nodes represent variables that are either observed or assumed to be known, while shaded nodes indicate unobserved or latent variables. The left graph illustrates the causal relationships between features along with their exogenous parents, while the right graph utilizes exogenous variables for the TMI mapping of the SCM of inputs. In both figures, … view at source ↗
Figure 2
Figure 2. Causal graphs for experimental datasets: (a) COMPAS, (b) Synthetic, (c) AutoMPG [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Global attribution explanations - feature importances. Top left: AutoMPG dataset. Top right: COMPAS dataset. Bottom: Synthetic dataset. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.