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Causal Integrated Gradients: Scalable Feature Attribution Along Causal Paths

Joran Michiels, Anis Ismail, Maarten De Vos, Alejandro Sifrim

arXiv:2610.22126v1cs.LGcs.AI

TL;DR

Feature-attribution methods that account for dependencies can be computationally expensive at high dimensions, while common gradient methods assume independence. CIG and CEG route attribution paths along a causal graph to represent direct and mediated effects. The paper reports strong causal-effect and transcription-factor recovery across synthetic and biological evaluations, with limitations tied to graph and surrogate assumptions and worst-case scaling.

  • Problem

    Dependency-aware feature attribution can be costly at high dimensions, while common gradient-based methods assume feature independence.

    Source: §1 · pp. 1–2

  • Method

    CIG and CEG incorporate causal structure into integrated-gradient paths, decomposing attribution into direct effects and effects through causal descendants.

    Source: §1 · p. 2, §3.1 · p. 4

  • Results

    CIG best matches synthetic ground-truth causal effects at a fraction of causal Shapley values’ cost and best recovers transcription factors across simulated and real single-cell data.

    Source: §1 · p. 2, §6 · p. 10

  • Takeaways & Limitations

    Causal structure can be incorporated into gradient-based attribution while scaling to thousands of genes, and the reported evaluations recover upstream drivers.

    Source: §6 · p. 10

    Worst-case scaling is quadratic in feature count, and attribution may concentrate on roots, potentially underweighting relevant intermediate nodes.

    Source: §5 · p. 9

Abstract

from arXiv · show

Existing feature attribution methods that account for feature dependencies do not scale to high dimensions. We propose causal integrated gradients (CIG) and causal expected gradients (CEG), which extend integrated gradients to incorporate prior causal structure among input features while retaining its computational efficiency and, under conditions on the causal graph, its axiomatic guarantees. In many settings, some causal structure between features is already known, for example, that a transcription factor causally regulates a target gene's expression. Yet existing gradient-based methods cannot exploit this structure, instead crediting a feature's downstream effects over its true upstream cause, such as ranking the target gene above the transcription factor that regulates it. Our key idea is to embed this causal structure by routing the integration path along the causal graph. The method thereby decomposes each feature's attribution into a direct effect and effects mediated through its causal descendants. We evaluate CIG in three settings of increasing realism. First, on a synthetic benchmark with a known causal model, CIG's attributions best match the ground-truth causal effect from Pearl's do-calculus, matching the accuracy of causal Shapley values at a fraction of the computational cost. Second, on simulated single-cell gene expression data with a known gene regulatory network (Dyngen), CIG most accurately recovers the transcription factors driving the simulation. Third, on real interferon-$\beta$ stimulation data, where only an approximate causal graph is available, CIG better prioritizes the transcription factors driving the measured response than DeepSHAP and integrated gradients. Together, these results show that incorporating causal structure into gradient-based attributions recovers upstream drivers that existing methods miss, without sacrificing scalability.

1 Introduction

Feature dependencies make independence-assuming explanations potentially misleading, while dependency-aware Shapley methods can be costly in high dimensions. CIG and CEG incorporate causal structure into gradient attributions, with reported evaluations showing causal attribution and gene-regulator recovery at lower computational cost.

1 Introduction

  • Local post-hoc explanations estimate individual input-feature contributions, but most existing methods assume feature independence.Gene expression is one example where feature dependencies reflect underlying biological processes.Source: §1 · p. 1
  • Dependency-aware Shapley methods account for feature dependencies but become prohibitively expensive at high dimensions, while integrated and expected gradients are efficient but assume independence.Ignoring dependencies can produce misleading attributions that fail to uncover the data-generating mechanism.Source: §1 · p. 2
  • CIG and CEG extend integrated gradients with causal structure, retaining its efficiency and, under stated conditions, its axiomatic guarantees.They require a causal graph and a deterministic surrogate rather than a full conditional distribution for sampling.Source: §1 · p. 2
  • On a synthetic benchmark, CIG and CEG are reported to improve causal attribution at substantially lower computational cost, reducing runtimes from hours to minutes.The paper also reports improved recovery of causal transcription factors across simulated and real single-cell datasets.Source: §1 · p. 2

2 Background and related work

Path methods attribute changes in a differentiable model’s output along a path from a baseline to the input. Prior dependency-aware attribution work centers on Shapley values, while existing integrated-gradient path modifications address numerical concerns rather than feature dependencies.

Path methods

  • Path methods integrate model gradients along a differentiable path from a baseline to the input, and their attributions sum to the output difference.Integrated gradients uses a straight-line path with a fixed baseline; expected gradients averages over baselines.Source: §2 · p. 3

Dependency-aware attribution

  • Conditional SHAP models statistical dependencies, whereas causal Shapley values use a causal graph to distinguish direct from mediated effects.The passage notes that incorporating dependencies is contested, but preferable when the goal is understanding the data-generating process.Source: §2 · p. 3

Integrated-gradients extensions

  • Prior integrated-gradients extensions modify paths to address attribution noise or baseline choice, not feature dependencies.Kapishnikov et al. [2021] avoids regions of high partial derivatives; Zhuo and Ge [2024] follows the model’s gradient.Source: §2 · p. 3

Attribution in RNA expression

  • Feature attribution has been used to interpret RNA-expression models, but dependency-ignoring methods may misrepresent biological structure and high-dimensional applications remain largely unexplored due to computational complexity.Examples include expected gradients on GTEx RNA-seq data and explainability methods applied to unsupervised models.Source: §2 · p. 3

3 Causal integrated and expected gradients

CIG and CEG adapt path methods to encode causal dependencies, separating direct effects from effects mediated through descendants. The construction uses an additive-residual causal surrogate, with a sparse implementation and conditions on gradient estimation and axiomatic guarantees.

3 Causal integrated and expected gradients

  • The method reparametrizes path functions to encode dependencies, then uses chain-rule gradients and a model of feature dependencies to compute attributions.The path-function choice is used to encode causal structure rather than numerical considerations.Source: §3 · pp. 3–4

3.1 Incorporating dependencies in the path function

  • Causal paths let an attribution include a feature’s direct effect and its indirect effects through dependent features.The formulation shown captures first-order, one-hop indirect effects; multi-hop effects are treated in an Appendix A extension.Source: §3.1 · p. 4, Figure 2 · p. 4

3.2 Determining the gradients of the path function

  • The path functions are modeled as causal mechanisms plus additive residuals, with a surrogate estimating each feature from its causal parents.The path derivative with respect to the residual parameter is approximated by a backward finite difference, and its quality depends on the surrogate.Source: §3.2 · p. 5

3.3 Modeling feature dependencies through a surrogate model

  • The sparse masked surrogate represents parent-to-child effects as edge-wise contributions on a directed graph and stores only active edges.A per-feature basis expansion supports nonlinear effects, while the sparse formulation reduces memory and computation for large sparse graphs.Source: §3.3 · p. 5

3.4 Extension to causal expected gradients

  • CEG extends the method to multiple baselines, but its nonlinear path derivatives make additional path points more useful than additional baselines under a fixed evaluation budget.Finite-difference and Riemann-sum errors compound when paths are under-sampled.Source: §3.4 · p. 6

3.5 Axiomatic properties

  • The methods retain completeness and direct-term sensitivity(a) unconditionally, while other listed axioms require precise conditions on the graph and surrogate.The dummy axiom is an exception: a feature absent from the model can receive nonzero indirect attribution if it is causally upstream of a feature that appears.Source: §3.5 · p. 6

4 Experiments

Across synthetic and biological evaluations, CIG performs well at identifying causal effects and upstream transcription-factor regulators. The experiments progress from a known causal model to simulated gene expression and real stimulation data, where the causal graph is approximate.

4 Experiments

  • The experiments assess a synthetic known-SCM benchmark against CATE rankings, followed by TF recovery on simulated and real single-cell data.The biological settings use global known drivers in Dyngen and an approximate graph-based evaluation for IFN-β stimulation.Source: §4 · p. 6, §4.1 · p. 6, §4.2 · p. 7

4.1 Comparison with causal attributions

  • CIG achieves near-perfect CATE correlation on synthetic root nodes and outperforms other methods on all nodes, while costing substantially less than causal Shapley values.Causal Shapley computation scales exponentially in feature count, compared with CIG’s worst-case quadratic scaling; standard integrated gradients is faster but captures no causal structure.Source: §4.1 · pp. 6–7, Figure 3 · p. 7

4.2 Causal gene recovery in a simulated biological dataset

  • On Dyngen, expanded-basis second-order CIG achieves the strongest backbone-TF recovery and remains best overall at DCG thresholds of 50 and 250.Method variants are near-optimal at a threshold of 10 genes, and CIG beats DeepSHAP and integrated gradients in every transition state.Source: §4.2 · p. 7, Figure 4 · p. 8

4.3 Perturbed gene discovery in the IFN-β stimulation dataset

  • On the IFN-β dataset, CIG ranks root regulators higher than DeepSHAP and integrated gradients, with a smaller advantage one hop downstream on the OmniPath prior and the same root-focused pattern on SCENIC.Across cell types, its rank-enrichment score for responsive TFs is two- to three-fold larger; its top-k enrichment persists at every cutoff, unlike the other methods.Source: §4.3 · pp. 8–9, Figure 5 · p. 9

5 Limitations and future work

CIG depends on a supplied directed causal graph and a surrogate that models parent effects additively, while scaling quadratically in the worst case. The current formulation may underweight relevant intermediate nodes, and extending it to undirected or partially directed graphs remains future work.

5 Limitations and future work

  • CIG requires prior knowledge of a directed causal graph; removing or reversing genuine edges degrades performance, while spurious additions are mostly tolerated.The graph need not be exactly correct to retain most of the benefit, and CIG tracks instance-specific cell transitions rather than relying on the static graph alone.Source: §5 · p. 9
  • The chosen surrogate cannot capture non-additive parent interactions, and unmodeled confounders bias attribution magnitudes for affected nodes.Any differentiable surrogate can replace the chosen one for parent interactions, but a different surrogate cannot fix an unmodeled confounder.Source: §5 · p. 9
  • CIG and CEG scale quadratically in the worst case, slower than integrated gradients but substantially faster than Shapley-based alternatives.The current formulation may underweight relevant intermediate nodes by concentrating attribution on root nodes; future work could distribute indirect effects more evenly along the causal path.Source: §5 · p. 9

6 Conclusion

CIG and CEG extend integrated gradients with known causal structure to recover causal effects and transcription-factor drivers, with axiomatic guarantees under conditions on the causal graph and surrogate. The method preserves linearity for ensemble members sharing a dataset’s causal graph and mechanisms, while intentionally violating the dummy axiom to attribute absent upstream causes.

6 Conclusion

  • CIG best matches synthetic ground-truth causal effects at a fraction of causal Shapley’s cost and best recovers response-driving transcription factors on simulated and real single-cell data, scaling to thousands of genes.The conclusion reports these results across a synthetic benchmark, simulated Dyngen data, and real IFN-β data.Source: §6 · p. 10

AI use statement

  • The authors used generative AI for drafting, proofs, references, literature searches, and general-purpose coding, but not to implement CIG/CEG, and reviewed and verified its contributions.They state that the authors retain responsibility for the paper’s content.Source: AI use statement · p. 10

Ethics statement

  • The study used public, de-identified data and collected no new human-subject data; prioritized causal-gene candidates require experimental validation before biological or clinical conclusions.The method is intended to prioritize candidates for downstream experimental follow-up, not to make definitive causal or clinical claims.Source: Ethics statement · p. 10

A Implementation for longer dependency paths

  • At any tracked depth, untracked dependency effects are folded into the deepest layer’s full derivative rather than discarded; the implementation stops at two hops, sufficient for the experiments.The topological ordering is a bookkeeping device: any valid ordering of the same DAG gives the same parent-child relationships.Source: A Implementation for longer dependency… · p. 13

B Details of the basis expansion

  • The surrogate’s per-feature structured basis combines polynomial, truncated-power, and Fourier functions to capture smooth, localized, and periodic dependencies while preserving edge-level interpretability.The truncated-power component can model sharp transitions or regime-specific effects.Source: B Details of the basis expansion · pp. 13–14

Completeness

  • At every truncation depth, attributions sum exactly to the model-output difference when the deepest tracked layer uses the full derivative.Practical residuals arise from integration-step discretization and finite differences; CIG’s residual is consistently larger than IG’s, and real-data residuals also reflect surrogate fitting error, but the paper says this is unlikely to affect ranking-based metrics.Source: Completeness · pp. 14–16

C Axiomatic properties

  • With the same graph and mechanisms, CIG preserves implementation invariance and linearity, while its direct attribution component satisfies sensitivity(a).Implementation invariance compares models under one shared explanation setup; linearity likewise requires the component models to share the causal graph and mechanism input-output relations.Source: Implementation invariance · p. 16, Sensitivity(a) · p. 17, Linearity · p. 18

Symmetry

  • Symmetric features receive equal attributions only when they have identical parents and children and exchangeable mechanisms, including symmetric mechanisms for shared children; different graph positions are asymmetric by design.These conditions also suffice for symmetry at every truncation depth.Source: Symmetry · pp. 17–18

Linearity

  • For ensembles trained on one dataset, linearity holds automatically when members share its graph and mechanisms; it may fail if those differ.The graph and mechanisms are properties of the data rather than individual ensemble members.Source: Linearity · p. 18

Dummy

  • A feature absent from the model can receive non-zero indirect attribution when it causally influences a feature the model uses; conditional SHAP and causal Shapley values behave similarly.This is a deliberate choice to capture upstream causal influence.Source: Dummy · p. 19

D Computational complexity analysis

CIG’s cost depends on the causal graph: dense graphs require a dense surrogate Jacobian, while sparse graphs allow computation over active edges. Its polynomial per-instance cost contrasts with sampling-based Shapley methods and their higher computational costs.

D Computational complexity analysis

  • With a sparse causal graph, CIG computes the surrogate Jacobian over active edges in O(d); a dense graph requires O(d^2) work.The dense cost is generic to computing a dense d-to-d Jacobian, while the sparse calculation skips the d-fold blow-up.Source: D Computational complexity analysis · p. 19
  • Adding second-order indirect effects leaves CIG’s complexity class unchanged, with a roughly 2× increase in the first-order-only cost.The extension applies the Jacobian-and-contraction step a second time.Source: D Computational complexity analysis · p. 19
  • Causal Shapley values and conditional SHAP have quadratic and cubic sampling costs, respectively, while exact Shapley values cost O(2^d).At the scale of the real Dyngen and IFN-β experiments, both sampling-based methods are computationally infeasible.Source: D Computational complexity analysis · pp. 19–20
  • Surrogate training is a one-time offline cost that is amortized over explained instances.For a sparse graph, the closed-form sparse surrogate costs O(T · n · d) to train; for a dense graph, it costs O(T · n · d^2).Source: D Computational complexity analysis · p. 20

E Experimental setup and end-to-end cost accounting

The end-to-end profiling separates surrogate fitting from attribution for the Dyngen and IFN-β test sets. Which stage dominates depends on the surrogate, and CEG’s repeated baseline integrations increase attribution time.

E Experimental setup and end-to-end cost accounting

  • CEG attribution is two orders of magnitude slower than CIG because it repeats integration for each reference baseline.Its peak memory is essentially the same as CIG’s.Source: E Experimental setup and end-to-end… · p. 20, Table E.1 · p. 20
  • Moving from first- to second-order indirect effects roughly doubles CIG attribution time, with a smaller increase in memory.Peak memory is set by attribution and grows mainly with the order of indirect effects.Source: E Experimental setup and end-to-end… · p. 20
  • The basis expansion raises surrogate fitting time and training memory by an order of magnitude, while increasing attribution time only moderately.The fitting surcharge is one-off and amortizable rather than recurring per explanation.Source: E Experimental setup and end-to-end… · p. 20

F Details of the synthetic causal attribution benchmark

The synthetic benchmark compares local attributions with individual CATEs in randomly generated causal models. It illustrates how attribution methods differ when features are causally dependent and tests CIG with both true and learned mechanisms.

F Details of the synthetic causal attribution benchmark

  • Each benchmark run generates a random causal graph of maximum depth 2, with expected node degree 5 and non-root mechanisms drawn from linear, tanh, or interaction types.The interaction mechanism uses only the first two parents, assigning any additional parents a structural weight of zero.Source: F Details of the synthetic causal… · pp. 20–21
  • Attributions are evaluated by sample-wise Spearman correlation with individual CATEs, averaged over 100 samples, across all features or root nodes.Figure F.1 reports the all-node comparison as a function of feature count.Source: F Details of the synthetic causal… · p. 21, Figure F.1 · p. 22
  • For experiments with 464 or more features, SHAP baselines use 4 permutations and 10 background samples, down from 10 and 50.The reduced sampling budget means the runtime plot underestimates the true asymptotic gap, since comparable approximation quality would require increasing these parameters.Source: F Details of the synthetic causal… · p. 21
  • For x1 → x2 → F with x1 → F, CIG assigns R2 = 0, causal Shapley gives R1 > R2, and conditional SHAP gives R1 = R2.CIG identifies x2 as having no independent effect beyond x1; causal Shapley still credits x2 because it considers settings where x1 is not set to a value.Source: F Details of the synthetic causal… · p. 21
  • Using the learned expanded-basis surrogate, CIG tracks the true-mechanism results across 100–1000 features, and both remain far above graph-agnostic IG.The reported gap between the learned- and true-mechanism settings is negligible and does not materially change attribution values or rankings.Source: F Details of the synthetic causal… · pp. 21–22, Figure F.2 · p. 22

G Details of the Dyngen experiment

The Dyngen appendix documents the generated datasets, model configuration, attribution methods, and GRN-based gene relevance scoring. The regression datasets are reported across five simulation seeds with nested train, validation, and test splits.

G Details of the Dyngen experiment

  • The Dyngen regression datasets are reported across five seeds, with cells and genes counted before and after quality-control filtering.The filtered cells use a nested 70/30 split, yielding 49% training, 21% validation, and 30% test data.Source: Table G.2 · p. 22
  • The appendix provides summary statistics and a backbone module network for the generated Dyngen linear dataset.It also lists surrogate-model hyperparameters and preprocessing and prediction-model hyperparameters.Source: Table G.1 · p. 23, Figure G.1 · p. 23, Table G.3 · p. 23
  • Gene relevance is scored using the simulated gene regulatory network and DCG at cutoff K.The appendix also lists the feature attribution methods evaluated in the simulated single-cell experiment.Source: Table G.5 · p. 23, Table G.6 · p. 24

H Details of the IFN-β experiment

The IFN-β analyses construct prior and SCENIC regulatory graphs and evaluate attribution rankings by distance from graph roots, alongside enrichment of stimulation-responsive regulons. CIG ranks root genes higher than DeepSHAP and integrated gradients on the prior graph, with reproducible rankings across seeds.

H Details of the IFN-β experiment

  • The OmniPath prior is restricted to measured genes and transcriptional interactions, then converted from a cyclic graph to a DAG, yielding 11,737 edges over 3,792 connected genes.The conversion removed 399 of 12,136 edges, or 3.3% of edges and 3.6% of total confidence mass.Source: H Details of the IFN-β experiment · p. 24
  • The SCENIC evaluation begins with PBMC single-cell RNA-seq from eight lupus patients, using six patients for training and retaining 17,448 cells after quality control.The filtered matrix contained 24,562 cells and 6,941 genes; two patients were left out of training.Source: H Details of the IFN-β experiment · p. 25
  • SCENIC networks are inferred across ten random seeds and combined into a consensus requiring an edge in at least eight seeds, producing 23,451 edges from 155 TFs to 5,442 targets.The consensus activity set contains 155 regulons; the edge weights are medians across seeds containing each edge.Source: H Details of the IFN-β experiment · p. 25

I.2 Top-k enrichment of stimulation-responsive SCENIC regulons

  • The responsive-regulon analysis selects, per cell type, 50 regulons with the largest absolute stimulated-versus-control ΔAUC among regulons significant at BH-FDR < 0.05.Figure I.2 reports top-k enrichment z-scores for responsive TFs across methods and cell types, averaged over five seeds, with standard-deviation error bars and shading for significant bars.Source: H Details of the IFN-β experiment · p. 25, Figure I.2 · p. 27

I.1 Attribution rankings by hop distance from the roots

  • CIG ranks root genes higher than DeepSHAP and integrated gradients on the OmniPath prior, with a smaller advantage one hop downstream; its SCENIC-root rankings are the most stable of the three methods.Across training seeds, percentile-rank spread remains below what independent seed-to-seed rankings would produce.Source: §I.1 · p. 25

I.3 Recovery of pseudobulk DEGs

Against pseudobulk DEGs, DeepSHAP and integrated gradients align more closely with differential-expression statistics at shallow cutoffs, while CIG prioritizes upstream regulators. In Dyngen, CIG’s transition-specific attributions significantly prioritize causal TFs across all six transitions.

I.3 Recovery of pseudobulk DEGs

  • CIG is the least correlated with the DE statistic in all seven cell types, and none of the methods tracks that statistic closely enough to be considered equivalent to a differential-expression test.The whole-panel correlations are stable across seeds.Source: §I.3 · p. 27, Table I.3 · p. 28
  • CIG’s precision is non-monotone in k, peaking at k = 100 in every cell type except NK cells, where it peaks at k = 50.DeepSHAP and integrated gradients are at or near their maxima at k = 10 and decline thereafter; CIG’s first few dozen genes are less likely to be DEGs than genes immediately below them.Source: §I.3 · p. 28, Table I.7 · p. 32, Table I.8 · p. 33
  • The results are consistent with DeepSHAP and integrated gradients ranking downstream response genes first, whereas CIG prioritizes upstream regulators that are rarely strong DEGs themselves.Source: §I.3 · p. 28
  • Across the full DEG sets, CIG has the lowest F1 in six of seven cell types and the lowest recovery AUC in six of seven; its NDCG is lower in every cell type.The exceptions are a dendritic-cell F1 tie with integrated gradients and higher CIG recovery AUC than DeepSHAP in CD4 T cells.Source: §I.3 · p. 27, Table I.4 · p. 29
  • Among the first ten genes, DeepSHAP and integrated gradients reach 94–100% precision and NDCG of 64–91, versus 34–52% and 22–32 for CIG.The methods separate in the same direction in every cell type at this cutoff.Source: §I.3 · p. 27, Table I.5 · p. 30

J Isolating the value of CIG’s formulation beyond graph access

  • CIG significantly prioritizes known causal TFs in all six Dyngen transitions, compared with significant prioritization in two transitions for DeepSHAP and none for integrated gradients.Because the graph is shared across cells, the transition-specific differences indicate that CIG explanations depend on the cells being explained, not only on the static graph.Source: J Isolating the value of CIG’s… · p. 38, Figure J.1 · p. 38

K Robustness to causal graph misspecification

CIG is tested under edge removal, reversal, and spurious-edge addition at three corruption levels on synthetic and Dyngen graphs. Its performance degrades substantially with removed or reversed edges, but is comparatively tolerant of spurious edges and corruption confined to housekeeping genes.

K Robustness to causal graph misspecification

  • The synthetic benchmark measures Spearman correlation with ground-truth CATE rankings for root-cause features across 100–1,000 features and five seeds.The robustness tests corrupt graphs by removing, reversing, or adding 25%, 50%, or 75% of edges; for spurious additions and reversals, a learned surrogate is refit for each perturbed graph.Source: K Robustness to causal graph… · p. 38
  • The Dyngen analysis reports normalized TF-recovery AUC separately for backbone and housekeeping genes, with an uncorrupted baseline of 97.76 ± 3.18.Figure K.1–K.3 show synthetic root-cause Spearman correlation across feature counts for edge removal, reversal, and spurious addition, respectively.Source: K Robustness to causal graph… · p. 38, Table K.1 · p. 39, Figure K.1 · p. 39
  • Removing or reversing edges substantially degrades CIG, while adding spurious edges barely decreases performance across the synthetic and Dyngen evaluations.In Dyngen, housekeeping-gene corruption does not reduce causal-TF recovery, whereas removing or reversing backbone edges degrades recovery increasingly with severity.Source: K Robustness to causal graph… · pp. 39–40

L Residual-dependence diagnostics under non-additive mechanisms

The diagnostics compare CIG indirect-effect attributions from an oracle mechanism and a learned surrogate for additive and non-additive mechanisms. The discrepancy is systematically associated with the product of the two parents for interactions, although relative-error magnitude does not cleanly distinguish interaction from linear controls.

L Residual-dependence diagnostics under non-additive mechanisms

  • The surrogate models each feature mechanism as additive across parents, so non-additive true mechanisms can misallocate attribution between direct and indirect effects.The diagnostic compares oracle and learned-surrogate indirect effects on eligible nodes with an interaction or matched linear-control mechanism.Source: L Residual-dependence diagnostics… · p. 40
  • The discrepancy is tested against the product of the two parents: correlation is r = 0.83 for that product, versus r = −0.16 for the co-parent alone.The relative error is RMSE(discrepancy) divided by RMSE(oracle indirect effect); both statistics use the same discrepancy observations.Source: L Residual-dependence diagnostics… · p. 40
  • For non-additive mechanisms, CIG’s indirect-effect misallocation is systematic and tied to interaction structure, while relative-error magnitude does not cleanly separate interactions from matched linear controls.The discrepancy’s mean absolute correlation and significance rate are 3–7× and 4–5× higher for interactions; in-graph relative errors are 0.286 versus 0.288.Source: L Residual-dependence diagnostics… · p. 41
  • Despite the diagnosed bias, the learned surrogate achieves close to oracle-level Spearman correlation with true CATE on the main benchmark.Source: L Residual-dependence diagnostics… · p. 41

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