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

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

cs.LGOct 1, 2026

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

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

Organizations: Department of Informatics, University of Oslo Oslo, Norway · Department of Electronic Systems, Aalborg University Aalborg, Denmark

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1 : Conceptual view of the PCIR baseline. PCIR evaluates global feature–output association without explicitly conditioning on the dependence neighbourhood $X_{\Phi(i)}$ . The connecting lines represent statistical association, not causal direction.

Panel of Figure 2 — Figure 2 : MCIR intuition. The target feature $X_{i}$ and its selected dependence neighbourhood $X_{\Phi(i)}=\{X_{\phi_{1}},X_{\phi_{2}}\}$ may carry shared predictive information about the model-output representation $Q$ . MCIR conditions on $X_{\Phi(i)}$ to isolate the unique predictive information contributed by $X_{i}$ relative to that neighbourhood and normalizes it to $[0,1]$ . The arrows represent predictive dependence, not causal effects.

Data regime	Estimator	Scope and conditions
Continuous	Gaussian–copula after rank Gaussianization, $\widehat{F}\mapsto\Phi_{\mathcal{N}}^{-1}(\widehat{F})$	Primary estimator for continuous variables under a Gaussian-copula approximation. Rank-based invariance holds for strictly monotone transformations, with a fixed rule for ties.
Continuous, nonparametric	$k$ NN/KSG MI and conditional extensions	Uses local neighbour distances without a Gaussian-copula assumption. Applied in low- or moderate-dimensional conditioning blocks; accuracy may deteriorate as $\|X_{\Phi(i)}\|$ increases.
Discrete	Plug-in frequency estimator	Computes MI and CMI from empirical probability masses. Requires adequate cell counts; sparse contingency tables may introduce finite-sample bias.
Mixed continuous–discrete	Not evaluated empirically	The population definition remains valid, but this study does not implement or report a general mixed-type MI/CMI estimator. Application to such blocks requires a fully specified mixed-type estimator and its corresponding regularity assumptions.
High-dimensional conditioning	Dimension diagnostic	No uniform estimator guarantee is assumed. Results are evaluated as a function of $\|\Phi(i)\|$ , and large conditioning blocks are reported as an estimator-limitation regime.

Table 3 : Estimator selection by variable type and conditioning regime.

\widehat{R}(r)\leftarrow\widehat{V}(r)+\lambda_{B}\widetilde{B}(r)+\lambda_{D}\frac{r}{M_{\max}}.

Algorithm 2 Auto-

\Phi

Shared-Neighbourhood-Size Selection

Figure 3 : PCIR–MCIR rank overlays for UCI HAR (left) and HouseEnergy-Sim (right). Each point represents one feature, with its PCIR rank on the horizontal axis and MCIR rank on the vertical axis. Proximity to the diagonal therefore indicates agreement between the two methods. Full and lightweight results are distinguished by marker type; the displacement between corresponding markers indicates sensitivity to lightweight subsampling. Full-versus-lightweight agreement is quantified separately using Kendall–

\tau_{b}

and Jaccard@

K

Category	Specification
Software environment	Python 3.10; NumPy 1.26; SciPy 1.11; scikit-learn 1.3; pandas 2.1; PyTorch; Matplotlib; and Seaborn. Exact package versions are pinned in the repository environment file.
Execution environment	Linux-based Google Colab Pro sessions. Deep-model training uses the GPU assigned to the recorded session; attribution computations run on CPU. The exact CPU, GPU, memory, and runtime configuration is recorded with each experiment log.
Randomness control	Seed 7 for train/test partitions and seed 42 for subsampling, bootstrap, estimator, NumPy, scikit-learn, and PyTorch randomness. Deterministic operations are used where supported.
Gaussian–copula estimator	Rank transformation followed by empirical-CDF Gaussianization $\widehat{F}\mapsto\Phi_{\mathcal{N}}^{-1}(\widehat{F})$ , deterministic tie handling, and ridge-regularized covariance estimation. Multivariate MI and CMI are computed through covariance determinant or Schur complement formulas. Ridge values are reported per experiment in Table 6 .
$k$ NN estimator	KSG-type MI and conditional extensions for continuous, low- or moderate-dimensional blocks; $k_{\mathrm{NN}}\in\{5,10,20\}$ with Euclidean distance. Results are reported as a function of $\|\Phi(i)\|$ because estimator behaviour may degrade with conditioning dimension.
Discrete estimator	Plug-in MI and CMI estimation from empirical probability masses. Finite-sample corrections and minimum cell-count requirements are reported with the corresponding experiment.

Table 5 : Experimental configuration and reproducibility summary.

Figure 6 : Population behaviour under a Gaussian redundancy construction. Let

X_{1},X_{3},E_{2}

be independent standard Gaussian variables,

X_{2}=\alpha X_{1}+\sqrt{1-\alpha^{2}}E_{2}

, and

Q=X_{1}+0.5X_{3}+\varepsilon

, where

\varepsilon\sim\mathcal{N}(0,0.25)

is independent. Left: because

Q\perp\!\!\!\perp X_{2}\mid X_{1}

, the redundant proxy has

C_{2}=0

for every

\alpha

when

\Phi(2)=\{1\}

. Middle: the unique predictive information of

X_{1}

relative to

X_{2}

decreases as the proxy becomes more accurate and vanishes at the exact-duplicate limit

\alpha=1

. Right: the normalized marginal information

I(Q;X_{2})/I(Q;X_{1})

increases with

\alpha

, illustrating how an unconditioned association measure can credit a redundant proxy. At

\alpha=1

, neither feature can be identified as the “original” from statistical dependence alone when each is conditioned on the other.

Figure 7 : Sensitivity to neighbourhood size. The case

|\Phi(i)|=0

is shown only as a degenerate endpoint: whenever

I(Q;X_{i})>0

, Definition 3 gives

C_{i}=1

. Nonempty neighbourhoods enable MCIR to distinguish unique from shared information. Moderate sizes provide the best empirical balance between residual redundancy and CMI estimation variability in this experiment.

$n$	$r^{\star}$	Kendall $\tau_{b}$	Jaccard@10	Driver Recall@10
500	1	0.391	0.514	0.538
1000	1	0.445	0.583	0.638
2000	1	0.514	0.656	0.650

Table 7 : Auto-

\Phi

penalty sensitivity on HouseEnergy-Sim. Values are means over 20 repetitions. Every evaluated penalty pair selected

r^{\star}=1

in every repetition.

$r$	MCIR		MI		PCIR
	$\Delta_{\rm rank}$	$\Delta_{\rm alloc}$	$\Delta_{\rm rank}$	$\Delta_{\rm alloc}$	$\Delta_{\rm rank}$	$\Delta_{\rm alloc}$
0	0.0000	0.0105	0.0000	0.0103	0.0000	0.0015
5	0.4667	0.4473	0.0000	0.4495	1.0000	0.2178
10	0.4667	0.5261	1.3333	0.5288	1.0000	0.2499
20	0.4667	0.5785	1.2667	0.5789	1.0000	0.2747
40	0.5333	0.6095	1.3333	0.6077	1.0000	0.2910

Table 8: HouseEnergy-Sim redundancy stress test. Values are means over the stated repetitions; lower is better.

Figure 8 : Redundancy stress test on HouseEnergy-Sim. Left: mean ordinal instability $\Delta_{\mathrm{rank}}=1-\tau_{b}$ on the original features. Right: mean allocation drift $\Delta_{\mathrm{alloc}}$ on the same features. Lower values indicate greater stability. MCIR exhibits lower ordinal instability than MI and PCIR at the larger redundancy levels, while MCIR and MI show similar allocation drift.

Setting	Training rows	Accuracy	Macro-F1	$\bm{D_{\mathrm{KL}}}$	Jaccard@30	F1 ratio
Full training	7,352	0.930	0.928	0.000	1.000	1.000
Reduced training	3,676	0.917	0.914	7.174	0.622	0.985

Table 9 : UCI HAR training-subsample sensitivity. The reduced configuration uses half of the training rows and therefore evaluates a different fitted model. This table is reported separately from the fixed-model lightweight attribution analysis.

Figure 9 : Full-versus-lightweight attribution comparison. The first row reports PCIR results for UCI HAR and HouseEnergy-Sim, and the second row reports MCIR results. Each panel presents full and lightweight score distributions, paired feature-level scores, and mean confidence intervals. The $\Delta$ histograms show $\widehat{C}_{i,n}-\widehat{C}_{i,N}$ . Inset badges report Spearman’s $\rho$ , Kendall’s $\tau_{b}$ , and Jaccard@ $20$ . These quantities measure different aspects of score and ranking agreement.

Figure 10 : Group-Jaccard@ $30$ under feature dependence. MCIR obtains the highest group-level agreement at the $20\%$ and $40\%$ lightweight fractions, while HSIC obtains the highest value at $60\%$ .

Metric	Mean	2.5%	97.5%
Kendall $\tau_{b}$	0.58	0.51	0.64
Spearman $\rho$	0.73	0.67	0.78
Jaccard@10	0.40	0.30	0.50

Table 13 : Bootstrap agreement between Gaussian–copula and

k

NN MCIR estimates on HouseEnergy-Sim.

Figure 11 : HouseEnergy-Sim estimator sensitivity: (a) Gaussian–copula versus

k

NN rank overlay and (b) Jaccard@

K

as the selection size varies.

Figure 12 : HouseEnergy-Sim attribution profiles. MCIR reports unique predictive information relative to each selected neighbourhood, normalized to

[0,1]

. PCIR reports an unconditioned dispersion-based score whose value depends on the stated preprocessing.

Method	Core Recall@4	Original mass	Proxy mass	$R^{2}$ -drop AUC	Runtime (s)
MCIR	$0.825\pm 0.206$	$0.983\pm 0.016$	$0.013\pm 0.013$	$0.660\pm 0.122$	$0.071\pm 0.038$
LOCO	$0.775\pm 0.079$	$\mathbf{0.984}\pm 0.016$	$\mathbf{0.009}\pm 0.011$	$0.481\pm 0.118$	$1.801\pm 0.892$
Region-weighted LOCO	$\mathbf{0.850}\pm 0.129$	$0.977\pm 0.020$	$0.012\pm 0.011$	$0.535\pm 0.147$	$2.029\pm 0.917$
Shapley-effects plug-in	$0.375\pm 0.132$	$0.331\pm 0.061$	$0.661\pm 0.057$	$\mathbf{0.680}\pm 0.121$	$14.337\pm 3.693$

Table 14 : HouseEnergy-Sim comparison of distinct importance functionals. Values are mean

\pm

standard deviation over ten independent runs. Original and proxy mass are computed after applying the positive-part normalization to each estimated attribution vector. Higher Recall@

4

and cumulative

R^{2}

-drop AUC indicate stronger recovery and faster predictive degradation, respectively. Runtime includes method-specific estimation after

Q

is available.

Method	HouseEnergy-Sim		UCI HAR
Method	Time (s)	Setting	Time (s)	Setting
MCIR (copula)	0.4487	rank Gaussianization	451.412	rank Gaussianization
MCIR ( $k$ NN)	4.3769	$k_{\rm NN}=5$	139.738	$k_{\rm NN}=5$
HSIC (RBF)	7.0033	median bandwidth	207.823	median bandwidth

Table 17 : Observed attribution runtime on HouseEnergy-Sim (

n=2000

k=20

) and UCI HAR (

n=2000

k=561

). Times include the implemented scoring pipeline.

Method	Deletion AUC	Accuracy after top-128 deletion
MCIR	0.887	$0.96\rightarrow 0.84$
PCIR	0.912	$0.96\rightarrow 0.87$
HSIC	0.938	$0.96\rightarrow 0.89$
MI	0.951	$0.96\rightarrow 0.90$

Table 18 : CIFAR-10 deletion results. Lower AUC indicates stronger deletion faithfulness under the stated protocol.

Area	Test $R^{2}$	$\bm{\rho}$	$\bm{\tau_{b}}$	J@8	Group-J@8	$\bm{W_{1}}$
NO1	0.9924	0.666	0.508	0.639	0.623	64.57
NO2	0.9919	0.709	0.558	0.799	0.814	37.04
NO3	0.9924	0.659	0.507	0.671	0.757	22.30
NO4	0.9937	0.640	0.478	0.560	0.617	14.40
NO5	0.9903	0.618	0.471	0.600	0.713	14.13

Table 19 : Predictive performance and full–lightweight MCIR agreement in the Norwegian residential-demand experiment. Values are means over ten explanation samples. Correlation groups are connected components of the training-data correlation graph at

|\rho|\geq 0.85

W_{1}

is reported in MWh and is interpreted only within each area.

Panel of Figure 14 — Figure 14 : Norwegian residential-demand results. (a) Full–lightweight exact and group-level top-eight agreement for MCIR, averaged over ten explanation samples. (b) Cumulative $R^{2}$ -drop AUC under the common training-mean replacement protocol; higher values indicate faster cumulative predictive degradation.

	MCIR versus TreeSHAP				Cumulative $R^{2}$ -Drop AUC
Area	$\bm{\rho}$	$\bm{\tau_{b}}$	J@8	Group-J@8	MCIR	TreeSHAP	MI	HSIC
NO1	0.377	0.254	0.333	0.500	1.695	1.798	1.787	1.788
NO2	0.552	0.397	0.600	0.667	1.276	1.388	1.401	1.407
NO3	0.257	0.169	0.333	0.429	1.881	2.017	2.030	2.025
NO4	0.273	0.228	0.455	0.429	2.747	3.188	3.211	3.127
NO5	0.247	0.169	0.333	0.429	1.418	1.591	1.582	1.581

Table 20 : Cross-method agreement and deletion performance in the Norwegian residential-demand experiment. Agreement is measured between MCIR and TreeSHAP. Cumulative

R^{2}

-drop AUC is defined as

\int_{0}^{1}[R^{2}(0)-R^{2}(t)]\,dt

, where

t

is the fraction of features replaced by their training-set means. Higher values indicate faster cumulative predictive degradation.

Rank	NO1	NO2	NO3	NO4	NO5
1	sin_hour	sin_hour	relative_humidity_2m	load_lag_1h	cos_hour
2	cos_hour	cos_hour	load_lag_1h	wind_speed_10m	load_lag_1h
3	load_lag_1h	load_lag_1h	cos_hour	cos_hour	relative_humidity_2m
4	load_roll_mean_24h	relative_humidity_2m	load_roll_mean_24h	load_roll_mean_24h	load_roll_std_6h
5	load_lag_24h	load_lag_24h	precipitation	load_roll_std_6h	load_roll_mean_24h
6	wind_speed_10m	load_roll_mean_24h	load_roll_std_6h	sin_dow	load_lag_24h

Table 21 : Top-eight MCIR features from the full-data reference ranking. Feature names follow the implementation. These rankings describe predictive association with the fitted XGBoost outputs.

Method	Minimum (s)	Maximum (s)
Marginal MI	0.0559	0.0675
MCIR	0.0810	0.0983
HSIC	0.1108	0.1801
TreeSHAP	0.2280	0.2817

Table 22 : Range of mean attribution runtimes across NO1–NO5 for

n=200

explanation observations. Times exclude model fitting and are hardware- and implementation-specific.

Property	Formal result	Evidence and scope
Boundedness and endpoints	Equations 26 and 27 ; Eq. equation 28	Population guarantee. $0\leq C_{i}\leq 1$ , with the endpoints characterized by the corresponding information quantities. Empirical clipping in Equation 37 enforces this range and is therefore not independent empirical validation.
Exact conditional-redundancy collapse	Equation 29	Theory with sensitivity evidence. Measurability of $X_{i}$ with respect to $\sigma(X_{\Phi(i)})$ implies $C_{i}=0$ . The noisy-duplicate results in Figures 8 and 8 and Tables 26 , 16 and 17 assess finite-sample sensitivity, not exact deterministic collapse.
Weak-dependence reduction	Proposition 2	Theoretical result. MCIR approaches a normalized marginal-information score under vanishing conditioning effects and denominator separation. No controlled zero-dependence experiment or ordering equivalence with PCIR is claimed.
Population reparameterization invariance	Proposition 1	Population guarantee. MCIR is invariant under bimeasurable bijections of $Q$ , $X_{i}$ , and $X_{\Phi(i)}$ . The separate rank–Gaussianized estimator result is given in Proposition 7 ; no transformation experiment is reported.
Consistency and rank stability	Equations 38 and 1	Conditional guarantee; indirect evidence. The result requires estimator-error, denominator-separation, and population-margin conditions. Subsampling results in Tables 11 and 10 use a full-data empirical reference rather than the unavailable population ranking. The conditioning-dimension stress test in Figures 13 and 31 additionally shows that empirical score error and ranking agreement can diverge as $\|\Phi\|$ increases.
Estimator switching	Equation 51	Empirical heuristic. Fixed-estimator sensitivity is reported in Figures 11 and 13 , with repeated-subsampling results in Appendix Table 37 . Performance depends on $n$ and $\|\Phi(i)\|$ ; no oracle inequality is claimed.

Table 23 : Traceability between MCIR’s formal results and supporting evidence. Empirical consistency denotes compatibility, not proof.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 15 — Figure 15 : MCIR workflow. Candidate screening, neighbourhood selection, estimator choice, scoring, and full–lightweight diagnostics are treated as distinct stages.

	OrigMass			FamilyShare
$r$	MCIR	TreeSHAP	PFI	MCIR	TreeSHAP	PFI
1	$0.635\pm 0.018$	$0.414\pm 0.032$	$0.475\pm 0.028$	0.908	0.784	0.834
2	$0.610\pm 0.019$	$0.312\pm 0.035$	$0.398\pm 0.031$	0.904	0.687	0.776
4	$0.568\pm 0.021$	$0.178\pm 0.038$	$0.272\pm 0.034$	0.887	0.503	0.651
8	$0.509\pm 0.023$	$0.052\pm 0.019$	$0.187\pm 0.029$	0.852	0.150	0.387

Appendix

Table 26 : HouseEnergy-Sim redundancy diagnostics. OrigMass is the normalized attribution assigned to the designated anchor; FamilyShare is its share within the duplicate family.

$r$	TreeSHAP	PFI
0	0.0000	0.0000
1	0.6667	0.2500
2	0.6667	0.6667
4	0.6667	0.6667
8	0.6667	0.6667
16	0.6667	1.3333

Appendix

Table 27 : Original-feature ranking instability on HouseEnergy-Sim:

\Delta_{\mathrm{rank}}=1-\tau_{b}

; lower is better.

Figure 16 : OrigMass under increasing redundancy in HouseEnergy-Sim. MCIR shows a smaller decline than TreeSHAP and PFI under the reported protocol.

$r$	MCIR	TreeSHAP	PFI
1	$0.639\pm 0.016$	$0.439\pm 0.034$	$0.466\pm 0.027$
2	$0.614\pm 0.018$	$0.331\pm 0.036$	$0.389\pm 0.030$
4	$0.572\pm 0.020$	$0.189\pm 0.041$	$0.265\pm 0.033$
8	$0.503\pm 0.022$	$0.055\pm 0.021$	$0.189\pm 0.031$

Appendix

Table 28 : OrigMass under duplicate injection on UCI HAR.

Figure 17 : OrigMass under redundancy on HouseEnergy-Sim and UCI HAR. MCIR retains more mass on the designated anchor than TreeSHAP and PFI under the evaluated constructions.

Method	Time (s)	Relative time
MCIR (lightweight)	0.0341	$1.0\times$
TreeSHAP	0.9114	$26.7\times$
PFI (5 repeats)	0.2496	$7.3\times$

Appendix

Table 29 : Runtime at

r=8

using 400 observations and 20 features. Values are means over ten runs.

Figure 18 : Conceptual summary of allocation under the evaluated near-duplicate protocol. Left: MCIR retains more score on the designated anchor. Middle: TreeSHAP distributes score across correlated copies. Right: the marginal effect measured by PFI can decrease while predictive substitutes remain. This diagram summarizes the experiment and is not a general theorem about the methods.

Figure 19 : Synthetic common-cause sensitivity experiment. Left: marginal MI with $Q$ as $a$ increases. Right: MCIR under the stated neighbourhoods. Conditioning on $X_{1}$ reduces the additional statistical information assigned to $X_{2}$ . This observed-variable dependence result does not recover the latent variable or establish causal importance.

Figure 20 : HouseEnergy-Sim comparison across distinct attribution functionals. Left: recovery of the four prespecified functional drivers. Middle: normalized positive attribution retained on those drivers. Right: cumulative

R^{2}

-drop AUC under the common training-mean replacement protocol. Values are means over ten independent runs; the numerical uncertainty summaries are reported in Table 14 .

Method A	Method B	$\bm{\tau_{b}}$	$\bm{\rho}$	Jaccard@4
MCIR	LOCO	0.436	0.568	0.560
MCIR	Region-weighted LOCO	0.449	0.594	0.667
MCIR	Shapley-effects plug-in	$-0.071$	$-0.122$	0.303
LOCO	Region-weighted LOCO	0.764	0.865	0.720
LOCO	Shapley-effects plug-in	$-0.262$	$-0.377$	0.205
Region-weighted LOCO	Shapley-effects plug-in	$-0.267$	$-0.377$	0.195

Appendix

Table 30 : Mean cross-method ranking agreement over ten independent HouseEnergy-Sim runs. The statistics compare distinct importance targets and do not measure attribution correctness.

$n$	$\|\Phi\|$	Estimator	Score MAE	$\tau_{b}$	J@4	Negative components	Runtime (s)
500	1	copula	0.163	0.516	0.295	0.000	0.08
500	1	knn	0.204	0.136	0.268	0.365	0.26
500	2	copula	0.101	0.441	0.533	0.000	0.10
500	2	knn	0.149	0.134	0.281	0.380	0.33
500	3	copula	0.067	0.394	0.433	0.000	0.12
500	3	knn	0.115	0.118	0.295	0.333	0.37

Appendix

Table 31 : High-dimensional CMI ablation on HouseEnergy-Sim. Values are means over eight repetitions. Agreement and score MAE are computed against an estimator-specific reference based on 5000 observations; the results therefore characterize finite-sample sensitivity rather than population accuracy. Negative-component frequency is the proportion of unprojected

k

NN MI/CMI components below zero; Gaussian–copula components are non-negative by implementation. Decreasing score MAE at large

|\Phi|

should be interpreted jointly with rank agreement because it may reflect score compression rather than improved ranking recovery.

Method	20%	40%	60%
Exact Jaccard@ $30$
MCIR	0.585	0.624	0.627
PCIR	0.673	0.710	0.767
SHAP	0.644	0.698	0.747
HSIC	0.674	0.711	0.787
Group-Jaccard@ $30$

Appendix

Table 32 : Exact and group-level Jaccard@

30

under subsampling.

Figure 21 : Deletion curves for MCIR, PCIR, SHAP, and HSIC under the same model, test set, masking baseline, and deletion schedule. A sharper early performance decrease and lower deletion AUC indicate stronger faithfulness under this protocol.

Figure 22 : Full–lightweight rank overlays at the $40\%$ subset fraction. Each panel compares feature rankings under full and subsampled data.

$\gamma$	MCIR	PCIR	SHAP	HSIC
0.60	0.6405	0.5036	0.9200	0.4284
0.70	0.6261	0.4648	0.9667	0.4334
0.75	0.5792	0.4204	0.9741	0.4582
0.80	0.5946	0.4720	0.9788	0.5201
0.85	0.5821	0.4321	0.9662	0.4994
0.90	0.5732	0.4372	0.9490	0.5006

Appendix

Table 33 : Group-Jaccard@

30

sensitivity to the correlation-group threshold on UCI HAR. Larger values indicate greater overlap under the grouping induced by that row’s threshold.

	GT Recall@ $10$		Runtime
$\|\Phi(i)\|$	Copula	$k$ NN	Copula	$k$ NN
1	0.4615	0.6154	0.0794	56.0264
2	0.6154	0.7692	0.0474	13.6651
3	0.5385	0.7692	0.0408	15.9895
5	0.6923	0.7692	0.0441	22.2782

Appendix

Table 34 : Full-data HouseEnergy-Sim results. GT Recall@

10

evaluates recovery of the prespecified relevant set; runtime is measured in seconds.

$n$	$\|\Phi\|=1$	$\|\Phi\|=2$	$\|\Phi\|=3$	$\|\Phi\|=5$
200	4 (13.3)/26 (86.7)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)
500	26 (86.7)/4 (13.3)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)
1000	30 (100.0)/0 (0.0)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)
2000	30 (100.0)/0 (0.0)	4 (13.3)/26 (86.7)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)
3000	30 (100.0)/0 (0.0)	20 (66.7)/10 (33.3)	0 (0.0)/30 (100.0)	0 (0.0)/30 (100.0)

Appendix

Table 36 : Estimator selections over 30 repetitions. Each entry reports copula count (%)/

k

NN count (%).

Features	Screening	Calibration	Final MCIR	Total
20	0.001	0.179	0.004	0.184
50	0.002	0.418	0.012	0.432
100	0.002	0.869	0.029	0.900
200	0.003	1.939	0.069	2.011
400	0.005	4.006	0.116	4.127
561	0.007	5.566	0.188	5.762

Appendix

Table 38 : Runtime decomposition on UCI HAR with

n=2000

. Entries are mean seconds over five repetitions.

Figure 25 : Measured total runtime of the adaptive MCIR implementation versus feature count. Error bars denote standard errors over five repetitions.

Figure 26 : Measured runtime of screening, threshold calibration, and final MCIR evaluation.

Figure 27 : Stacked decomposition of the measured adaptive-pipeline runtime. Error bars denote standard errors of total runtime.

Panel of Figure 28 — Figure 28 : Higher-order dependence diagnostic: (a) recovery of the relevant feature set and (b) ranking agreement with interaction-informed MCIR.

Observation	Interpretation
Pairwise screening omits relevant structure	The association between $X_{1}$ , $X_{2}$ , and $Q$ is mediated by their product, which is weakly visible to pairwise linear correlation.
Low all-relevant recall for correlation- $\Phi$ MCIR	The limitation arises from constructing $\Phi(i)$ , not solely from the bounded MCIR normalization.
High recall for MI, HSIC, and RF-PFI	These methods recover marginal or predictive relevance but do not estimate the same unique conditional contribution as MCIR.
Reference- $\Phi$ is not deployable	It is supplied manually only to show how the ranking changes when interaction-informed neighbourhoods are imposed.

Appendix

Table 40 : Interpretation of the higher-order failure-case diagnostic.

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Jan 10, 2026cs.LG

Explainability of Complex AI Models with Correlation Impact Ratio

Complex AI systems make better predictions but often lack transparency, limiting trustworthiness, interpretability, and safe deployment. Common post hoc AI explainers, such as LIME, SHAP, HSIC, and SAGE, are model agnostic but are too restricted in one significant regard: they tend to misrank correlated features and require costly perturbations, which do not scale to high dimensional data. We introduce ExCIR (Explainability through Correlation Impact Ratio), a theoretically grounded, simple, and reliable metric for explaining the contribution of input features to model outputs, which remains stable and consistent under noise and sampling variations. We demonstrate that ExCIR captures dependencies arising from correlated features through a lightweight single pass formulation. Experimental evaluations on diverse datasets, including EEG, synthetic vehicular data, Digits, and Cats-Dogs, validate the effectiveness and stability of ExCIR across domains, achieving more interpretable feature explanations than existing methods while remaining computationally efficient. To this end, we further extend ExCIR with an information theoretic foundation that unifies the correlation ratio with Canonical Correlation Analysis under mutual information bounds, enabling multi output and class conditioned explainability at scale.

Poushali Sengupta, Rabindra Khadka, Sabita Maharjan +5

Institute of Informatics, University of Oslo, Oslo, Norway · Department of Computer Science, Oslo Metropolitan University, Oslo, Norway · Aalborg University, Denmark

Explainability Interpretability Spurious Correlations Artificial Intelligence Models

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Sep 17, 2026cs.LG

Evaluating Explanation Methods by the Predictors They Induce

Explanations of machine learning models are usually judged by criteria that are hard to compare. We propose a simpler test: if an explanation really describes how a model uses its features, it should be possible to rebuild the model's predictions from it. We turn each explanation into a predictor by reading each feature's effect and adding them up, and measure how well that predictor reproduces the model on unseen data. Nothing is fitted, so the score reflects the explanation itself. The test applies to any explanation that can be written as a function of the features; we demonstrate it on partial dependence plots (PDP), accumulated local effects (ALE), SHAP and LIME. We prove that summing partial dependence curves gives the best possible additive summary of a model when its features are independent, and that this fails when they are dependent. Across 13 real datasets and 9 synthetic designs and four model families, which method scores best depends entirely on feature dependence: where features are independent SHAP is slightly worse than PDP, exactly as the theory predicts; on dependent real data SHAP leads. Some widely used quality metrics even prefer a damaged explanation to an intact one.

Jacob Selbæk, Hugo L. Hammer

Department of Computer Science, Oslo Metropolitan University, Oslo, Norway · Department of Holistic Systems, SimulaMet, Oslo, Norway · Department of Plastic and Reconstructive Surgery, Oslo University Hospital, Oslo, Norway

Shapley Additive Explanations Explainable AI Methods Model Evaluation Simple

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Aug 11, 2026cs.LG

RelShap: Relationally Consistent Shapley Explanations

Machine learning pipelines commonly flatten relational data into single-table representations, discarding structural constraints. Widely used Shapley value-based feature attributions then rely on feature independence, evaluating the model on combinations that could never arise in the underlying data, producing misleading explanations. We propose RelShap, a framework that incorporates relational constraints and data provenance into Shapley value computation, restricting both background data and coalition evaluation to relationally valid configurations. The framework is estimator-agnostic and composes with Kernel SHAP, Monte Carlo, and Leverage SHAP without altering their sampling or weighting properties. Functional dependencies further induce equivalence classes over feature coalitions, which RelShap exploits to reduce runtime without changing Shapley values; we provide a combinatorial characterization of the expected speedup. Experiments across multiple datasets, models, and estimators show that RelShap produces explanations that are more faithful to the data-generating process, correctly identifying the dominant feature in controlled settings where existing methods, including Conditional SHAP and ManifoldShap, do not. Our code is available at: https://github.com/duneag2/relshap.

Seungeun Lee, Joao Fonseca, Julia Stoyanovich

New York University, New York, USA · INESC-ID, Lisbon, Portugal

Shapley Additive Explanations Shapley Value Shaping Explainable AI Methods

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