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MCIR: A Feature Dependence-Aware Explainability Method with Reliability Guarantees

Poushali Sengupta, Sabita Maharjan, Frank Eliassen, Shashi Raj Pandey, Yan Zhang

arXiv:2610.01641v1cs.LGcs.AIstat.CO

TL;DR

Correlated predictors make global attribution depend on how methods treat shared information. MCIR-M scores each feature’s unique predictive information beyond a selected dependence neighbourhood; it shows its clearest gains in controlled redundancy experiments, while real-data comparisons are mixed.

  • Problem

    With correlated or redundant predictors, feature attributions depend on the target functional and how feature dependence is treated.

    Source: §1 · p. 2

  • Method

    MCIR-M conditions each feature on a selected dependence neighbourhood and normalizes its remaining predictive information by the joint information in the feature block.

    Source: §1 · pp. 2–3, §4.1.3 · p. 10

  • Results

    MCIR’s clearest gains occur in controlled redundancy experiments, while comparisons on real data are mixed across criteria.

    Source: §1 · p. 3, §9 · p. 45

  • Takeaways & Limitations

    MCIR-M is a dependence-aware diagnostic for explaining statistical dependence between input features and a fixed model-output representation, not a causal importance measure.

    Source: §7 · p. 44, §9 · p. 45

    MCIR-M does not remove CMI estimation challenges, which depend on estimator compatibility, sample size, regularity conditions, and conditioning dimension.

    Source: §4.4.1 · p. 17

Abstract

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Modern machine-learning models often contain strongly dependent or redundant features, making feature attribution difficult because shared predictive information can be distributed across correlated predictors. Existing methods such as SHAP, LIME, HSIC, MI/CMI, and SAGE may therefore produce unstable rankings under multicollinearity or near-duplicate predictors. We propose the Mutual Correlation Impact Ratio Method (MCIR-M), a dependence-aware global feature-importance approach that quantifies the unique predictive information contributed by each feature beyond a selected dependence neighbourhood. MCIR-M introduces the Mutual Correlation Impact Ratio (MCIR), which conditions each feature on strongly dependent neighbours and computes a normalized ratio of conditional to block-level information. The population score lies in [0,1] and equals zero under exact conditional redundancy. We also introduce a lightweight estimation procedure that computes MCIR using a fraction of the available data and evaluates agreement with full-data explanations. Across controlled synthetic redundancy experiments and the UCI HAR benchmark, MCIR shows dependence-aware ranking behaviour, with its clearest advantage under injected near-duplicate predictors. Comparisons with independent and conditional SHAP, SAGE, HSIC, MI-based scores, and CIR-family baselines are mixed across real-data criteria. Reduced explanation samples lower computational burden in the evaluated configurations, while agreement with full-data explanations is assessed separately through ranking, head-set, and faithfulness diagnostics. Overall, MCIR-M provides a practical dependence-aware diagnostic for global explanation under strong feature dependence.

1 Introduction

Correlated and redundant predictors make global attribution sensitive to the target and dependence treatment. MCIR-M measures each feature’s unique predictive information beyond a selected dependence neighbourhood, with bounded population scores and a lightweight evaluation strategy.

1 Introduction

  • MCIR-M measures each feature’s predictive information beyond a selected neighbourhood and normalizes it by the information in the enlarged feature block.Its population score lies in [0, 1] and is zero under exact conditional redundancy.Source: §1 · pp. 2–3
  • The lightweight strategy computes MCIR from a reduced explanation sample and assesses its fidelity separately through representation, ranking, top-K, and behavioural diagnostics.The empirical evaluation compares the computational and attributional effects of reducing the explanation sample.Source: §1 · pp. 2–3
  • The proposed population functional accommodates continuous, discrete, and mixed-type variables, while the method’s contribution is not a new general-purpose CMI estimator.Estimation remains sensitive to variable types, sample size, regularity conditions, and conditioning dimension.Source: §1 · pp. 2–3
  • The paper establishes population properties and conditional finite-sample ranking bounds under explicit assumptions, while presenting Auto-Φ and estimator switching as practical heuristics.The bounds depend on component-error, denominator-separation, and ranking-margin assumptions.Source: §1 · pp. 2–3
  • Across controlled redundancy, UCI HAR, HouseEnergy-Sim, CIFAR-10 representations, and Norwegian load forecasting, the clearest gains occur in controlled redundancy and real-data comparisons are mixed.Reduced-sample experiments report runtime and agreement with full-data explanations as separate diagnostics.Source: §1 · p. 3

2 Background and Related Work

Feature-attribution methods can differ because they target distinct functionals and treat dependence differently. MCIR-M instead targets unique conditional information relative to a selected feature-specific neighbourhood, so different valid methods need not agree on rankings.

2 Background and Related Work

  • Shapley-based methods allocate contributions through a coalition value function, and their dependence treatment varies with the value function, background distribution, and conditional or interventional implementation.Different valid Shapley formulations can assign different contributions to correlated predictors; this reflects different estimands rather than a universal failure of Shapley values.Source: §2 · p. 3
  • Marginal MI and HSIC quantify unconditional dependence, whereas CMI measures dependence remaining after conditioning; nonparametric CMI estimation requires estimator-specific conditions.Perturbation-based ROAR and deletion results also depend on perturbation, replacement, retraining, and evaluation protocols.Source: §2 · p. 3
  • LOCO, regional LOCO-style procedures, Shapley effects, and MCIR target distinct quantities, including predictive-risk change, regional loss variation, explained-variance allocation, and unique conditional information.These attribution targets need not induce the same feature ordering.Source: §2 · p. 4
  • MCIR-M conditions on a feature-specific dependence neighbourhood, unlike grouping or class-conditioning approaches, and does not claim that different attribution targets should yield identical rankings.The paper compares its redundancy-aware functional with alternatives as distinct attribution estimands.Source: §2 · p. 4

3 Preliminaries

The preliminaries define MCIR in relation to a fixed model-output representation and distinguish marginal from conditional information. They also clarify the scope of lightweight comparisons and the assumptions and diagnostics needed to interpret them.

3 Preliminaries

  • The continuous experiments use a fixed predictor M to produce model-output representation Z, and define neighbourhood-conditioned information relative to the joint feature–output law.I(Z; Xi) is marginal information, while I(Z; Xi | XΦ(i)) is the information remaining after accounting for the selected neighbourhood.Source: §3 · pp. 4–5
  • MCIR explains statistical dependence between input features and model outputs, not causal dependence on the ground-truth target.The paper frames it as a model-explanation method rather than a causal variable-importance procedure.Source: §3 · p. 5
  • The lightweight environment may reduce the observation count and, separately, the output-representation dimension; representation similarity alone does not guarantee attribution similarity.The paper therefore reports representation, ranking, and head-ranking agreement as separate diagnostics.Source: §3 · pp. 6–8
  • When the output dimension is reduced, the projection may discard predictive information, so MCIR preservation and attribution agreement are not automatic.The paper states that information loss and attribution discrepancy must be analyzed explicitly.Source: §3 · p. 7
  • PCIR evaluates global feature–output association without conditioning on the dependence neighbourhood, whereas MCIR measures conditional information beyond that neighbourhood.Marginal MI and HSIC do not isolate conditional effects, while CMI conditions on specified variables.Source: §3 · pp. 5, 8

4 MCIR: Formal Definition and Analysis

This section defines MCIR as a population measure of unique predictive information conditioned on a selected dependence neighbourhood, and analyzes finite-sample guarantees for its scores and rankings across estimation regimes. It also covers neighbourhood selection, lightweight-fidelity diagnostics, and computational costs, while noting the associational nature of MCIR and its sensitivity to latent confounding and distribution shift.

4.1.1 Environment-Specific Output Representation; 4.1.2 Standard-Borel Well-Posedness

  • The population formulation applies to full and lightweight model-output representations under their corresponding joint laws, with standard Borel spaces and finite block information ensuring well-posedness.These assumptions support regular conditional probability kernels and accommodate continuous, discrete, and mixed-type variables.Source: §4.1.1 · p. 8, §4.1.2 · p. 9

4.1.3 Mixed-Variable Conditional Mutual Information; 4.1.5 Information Components and MCIR Score

  • Mixed-variable conditional mutual information measures predictive information contributed by Xi beyond its selected neighbourhood; this is statistical, not causal, uniqueness.It uses the established population CMI functional across continuous, discrete, or mixed variables; MCIR’s contribution is the normalized attribution score.Source: §4.1.3 · p. 10, §4.1.5 · p. 12

4.1.4 Joint Mutual Information; 4.1.5 Information Components and MCIR Score

  • MCIR normalizes a feature’s conditional mutual information by the joint information in its enlarged feature block, expressing unique contribution relative to neighbourhood-level information.The score is defined as zero when both information quantities vanish; unlike raw conditional mutual information, it is bounded.Source: §4.1.4 · p. 10, §4.1.5 · p. 11

4.1.5 Information Components and MCIR Score; 4 MCIR: Formal Definition and Analysis

  • MCIR has a narrower aim than a full partial-information decomposition: it scores one feature relative to a selected neighbourhood and does not separate synergistic components.Approximate redundancy and estimator-dependent behaviour are addressed separately from these population properties.Source: §4 · p. 8, §4.1.5 · p. 12

4.2 Population Properties; 4.2.1 Boundedness and Endpoints

  • MCIR lies in [0, 1]: it is zero when Xi adds no information beyond its neighbourhood and one when Xi contributes information while the neighbourhood alone contributes none.The common numerical range does not imply identical calibration across data distributions or tasks.Source: §4.1.5 · p. 11, §4.2.1 · pp. 12–13

4.2.2 Exact Conditional-Redundancy Collapse; 4.2 Population Properties

  • If Xi is measurable from its selected neighbourhood, including when it is an exact duplicate, MCIR assigns it zero population attribution.Which feature retains positive attribution depends on the specified neighbourhoods; dependence alone cannot identify an “original” among exact duplicates.Source: §4.2 · p. 12, §4.2.2 · p. 13

4.2.3 Population Invariance

  • Bimeasurable bijections of the output, target feature, and neighbourhood leave the population information components and MCIR score unchanged.This population invariance is distinct from estimator-specific invariance.Source: §4.2.3 · p. 14

4.2.4 Weak-Dependence Reduction

  • When conditioning changes feature information by a vanishing amount and the denominator stays away from zero, MCIR reduces to a normalized marginal-information score.This result does not establish ordering equivalence with PCIR or another marginal attribution method.Source: §4.2.4 · p. 14

4.3.1 Empirical Clipping

  • Empirical MCIR clips estimated information components and projects the ratio onto [0, 1] as a numerical safeguard, while finite-sample guarantees require estimator-specific component-error assumptions.The projection does not restore the population information identity when separately clipped estimates violate the population chain inequality.Source: §4.3.1 · p. 15, §4.3.2 · p. 15

4.3.3 Ratio Perturbation

  • With consistent MI and CMI components and a positive population denominator, the empirical score is consistent; score perturbations are better conditioned when that denominator is separated from zero.The population results use no KL-continuity assumption, and approximate-redundancy behaviour is assessed through empirical sensitivity analysis.Source: §4.3.3 · pp. 15–16, §4.3.2 · p. 15

4.3.4 Conditional Rank-Stability Theorem

  • Under componentwise error bounds and a population-score margin, the theorem bounds discordant ranking pairs by 16Mδ/c0 with probability at least 1 − mIα.The guarantee applies to features whose MCIR denominators are at least c0 and depends on estimator-specific conditions; no common concentration rate is asserted across estimator families.Source: Assume further that the population… · pp. 16–17, §4.3.4 · p. 16

4.4 Estimator Selection

  • Population definitions cover continuous, discrete, and mixed variables, but empirical MI/CMI estimators must be chosen for the data regime, and the experiments make no general mixed-type estimator claim.Bootstrap variability can select the least variable compatible estimator, but it does not estimate bias or provide an oracle risk guarantee.Source: §4.4.1 · p. 17, §4.4.2 · p. 18

4.5 Dependence-Neighbourhood Selection

  • Auto-Φ searches nested candidate neighbourhoods rather than all subsets and selects using ranking stability, residual dependence, and a penalty for larger conditioning blocks.The objective is an algorithmic selection criterion, not a population guarantee.Source: §4.5 · p. 18, §4.5.1 · p. 18, §4.5.2 · p. 19

4.5.3 Misspecification and Computational Cost

  • Neighbourhood-based unique information is relative to Φ(i): omitting dependent features can leave shared information in Ui, while unnecessary features increase CMI difficulty and computational cost.The nested search cost is linear in the number of evaluated sizes, not all possible feature subsets.Source: §4.5.3 · p. 19

4.6 Lightweight Fidelity Analysis

  • Reducing observations and reducing representation dimension are distinct operations, and similarity of marginal output laws alone does not establish MCIR attribution agreement.Attribution fidelity is evaluated separately using full-ranking, top-K, and deletion/insertion diagnostics.Source: §4.6 · p. 19, §4.6.1 · p. 20, §4.6.2 · p. 20

4.6.3 Operational Acceptance Profile

  • The lightweight fidelity profile separately measures representation, complete-ranking, head-ranking, and behavioural-faithfulness agreement; acceptance requires each discrepancy to meet its prespecified tolerance.Ranking and curve agreement are measured outcomes, not assumptions or consequences guaranteed by output-law similarity alone.Source: §4.6.2 · p. 20, §4.6.3 · p. 21
  • A nonrepresentative lightweight sample can alter the output representation, MCIR ranking, and deletion or insertion curves, so these discrepancies are recorded separately.The passage illustrates this with an electricity-consumption sample dominated by weekdays that may underrepresent weekend behaviour.Source: §4.6.3 · pp. 21–22
  • Auto-Φ selects one shared neighbourhood size from nested candidates using ranking dispersion, residual dependence, and a conditioning-dimension penalty; it is an empirical heuristic, not a population-optimality result.The calibration estimator is fixed before selection; reported Auto-Φ experiments use the Gaussian–copula estimator.Source: §4.6.3 · pp. 21–22

4.7 Computational and Statistical Analysis

  • For fixed neighbourhoods, MCIR scoring costs TMI(s, ri) + k log k, with local dimension ri = dQ + mi + 1 and estimation cost dependent on conditioning-set and output-representation dimensions.The k log k term sorts feature scores.Source: §4.7 · p. 22
  • The end-to-end MCIR cost is O(Cscreen + kBA(Mmax + 1)TMI(s, rmax) + k log k), and depends on screening, estimator candidates, bootstrap replicates, and conditioning dimension.Dense pairwise screening generally costs O(sk2), while approximate dependence-graph construction may reduce screening cost; the method does not claim end-to-end linear complexity in k without specifying these factors.Source: §4.6.3 · p. 22, §4.7 · p. 23

4.8 Latent Confounding and Distribution Shift

  • MCIR scoring characterizes statistical dependence within PX,Q and is not causal variable importance; latent confounding may leave residual association after conditioning on the selected neighbourhood.Neighbourhood screening, estimator selection, and bootstrap diagnostics quantify aspects of finite-sample variability but do not remove latent confounding.Source: §4.8 · p. 23
  • A shift in the joint feature–output law can change dependence neighbourhoods, information quantities, and rankings, so MCIR does not guarantee robustness to arbitrary distribution shift.Interpret results alongside domain knowledge, estimator uncertainty, and lightweight-fidelity diagnostics.Source: §4.8 · p. 23

5 Experiments

The experiments compare MCIR with attribution baselines across four settings, examining dependence-aware attribution, ranking stability and lightweight fidelity, faithfulness, and computational cost.

5 Experiments

  • The evaluation spans HouseEnergy-Sim, UCI HAR, Norwegian load forecasting, and CIFAR-10 representations, comparing PCIR, BlockCIR, marginal MI, HSIC, SHAP variants, and SAGE.Source: §5 · p. 23

5.1 Experimental Setup and Evaluation Protocol

  • UCI HAR uses 561 sensor-derived features across six activities and a fixed random-forest classifier; HouseEnergy-Sim varies dependence and redundancy under a fixed random-forest regressor.Source: §5.1 · p. 24

5.2 Implementation Details and Reproducibility

  • Lightweight experiments distinguish observation subsampling from output-dimension changes, using the different-dimensional comparison only when an explicit representation map changes output dimension.Source: §5.1 · p. 24, §5.2 · p. 26

5.1 Experimental Setup and Evaluation Protocol

  • Neighbourhood sizes 0, 1, 3, and 5 are sensitivity-tested; estimator choices vary by variable type and block dimension, with bootstrap and resampling diagnostics for stability.Source: §5.1 · p. 25
  • Deletion and insertion tests assess performance changes under attribution rankings, while Group-Jaccard complements feature-level overlap for correlated predictors grouped at an absolute Pearson-correlation threshold of 0.85.Source: §5.1 · pp. 25–26

5.2 Implementation Details and Reproducibility

  • Runtime is measured separately for pipeline stages and remains implementation- and hardware-specific; comparisons use fixed data partitions and recorded random seeds.Source: §5.1 · p. 26, §5.2 · p. 26

6 Results

The results assess MCIR’s dependence-aware attribution, lightweight estimation and agreement with full-data explanations, predictive faithfulness, estimator sensitivity, and computational evidence across several experiments. They also describe conditions and limits on its guarantees, without establishing population-ranking optimality or linear scaling in neighbourhood size.

6.1.1 Dependence-Aware Attribution under Strong Dependence (RQ1)

  • In the Gaussian redundancy construction, the proxy’s conditional score C2 is zero for every α, while its normalized marginal information increases as α grows.As α approaches 1, the anchor’s unique information relative to the proxy vanishes; when both features are conditioned on each other, dependence alone cannot identify an original feature.Source: §6.1.1 · p. 29, Figure 6 · p. 30
  • With r ≥ 5 duplicates, MCIR’s ordinal instability was 0.4667–0.5333 versus PCIR’s 1.0; at r = 40, MCIR and MI allocation drift were 0.6095 and 0.6077.PCIR’s allocation drift was 0.2910, showing that allocation drift and ranking instability capture distinct behaviours; MCIR reduced ordinal instability but did not eliminate allocation drift.Source: Table 8 · p. 31, §6.1.1 · pp. 31–32

6.1.2 Lightweight Agreement and Runtime Profile (RQ2)

  • Across lightweight configurations, MCIR Kendall-τb was 0.72–0.89 for head rankings and 0.55–0.76 for complete rankings.At n = 2000, Gaussian–copula MCIR took 0.4487 s on HouseEnergy-Sim and 451.412 s on UCI HAR; kNN took 4.3769 s and 139.738 s, respectively, so runtime ordering depended on dataset and estimator.Source: §6 · pp. 26–27, §6.1.2 · p. 32, Figure 9 · p. 33
  • MCIR had the highest Group-Jaccard at 20% and 40% lightweight fractions, HSIC at 60%, while MCIR’s Group-Jaccard ranged from 0.579 to 0.641 across thresholds.MCIR did not lead Exact Jaccard in every configuration, and SHAP had the highest absolute Group-Jaccard across the evaluated threshold sensitivity analysis.Source: Table 11 · p. 34, §6.1.2 · p. 34, Figure 10 · p. 35

6.1.3 Predictive Faithfulness and Estimator Sensitivity (RQ3)

  • PCIR and MCIR had similar rankings: on UCI HAR, ρ = 0.83 and τb = 0.66; on HouseEnergy-Sim, ρ = 0.99 and τb = 0.98.Their lower agreement with KernelSHAP, SAGE, and HSIC reflects different ranking behaviour, not which method is correct or superior.Source: Table 12 · p. 35, §6.1.3 · p. 36
  • Removing or perturbing highly ranked MCIR features decreased fixed-model performance on UCI HAR and HouseEnergy-Sim, measuring predictive faithfulness rather than causal importance.The evaluations used test accuracy on UCI HAR and test R2 on HouseEnergy-Sim under the specified perturbation and deletion protocols.Source: §6 · pp. 27–28, §6.1.1 · p. 28, §6.1.3 · p. 36
  • On HouseEnergy-Sim, Gaussian–copula and kNN MCIR estimates had ρ = 0.73, τb = 0.58, and Jaccard@10 = 0.40.Agreement was moderate across the complete ranking, with a larger estimator effect on the top-ranked features.Source: Table 13 · p. 36, §6.1.3 · p. 36
  • Region-weighted LOCO leads core-feature Recall@4, Shapley effects lead cumulative R2-drop AUC, and MCIR and LOCO retain about 98% of positive attribution mass on functional drivers.MCIR’s mean Kendall agreement is 0.436 with LOCO, 0.449 with region-weighted LOCO, and −0.071 with Shapley effects. The largest held-out region has mean weight 0.999, while the Shapley-effects plug-in produces some negative finite-sample marginal contributions.Source: Table 14 · p. 37, §6.1.3 · p. 38
  • MCIR scores Space_heater, Water_heater, Washing_machine, and HVAC_load highly while assigning lower scores to correlated proxies; these scores are not causal effects.The attribution profile depends on the selected neighbourhood, and estimator choice measurably affects MCIR, particularly top-K selection. The reported estimator agreement applies to low-dimensional HouseEnergy-Sim neighbourhoods and does not establish high-dimensional kNN robustness.Source: §6.1.3 · pp. 36–37, Figure 12 · p. 37
  • At n = 2000, Gaussian–copula Kendall agreement falls from 0.640 at |Φ| = 1 to 0.311 at |Φ| = 12, while kNN agreement is 0.688 at |Φ| = 12.At |Φ| = 12, kNN takes 5.982 seconds versus 0.444 seconds for Gaussian–copula, and 21.6% of kNN pre-clipping component estimates are negative. Ranking agreement, score error, component behaviour, and runtime need not move together as conditioning dimension grows.Source: Figure 13 · p. 38, §6.1.3 · p. 38
  • The switching rule’s ranking agreement depends on sample size and neighbourhood size, and it does not uniformly select the fixed estimator with the highest agreement.Agreement is zero for |Φ(i)| = 5 and remains zero for |Φ(i)| = 3 until n = 2000; at |Φ(i)| = 5, switching and kNN nevertheless obtain Group-Jaccard@10 of 1.000 and ground-truth Recall@10 of 0.769. At n = 3000, switching takes 82.000–296.926 seconds, compared with under 0.06 seconds for fixed Gaussian–copula and 9.048–32.963 seconds for fixed kNN.Source: Table 15 · p. 39, §6.1.3 · pp. 39–40

Runtime in Lightweight Environments

  • Using a lightweight explanation sample reduces observations processed without changing the trained model or feature space, while runtime still depends on sample size, feature count, estimator, and local conditioning dimension.The reported wall-clock measurements illustrate implementation-specific settings rather than proving a universal scaling law; attribution fidelity is evaluated separately.Source: Runtime in Lightweight Environments · p. 40, Table 17 · p. 41

6.2.1 CIFAR-10/ResNet-50 Representation Experiment

  • Deleting the 128 MCIR-ranked CIFAR-10 representation coordinates reduces probe accuracy from 0.96 to 0.84 and yields the experiment’s lowest deletion AUC, 0.887.The result supports predictive influence for the fixed probe under the specified deletion baseline, not causal importance or general stability in arbitrary 2048-dimensional problems.Source: Table 18 · p. 41, §6.2.1 · p. 41

6.2.2 Norwegian Residential-Demand Case Study

  • Across the five Norwegian demand areas, full–lightweight MCIR agreement is moderate: mean Spearman ranges from 0.618 to 0.709 and exact Jaccard@8 from 0.560 to 0.799.Group-Jaccard@8 exceeds exact Jaccard@8 in four areas, suggesting some identity-level disagreement reflects selection of different members of correlated groups; this pattern is not universal, as NO1’s group-level value is slightly lower.Source: Table 19 · p. 42, Figure 14 · p. 42, §6.2.2 · p. 42
  • In Norwegian demand, MCIR does not achieve the largest cumulative R2-drop AUC, and its rankings differ from TreeSHAP, marginal MI, and HSIC.MCIR–TreeSHAP Spearman correlation ranges from 0.247 to 0.552, while marginal MI and HSIC each have exact Jaccard@8 = 0.231 against MCIR in every area. Marginal MI is fastest in the reported configuration and MCIR is second; scores describe association with fitted outputs, not physical or causal determinants.Source: §6.2.2 · pp. 42–43, Table 20 · p. 43
  • MCIR is invariant under bimeasurable bijections, while its conditional guarantee requires estimator-error, denominator-separation, and population-margin conditions; subsampling is compared with a full-data empirical reference, not a population ranking.As the conditioning dimension |Φ| increases, empirical score error and ranking agreement can diverge.Source: §6.2.2 · p. 44
  • Estimator-switching performance depends on n and |Φ(i)|, and the paper claims no oracle inequality; fixed-estimator sensitivity and repeated-subsampling results are reported separately.Auto-Φ selection is specified by Equation (55) and Algorithm 2.Source: §6.2.2 · p. 44
  • Figure 7 supports small neighbourhoods but does not prove population optimality; selection balances ranking dispersion, residual dependence, and conditioning dimension.The lightweight-fidelity diagnostic compares full and lightweight outputs, but output-law similarity alone does not guarantee ranking agreement; cross-method agreement is not treated as fidelity evidence.Source: §6.2.2 · p. 44
  • Computational measurements are reported across several tables, but end-to-end cost also depends on screening, estimator choice, bootstrapping, n, and k; linear scaling in k is not empirically asserted.Full–lightweight discrepancies are reported in Figure 9 and Tables 11 and 19.Source: §6.2.2 · p. 44

7 Ethics, Limitations, and Future Work

MCIR-M has finite-sample and neighbourhood-selection limitations, and the paper identifies practical boundaries for interpreting its runtime evidence and future development.

7 Ethics, Limitations, and Future Work

  • MCIR’s finite-sample behaviour depends on the estimator, sample size, output dimension, and conditioning dimension; it does not establish causal importance.The paper also warns against using attribution scores in high-stakes decisions without domain oversight.Source: §7 · p. 44
  • Pairwise screening may miss interaction-only dependence, while unnecessarily large neighbourhoods can increase finite-sample CMI bias, variance, and computational cost.The paper says the evaluated estimators do not provide a general solution to high-dimensional CMI estimation.Source: §7 · p. 44
  • Table 22 reports attribution-runtime ranges across NO1–NO5 for n = 200; times exclude model fitting and are hardware- and implementation-specific.Source: Table 22 · p. 45
  • Future work will investigate interaction-aware neighbourhoods, high-dimensional CMI estimators, distributionally robust estimation, principled sample-size selection, and temporal and structured-data extensions.Source: §7 · p. 45

8 Broader Impact

MCIR is a model-explanation diagnostic for dependence, not a causal measure, and its interpretation depends on the model, neighbourhood, estimator, and evaluation distribution.

8 Broader Impact

  • MCIR measures a feature’s unique predictive information relative to a selected local neighbourhood, with scores relative to the fitted model and observed feature set.Source: §8 · p. 45
  • A bounded score in [0, 1] does not guarantee identical calibration across datasets, models, or estimators, and MCIR is not evidence of causal effect.Source: §8 · p. 45
  • Responsible use calls for transparent reporting, uncertainty analysis, domain knowledge, and expert review, especially in high-stakes applications.Source: §8 · p. 45

9 Conclusion

MCIR-M provides bounded, dependence-aware attribution for a fixed model, with its clearest advantage under injected near-duplicates and mixed results across real-data criteria.

9 Conclusion

  • MCIR most clearly preserved designated predictive-variable ordering under injected near-duplicates; real-data criteria varied, with no method uniformly superior.Source: §9 · p. 45
  • The lightweight procedure reduced attribution cost in evaluated settings, but its adequacy was assessed empirically using separate representation, ranking, top-K, and faithfulness diagnostics.Source: §9 · p. 45
  • MCIR interpretation remains relative to the selected neighbourhood and joint feature–output law; finite-sample reliability depends on neighbourhood adequacy, estimator suitability, sample size, and ranking separation.Source: §9 · p. 45
  • Future work will examine higher-order and temporal neighbourhoods, improved high-dimensional and mixed-type CMI estimation, distribution-shift robustness, and broader importance-function comparisons.Source: §9 · p. 46

A Supplementary Notation and Scope

  • The notation distinguishes explanation sample sizes N and n < N from output-representation dimensions d and d′; reducing sample size changes empirical measure atoms, not ambient dimension.Source: A Supplementary Notation and Scope · p. 48
  • Output-law similarity alone does not imply MCIR agreement because MCIR depends on the joint feature–output law; representation, ranking, top-K, and deletion/insertion discrepancies are reported separately.Source: A Supplementary Notation and Scope · p. 48

B PCIR Implementation Used as a Baseline

This section establishes MCIR’s population properties and definitions, and specifies its estimators, regime-specific controls, and invariance procedures. It also describes bootstrap-based estimator switching as a calibration heuristic with mixed evidence and higher reported runtime than fixed estimators.

B PCIR Implementation Used as a Baseline

  • PCIR is included as an unconditioned dispersion-based baseline; its boundedness does not establish conditional, information-theoretic, or causal importance, and weak dependence does not imply ranking equivalence.PCIR can depend on relative parameterizations and preprocessing; in the weak-dependence reduction, the feature-dependent denominator term remains. Experiments use fixed preprocessing.Source: B PCIR Implementation Used as a Baseline · p. 48, §D.4 · p. 52

C Mixed-Variable Information Components

  • Mixed-variable CMI is the standard population CMI functional for mixed-type features, not a new measure or a guarantee that one estimator fits every regime.The workflow screens candidates, selects a conditioning neighbourhood, switches among compatible estimators by estimated variability, then reports bounded scores with uncertainty and stability diagnostics.Source: C Mixed-Variable Information Components · pp. 49–50

D.1 Boundedness and the Zero Endpoint

  • MCIR lies in [0,1] and equals zero exactly when the feature and output are conditionally independent given the selected neighbourhood.This characterization holds under Assumption 1.Source: §D.1 · p. 50

D.2 Exact Conditional-Redundancy Collapse

  • If a feature is measurable from its selected neighbourhood, its conditional information and MCIR are zero; this result does not establish convergence for noisy near-duplicates.No additional assumptions establish that noisy near-duplicates converge to the exact-duplicate case; estimator and neighbourhood sensitivity is evaluated empirically.Source: §D.2 · p. 51

D.3 The Upper Endpoint and Population Invariance

  • When unique information is positive, MCIR equals one if neighbourhood information is zero, and is invariant under bimeasurable bijections of the output, feature, and neighbourhood.Source: §D.3 · p. 51

D.5 Estimator Consistency, Ratio Perturbation, and Rank Stability

  • Under a positive denominator, convergence of estimated information components gives convergence of the projected empirical MCIR score; ranking guarantees require estimator-specific regularity conditions.The ranking bound concerns close population-score pairs, and no common convergence rate is established across MI/CMI estimators without verifying their regularity conditions.Source: §D.5 · pp. 52–53

E.1 Gaussian–Copula Estimation

  • Gaussian–copula estimation uses rank-Gaussianized local blocks and estimates unrestricted MI or CMI only under the stated copula specification and covariance conditions.For a general non-Gaussian copula it estimates a Gaussian-copula approximation; rank Gaussianization is invariant to strictly monotone componentwise transformations under the stated tie rule, not a universal unbiasedness guarantee.Source: §E.1 · p. 54

E.2 kNN Estimation

  • For continuous local blocks, KSG-type kNN estimators estimate MI and conditional extensions, but performance can deteriorate as neighbourhood size or output dimension grows.MCIR applies kNN CMI to (Q, Xi, XΦ(i)) and joint MI to (Q, XΦ(i)∪{i}), using the same sample, metric, kNN value, and preprocessing.Source: §E.2 · pp. 54–55
  • The kNN estimator is evaluated only in continuous, low- or moderate-dimensional regimes; no dimension-independent s^-1/2 error rate is assumed.Estimator variability is reported as a function of the conditioning-neighbourhood size |Φ(i)|.Source: §E.2 · p. 55

E.3 Discrete Plug-In Estimation

  • For discrete local blocks, empirical probability masses provide plug-in MI and CMI estimates, which are consistent for fixed finite alphabets but may become sparse and biased as complexity grows.Zero empirical joint-mass terms contribute zero; continuous variables require explicitly stated discretization rather than silent treatment as discrete.Source: §E.3 · p. 55

E.4 Mixed-Type Local Blocks

  • Mixed-variable CMI is a population functional, not a uniformly valid estimator; unsupported mixed-type local blocks are not assigned an MCIR score.A mixed-block estimator must specify an appropriate metric, conditional-density model, copula, or stratified calculation and its assumptions.Source: §E.4 · p. 55

E.5 Neighbourhood and Preprocessing Controls

  • Primary neighbourhoods use Pearson screening at |ρ| ≥ 0.60 with at most Mmax = 6 candidates, while Auto-Φ balances ranking dispersion, residual dependence, and conditioning dimension.Candidate neighbourhoods and preprocessing are determined without held-out outcomes; zero estimated denominators yield an empirical score of zero, while very small denominators are flagged.Source: §E.5 · p. 56

F Estimator-Specific Monotone Invariance

  • With componentwise rank Gaussianization and fixed tie handling, strictly monotone transformations leave the empirical Gaussian–copula MCIR unchanged; this invariance is estimator-specific.The result does not imply invariance for every finite-sample MI or CMI estimator.Source: F Estimator-Specific Monotone Invariance · p. 56

G.1 Bootstrap-Based Estimator Switching

  • Switching does not uniformly outperform fixed estimators and is costliest: at n = 3000, runtime is 82.000–296.926 seconds, versus <0.06 seconds for fixed Gaussian–copula and 9.048–32.963 seconds for fixed kNN.The rule selects the compatible estimator with the smallest bootstrap standard error, which measures sampling variability rather than bias; fixed-estimator results are reported alongside it.Source: §G.1 · pp. 56–57

G.2 Implementation Details for Auto-Φ

Auto-Φ selects a shared neighbourhood size through bootstrap ranking dispersion, residual conditional dependence, and a complexity penalty. The surrounding analysis stresses that MCIR and lightweight fidelity remain conditional on neighbourhood choice, the joint feature–output law, estimator assumptions, and the evaluation protocol.

G.2 Implementation Details for Auto-Φ

  • Auto-Φ selects the smallest candidate size minimizing a criterion combining bootstrap ranking dispersion, normalized residual conditional dependence, and a size penalty.The reported default penalty weights are (λB, λD) = (1, 0.10); the criterion is not claimed to recover a population-optimal neighbourhood.Source: §G.2 · pp. 57–58

H Role and Misspecification of the Neighbourhood

  • MCIR removes redundancy only to the extent that the screened neighbourhood captures dependence relevant to the model output.Under-conditioning can leave shared information in the conditional term, while over-conditioning can increase finite-sample bias, variance, and computational cost.Source: H Role and Misspecification of the… · p. 58

I Latent Confounding and Distribution Shift

  • MCIR measures unique predictive information relative to observed variables and does not identify causal effects or guarantee robustness under distribution shift.A latent common cause can leave conditional association even without a direct causal effect; shift can also alter candidate pools, neighbourhoods, and estimator behaviour.Source: I Latent Confounding and Distribution… · p. 58

J Supplementary Details for Lightweight Fidelity

  • The lightweight procedure can reduce observation count and model-output representation dimension, which are distinct changes to sample size and ambient space.The fidelity profile and its acceptance rule are defined earlier; these supplementary passages provide interpretation and implementation details.Source: J Supplementary Details for… · p. 59

J.1 Separation of Representation and Attribution Discrepancies

  • Equal marginal output laws do not ensure similar MCIR scores or rankings, because attribution depends on joint feature–output laws.The example gives identical marginal output laws but different feature-attribution rankings; without assumptions on joint laws, representation discrepancy alone cannot bound ranking discrepancy.Source: §J.1 · p. 59

J.2 Ranking and Head-Set Diagnostics

  • Complete-ranking discrepancy and top-K Jaccard discrepancy assess different forms of agreement, so preserved leading features can coexist with reordered lower ranks.Group-level Jaccard is reported separately when correlated predictors are treated as interchangeable.Source: §J.2 · pp. 59–60

J.3 Deletion and Insertion-Curve Diagnostic

  • Deletion- and insertion-curve discrepancies require the same model, test observations, replacement rule, metric, and schedule across environments.Otherwise, curve differences can reflect protocol changes rather than attribution changes; for unbounded metrics, the discrepancy retains the metric's scale.Source: §J.3 · p. 60

J.4 Operational Acceptance Criterion

  • The componentwise acceptance rule requires every prespecified tolerance to pass and is an application-specific diagnostic, not a theorem of automatic rank preservation.A small representation discrepancy cannot offset poor ranking agreement, and the tolerances do not establish a universal notion of lightweight fidelity.Source: §J.4 · p. 60

J.5 Population Shift and Estimator Error

The analysis separates estimator error from population change in the joint feature–output law, and treats computational and statistical guarantees as estimator- and regime-dependent. In particular, near-linear scoring applies only after fixing bounded neighbourhoods and excluding several pipeline costs.

J.5 Population Shift and Estimator Error

  • The score-vector difference separates estimator errors in each environment from the population MCIR change caused by changing the joint feature–output law.The theorem controls empirical ranking perturbations only under estimator-specific component-error, denominator, and population-margin conditions; marginal output discrepancy alone does not bound the population-change term.Source: §J.5 · p. 60

J.6 Fidelity-Diagnostic Cost

  • Fidelity-diagnostic cost combines Wasserstein calculation, ranking comparisons, and prescribed deletion/insertion evaluation, and is separate from MCIR scoring and Auto-Φ calibration.The lightweight profile does not infer fidelity from subsampling or marginal output-law similarity; its four components evaluate properties separately.Source: §J.6 · pp. 60–61

K Computational and Statistical Details

  • The computational and statistical analysis distinguishes observation count from representation dimension and supplies estimator-specific conditions rather than a universal rate.Here s denotes observations, dQ the model-output representation dimension, and mi the selected neighbourhood size.Source: K Computational and Statistical Details · p. 61, §K.3 · p. 62

K.1 Operational Complexity

  • Near-linear scoring in feature count applies only after fixing bounded local neighbourhoods and excluding dense screening, bootstrap calibration, and estimator switching.Fixed-neighbourhood scoring includes local information fits and sorting; candidate screening must be added separately, and dense pairwise screening prevents a general linear end-to-end claim.Source: §K.1 · p. 61

K.2 Estimator-Dependent Costs

  • Estimator costs depend on data regime and dimension: Gaussian–copula costs are approximately linear in sample count only for fixed dimension, while plug-in joint cells can grow exponentially.The commonly stated kNN O(s log s) cost also assumes fixed low dimension and effective spatial indexing.Source: §K.2 · pp. 61–62, Table 25 · p. 62

K.3 Estimator-Specific Statistical Conditions

  • Finite-sample conclusions require the support, regularity, and conditioning-dimension conditions of the selected estimator; no common rate or concentration result is assumed across estimator families.Continuity of the ratio transfers component convergence to MCIR but does not establish consistency of the underlying MI or CMI estimators.Source: §K.3 · p. 62

K.4 Conditional Delta-Method Result

The appendix gives a conditional delta-method result and practical boundaries for parallelization and lightweight sample selection. It also reports finite-sample redundancy experiments as sensitivity evidence under stated protocols, not as proof of population-level collapse.

K.4 Conditional Delta-Method Result

  • MCIR inherits asymptotic normality through the multivariate delta method only when the selected component estimators satisfy the joint central limit theorem and the denominator is positive.The result is conditional, not an assertion covering every estimator.Source: §K.4 · pp. 62–63

K.5 Memory and Parallelization

  • Feature-wise score calculations can be distributed across workers once neighbourhoods and estimator choices are fixed, but the idealized work bound does not guarantee near-linear speedup.Communication, scheduling, memory contention, unequal neighbourhood sizes, and estimator setup can affect realized speedup.Source: §K.5 · p. 63

K.6 Choice of Lightweight Sample Size

  • No universal lightweight sample fraction guarantees accurate MCIR estimation; the needed sample size depends on estimator, data, dependence, denominator, ranking margins, and target uncertainty.The authors recommend evaluating candidate sizes with fidelity diagnostics, estimator-specific bootstrap intervals, and repeated-subsampling rankings against prespecified tolerances.Source: §K.6 · p. 63

L Redundancy Robustness Experiments

  • The injected near-duplicate experiments provide finite-sample sensitivity evidence under stated neighbourhood and tie-handling protocols, not direct proof of exact population redundancy collapse.The designated anchor is known from experimental construction; MCIR does not infer which member of an exact-duplicate family is the original from dependence alone.Source: L Redundancy Robustness Experiments · p. 63

L.1 Experimental Protocol

Under the evaluated near-duplicate protocol, MCIR retained more attribution on designated anchors and better preserved original-feature ordering than TreeSHAP and PFI. The experiments also report protocol-specific runtime and ranking diagnostics.

L.1 Experimental Protocol

  • The protocol injects 1–8 noisy copies with anchor–copy correlations of approximately 0.98–0.995, gives each method the same augmented feature space, and holds each fitted model fixed.These are near-duplicates, not exact deterministic copies.Source: §L.1 · p. 64

L.2 HouseEnergy-Sim

  • At r = 8, MCIR retained 0.852 of duplicate-family mass on the HouseEnergy-Sim anchor, versus 0.150 for TreeSHAP and 0.387 for PFI.From r = 1 to r = 8, MCIR OrigMass changed from 0.635 to 0.509, compared with 0.414 to 0.052 for TreeSHAP and 0.475 to 0.187 for PFI.Source: §L.2 · p. 64, Table 26 · p. 64

L.2.1 Original-Feature Ranking Stability

  • MCIR preserves the same original-feature ordering in the reported HouseEnergy-Sim runs, while TreeSHAP and PFI show additional pairwise reversals as redundancy increases.The reported instability is the unnormalized ∆rank = 1 − τb, which can exceed one.Source: §L.2.1 · pp. 64–65, Table 27 · p. 65

L.3 UCI HAR

  • On UCI HAR, MCIR OrigMass changes from 0.639 at r = 1 to 0.503 at r = 8, compared with 0.439 to 0.055 for TreeSHAP and 0.466 to 0.189 for PFI.The result depends on the designated anchor, neighbourhood construction, estimator, and deterministic tie rule.Source: §L.3 · p. 65, Table 28 · p. 65

L.4 Cross-Dataset Degradation Analysis

  • From r = 1 to r = 8 on HouseEnergy-Sim, OrigMass changes by −0.126 for MCIR, −0.362 for TreeSHAP, and −0.287 for PFI; these are descriptive repeated-seed effects, not hypothesis-tested comparisons.The reported reproducibility record does not specify a complete inferential procedure for these comparisons.Source: §L.4 · p. 66

L.5 Runtime Comparison

  • At the reported sample and feature counts, MCIR is faster than TreeSHAP and repeated PFI; these hardware- and implementation-specific measurements do not establish a universal runtime ordering.MCIR avoids repeated model calls after Q has been obtained, unlike the comparison implementations.Source: §L.5 · p. 66, Table 29 · p. 66

L.6 Summary of the Redundancy Experiments

  • Across the evaluated datasets, MCIR retained more attribution on designated anchors and better preserved original-feature ordering than TreeSHAP and PFI under the reported near-duplicate protocol.The conclusion is limited to the reported constructions and does not establish that MCIR can identify an intrinsically original feature among statistically identical copies.Source: §L.2 · p. 64, Table 26 · p. 64, §L.3 · p. 65

M Latent-Confounding Sensitivity Experiment

The section examines observed dependence induced by a synthetic latent common cause, compares agreement across distinct importance targets, and reports finite-sample sensitivity to CMI estimation conditions.

M Latent-Confounding Sensitivity Experiment

  • Conditioning on observed correlated feature X1 reduces the additional statistical information assigned to X2 in the synthetic construction, but MCIR neither recovers the latent variable nor identifies causal effects.This is an observed-variable dependence result, not evidence that MCIR generally adjusts for unobserved confounding.Source: M Latent-Confounding Sensitivity… · p. 67, Figure 19 · p. 68

N Predictive-Risk and Variance-Allocation Comparisons

  • LOCO and regional LOCO show relatively high agreement because they share a predictive-loss target, while Shapley-effects plug-in agreement is lower given its variance-allocation target and finite-sample conditional-mean approximation.Agreement is descriptive across distinct importance functionals, not an attribution-correctness metric.Source: N Predictive-Risk and… · p. 68, Table 30 · p. 69

O High-Dimensional CMI Ablation

  • The high-dimensional CMI ablation evaluates finite-sample estimation sensitivity conditional on screened neighbourhoods, without re-estimating the screening graph.Its reference ranking is estimator-specific and empirical; negative-component frequencies compare unprojected kNN estimates, since the copula implementation is non-negative by construction.Source: O High-Dimensional CMI Ablation · pp. 69–70, Table 31 · p. 69

P.1 Experimental Protocol

The subsampling analyses separate exact feature overlap, correlation-group overlap, deletion faithfulness, and rank agreement. Threshold sensitivity also shows that group-overlap comparisons depend on how the evaluation groups are formed.

P.1 Experimental Protocol

  • The stability protocol uses a HouseEnergy-style dataset with 4,000 observations and 41 features, and reports identity- and group-level overlap alongside separate faithfulness and ranking diagnostics.All methods use the same trained model, partitions, test set, deletion schedule, and lightweight subsampling protocol.Source: §P.1 · p. 70, §P.2 · p. 70

P.2 Exact and Group-Level Stability

  • MCIR has lower exact feature overlap than PCIR, SHAP, and HSIC at all three fractions, but the highest Group-Jaccard at 20% and 40%; HSIC leads at 60%.The grouping rule does not establish functional or causal equivalence, so exact and group-level agreement remain complementary diagnostics.Source: §P.2 · p. 70, Table 32 · p. 70

P.3 Deletion-Based Faithfulness

  • Under the common deletion protocol, MCIR’s more gradual early performance decline is weaker deletion-faithfulness evidence than PCIR’s and HSIC’s sharper declines.Later-curve convergence may reflect compensation by remaining correlated variables, but the curves do not identify its source.Source: §P.3 · p. 70, Figure 21 · p. 71

P.4 Full–Lightweight Rank Overlays

  • At the 40% subset fraction, greater MCIR dispersion in full–lightweight rank overlays indicates lower exact feature-level stability, while PCIR and HSIC show tighter alignment.The overlay does not establish that different feature identities are informationally interchangeable.Source: §P.4 · p. 70, Figure 22 · p. 71

P.5 Stability Visualizations

  • MCIR leads group-level agreement at 20% and 40%, but not exact Jaccard or the 60% group comparison, showing metric-dependent rather than uniform stability dominance.Figures 23 and 24 visualize group-level and exact Jaccard, respectively.Source: §P.5 · p. 71, Figure 23 · p. 72, Figure 24 · p. 72

Q Additional Sensitivity and Ablation Studies

Q.1 Correlation-Group Threshold Sensitivity

  • MCIR Group-Jaccard@30 ranges from 0.5525 to 0.6405 across evaluated thresholds, while SHAP leads at every threshold; changing γ also changes the evaluation groups.The comparisons characterize threshold sensitivity, not a universally optimal grouping threshold.Source: §Q.1 · p. 72, Table 33 · p. 72

Q.2 Adaptive Conditional Mutual Information Estimation

The estimator analysis finds a recovery–runtime trade-off between kNN and Gaussian-copula CMI estimation, while bootstrap-based switching is not uniformly better than fixed estimators. The runtime section measures the calibration implementation separately from final MCIR scoring.

Q.2 Adaptive Conditional Mutual Information Estimation

  • kNN has higher GT Recall@10 at every evaluated neighbourhood size, reaching 0.7692 for |Φ(i)| ∈ {2, 3, 5}, while the copula implementation stays below 0.08 seconds in the reported measurements.The results do not identify a uniformly preferred estimator because recovery and runtime are distinct objectives.Source: §Q.2 · p. 73, Table 34 · p. 73
  • For |Φ(i)| = 2, kNN has Jaccard@10 = 1 and GT Recall@10 = 0.7692 at every sample size, while complete-ranking agreement rises from τb = 0 at n = 200 to 0.8965 at n = 3000.At n = 200, perfect top-10 overlap coexists with zero complete-ranking correlation, indicating differences in ordering or ties beyond the leading set.Source: §Q.2 · p. 73
  • The adaptive procedure matches kNN through n = 1000, then at |Φ(i)| = 2 its Jaccard@10 and GT Recall@10 fall to 0.9286 and 0.7436 at n = 2000, and 0.6437 and 0.6487 at n = 3000.At the two largest sample sizes, the switching rule increasingly selects the copula estimator for this conditioning size.Source: §Q.2 · p. 73, Table 36 · p. 74
  • At n = 3000 and |Φ(i)| = 2, adaptive calibration takes 81.9997 seconds, versus 9.0478 seconds for fixed kNN and 0.0534 seconds for fixed copula.This includes fitting and bootstrapping both candidates before selection; it is a calibration cost, not the runtime of one final MCIR evaluation.Source: §Q.2 · p. 73
  • The selected estimator depends jointly on sample size and conditioning dimension, and lower bootstrap variability need not imply higher GT Recall or ranking agreement.The rule selects kNN in every repetition for |Φ(i)| ∈ {3, 5}, while its preference at |Φ(i)| = 1 shifts from kNN to copula as n increases.Source: §Q.2 · p. 73
  • The switching rule avoids fixing one estimator globally but is not uniformly better than fixed candidates; bootstrap variability estimates variability, not bias.The authors treat switching as a heuristic and report it alongside fixed-estimator sensitivity analysis.Source: §Q.2 · pp. 72, 74

Q.3 Runtime Overhead of Adaptive Auto-Φ Selection

  • For the evaluated threshold-calibration implementation, total runtime rises from 0.184 seconds at 20 features to 5.762 seconds at 561 features, with calibration the dominant component.Screening remains below 0.01 seconds and final scoring below 0.2 seconds; this is not an empirical proof of a universal asymptotic rate or a direct evaluation of every term in the general Auto-Φ objective.Source: §Q.3 · pp. 74–75, Table 38 · p. 75
  • The selected neighbourhood structure can be reused only while feature definitions, preprocessing, screening rule, and relevant data regime remain unchanged; shifts may require recalibration.The measured runtime describes the reported UCI HAR implementation and setup.Source: §Q.3 · pp. 74–75

R Failure Case: Higher-Order Dependence Outside the Pairwise Neighbourhood

This failure-case diagnostic tests whether pairwise-correlation screening captures engineered higher-order dependence. Screened MCIR recovers only part of the relevant set, and the paper frames this as a neighbourhood-selection failure rather than a general advantage.

R Failure Case: Higher-Order Dependence Outside the Pairwise Neighbourhood

  • The experiment uses X3 = X1X2 + E3, X4 = sin(X1), and X5 = cos(X2), plus 25 independent noise features, over 30 seeds at n = 4000.Source: R Failure Case: Higher-Order… · p. 75
  • Screened MCIR recovers only part of the relevant set because pairwise screening does not expose all interaction structure.Table 39 reports means over 30 seeds with standard errors; its interaction-informed Φ row is a ranking reference, not a deployable recovery baseline.Source: R Failure Case: Higher-Order… · pp. 76–77, Table 39 · p. 77
  • Low ranking agreement for correlation-screened MCIR shows that neighbourhood misspecification changes its ranking, while negative agreement for marginal baselines indicates a different ordering.Source: R Failure Case: Higher-Order… · p. 77
  • The diagnostic compares recovery of the relevant feature set with ranking agreement against interaction-informed MCIR.Source: Figure 28 · p. 77, Table 39 · p. 77
  • Pairwise-correlation screening can miss higher-order structure, so dependence-aware MCIR remains conditional on the selected neighbourhood and alternative screens do not guarantee recovery of every relevant pattern.Source: R Failure Case: Higher-Order… · pp. 76–77
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