Mastering Model Fit: Applications of the InterModel Vigorish for Interpretable Machine Learning
Authors: Jiani Yan, Pengyun Wang, Klint Kanopka, Ben Domingue, Charles Rahal
Organizations: Leverhulme Centre for Demographic Science, University of Oxford · Nuffield College, University of Oxford · Data Science Institute, University of Chicago · NYU Steinhardt, New York University · Stanford Graduate School of Education, Stanford University
Accuracy alone gives little information about which features or model components improve probabilistic fit, which outcomes benefit, or how fit changes during training. We introduce the InterModel Vigorish (IMV) into the Machine Intelligence space in order to express differences in fit as an expected betting advantage over a reference model. SHAP-IMV attributes gains to features through Shapley-weighted subset comparisons, Multi-IMV evaluates one-vs-rest and pairwise class discrimination, and Ablate-IMV assesses architectural components and feature groups by ablation. We evaluate these applications over ten random seeds using tabular data, images, sensor sequences, and graphs. Feature rankings vary across estimators. High one-vs-rest accuracy can accompany negligible improvement over a null, while ablation can preserve or improve fit under a fixed training budget. On six graph-classification benchmarks, we time-profile IMV over 200 training epochs, reporting seed means and ranges. The open-source imvpy library and accompanying notebooks provide code to replicate our work and to use the IMV more widely.
Figures & tables
Figure 1: IMV calculation and applications. Steps 1–3 start with the probabilities that reference model f0 and candidate model f1 assign to the same observed outcomes. We aggregate these as geometric mean likelihoods. When both are at least 0.5 , Equation ( 2 ) gives equivalent coin weights w0 and w1 . The directional score IMV(f1,f0)=(w1−w0)/w0 measures the relative change in coin weight; positive values indicate higher mean log-likelihood for f1 . ( A ) SHAP-IMV attributes gains in fit to features through Shapley-weighted marginal contributions from refitted subset models, using a constant-only reference. ( B ) Multi-IMV reuses multiclass predictions for one-vs-all and pairwise comparisons, renormalizing probabilities within each class pair. ( C ) Ablate-IMV compares full and refitted ablated models on the same held-out observations to assess architectural components or feature groups. Scores depend on the reference and evaluation data. All data, probabilities, scores, and networks shown are schematic.
Figure 2: SHAP-IMV feature contributions for Titanic ( a. – c. ), Wisconsin Diagnostic Breast Cancer ( d. – f. ), and Adult Income ( g. – i. ). Columns show logistic regression, XGBoost, and LightGBM; each panel ranks eight predictors by mean signed SHAP-IMV. All 256 subsets are evaluated by five-fold stratified cross-validation against the same estimator fitted to a constant input. Shapley contributions use the mean IMV across folds. Bars and labels give means across ten seeds (42–51), with whiskers spanning one sample standard deviation rather than confidence intervals. Standardization uses the full dataset before cross-validation. Y-scales are shared within rows, and the contributions describe predictive fit rather than causal effects.
Methods
Titanic
Breast Cancer
Adult Income
Average
Accuracy
Precision
Accuracy
Precision
Accuracy
Precision
Accuracy
Precision
LR
SHAP-IMV
79.50
74.36
93.92
94.27
83.78
70.94
85.73
79.85
SHAP
79.74
73.95
94.10
94.46
83.78
70.94
85.87
79.78
LIME
79.74
73.95
94.10
94.46
84.16
71.35
86.00
79.92
Anchors
79.50
74.36
93.92
94.27
83.78
70.94
85.73
79.85
XGB
SHAP-IMV
81.16
79.33
93.76
93.11
85.97
79.42
86.96
83.95
Table 1: Predictive performance after top-5 feature selection by SHAP-IMV, SHAP, LIME, and Anchors. LR, XGB, and LGBM denote logistic regression, XGBoost, and LightGBM. Each dataset uses a fixed set of 8 variables drawn from the source data; Breast Cancer refers to the Wisconsin Diagnostic dataset. Rankings are averaged over seeds to select a fixed subset for refitting the same estimator. Accuracy and positive-class precision are percentages averaged over 10 stratified folds for each of 10 seeds; Average gives equal weight to the 3 datasets. Preprocessing and feature selection precede refit cross-validation, so the results describe these fixed subsets rather than a nested evaluation of the selection procedures. Within each estimator and column, bold and underlining mark the best and second-best distinct values at the displayed precision; ties share formatting, and columns in which all methods agree are unmarked.
Dataset
Model
Rank by mean SHAP-IMV
1
2
3
4
5
6
7
8
Titanic
LR
Sex (0.147)
Title (0.065)
Class (0.041)
Age × Class (0.035)
Fare (0.018)
Embarked (0.013)
Alone (0.011)
Age (0.004)
XGB
Title (0.113)
Sex (0.105)
Fare (0.060)
Class (0.045)
Age × Class (0.022)
Embarked (0.013)
Alone (0.009)
Age (-0.000)
LGBM
Title (0.113)
Sex (0.104)
Fare (0.053)
Class (0.046)
Age × Class (0.025)
Embarked (0.012)
Alone (0.009)
Age (0.001)
Breast Cancer
LR
Concave points (0.129)
Radius (0.124)
Concavity (0.096)
Compactness (0.072)
Texture (0.050)
Smoothness (0.031)
Symmetry (0.021)
Fractal dim. (0.016)
XGB
Concave points (0.129)
Radius (0.128)
Concavity (0.115)
Compactness (0.071)
Texture (0.051)
Smoothness (0.027)
Symmetry (0.013)
Fractal dim. (0.003)
Table 2: Complete SHAP-IMV feature rankings for the three binary classification examples. LR, XGB, and LGBM denote logistic regression, XGBoost, and LightGBM. Each dataset uses a fixed set of 8 variables drawn from the source data; Breast Cancer refers to the Wisconsin Diagnostic dataset. Predictors are ordered by decreasing mean exact Shapley contribution to model IMV; parentheses give means to three decimals. Attribution uses 5-fold stratified cross-validation for each of 10 seeds. The 10-fold refit evaluation in Table 1 is separate and evaluates the fixed subset in ranks 1–5 for its SHAP-IMV rows. Figure 2 plots the same means with one sample standard deviation across seeds. Rankings describe predictive contributions within each specification, not causal effects.
Figure 3: Multi-IMV on Dry Bean ( a. – f. ; seven classes, 16 features), Car Evaluation ( g. – l. ; four classes, six features), and Nursery ( m. – r. ; eight features, retaining not_recom, priority, and spec_prior). Columns show logistic regression, XGBoost, and LightGBM, each compared with its constant-input counterpart. Heatmaps restrict held-out observations to class pairs and renormalize both models’ probabilities without binary refitting. Matrices are symmetric with conventional zero diagonals and one colour scale per dataset. Bars show one-vs-rest IMV from unrenormalized class probabilities on all held-out observations; the two views have different populations and reference coin weights. Five-fold stratified scores are averaged within each of ten seeds (42–51). Cells, bars, and labels show seed means; whiskers show one sample standard deviation, not confidence intervals. Dashed lines show the ceiling of the One-vs-rest IMV for the targeted category.
Methods
Unacc
Acc
Good
Very good
Multi-IMV (LR)
0.15 ± 0.00
0.05 ± 0.00
0.00 ± 0.00
0.02 ± 0.00
Brier Loss (LR)
0.17 ± 0.00
0.16 ± 0.00
0.04 ± 0.00
0.03 ± 0.00
Accuracy (LR)
72.16 ± 0.31
74.30 ± 0.34
96.01 ± 0.00
95.48 ± 0.08
Precision (LR)
74.51 ± 0.19
34.99 ± 1.34
0.00 ± 0.00
33.91 ± 1.74
Recall (LR)
91.58 ± 0.33
18.23 ± 0.78
0.00 ± 0.00
21.23 ± 1.21
Multi-IMV (XGB)
0.40 ± 0.00
0.25 ± 0.00
0.03 ± 0.00
0.04 ± 0.00
Table 3: One-vs-rest Multi-IMV and classification performance on Car Evaluation. LR, XGB, and LGBM denote logistic regression, XGBoost, and LightGBM. Values are means ± one sample standard deviation across 10 seeds, each using 5-fold stratified cross-validation; these are not confidence intervals. IMV compares the model using the predictors with the same estimator fitted to a constant input only. We average IMV over folds within each seed and calculate the other metrics from pooled out-of-fold predictions. Accuracy, precision, and recall are percentages for each class versus the rest, based on the multiclass argmax prediction; undefined precision or recall is set to zero. Brier Loss is the Brier score loss on class probabilities. Rounding to two decimals can produce a displayed zero standard deviation even when runs differ. Unacc and Acc denote Unacceptable and Acceptable. Global metrics (LR): F1 Score: 63.84 ± 0.32; AUC-ROC: 77.29 ± 0.12; Log Loss: 0.69 ± 0.00; Hamming Loss: 0.31 ± 0.00; Cohen’s Kappa: 16.31 ± 0.78; Jaccard: 52.42 ± 0.31. Global metrics (XGB): F1 Score: 96.69 ± 0.29; AUC-ROC: 99.71 ± 0.05; Log Loss: 0.14 ± 0.00; Hamming Loss: 0.03 ± 0.00; Cohen’s Kappa: 92.79 ± 0.63; Jaccard: 93.76 ± 0.51. Global metrics (LGBM): F1 Score: 97.89 ± 0.18; AUC-ROC: 99.90 ± 0.04; Log Loss: 0.08 ± 0.00; Hamming Loss: 0.02 ± 0.00; Cohen’s Kappa: 95.38 ± 0.40; Jaccard: 95.94 ± 0.34. F1, one-vs-rest AUC-ROC, and Jaccard use support-weighted class averages. These three metrics and Cohen’s kappa are multiplied by 100; log loss and Hamming loss are unscaled.
Methods
Not recommend
Priority acceptance
Special acceptance
Multi-IMV (LR)
0.52 ± 0.00
0.31 ± 0.00
0.28 ± 0.00
Brier Loss (LR)
0.00 ± 0.00
0.13 ± 0.00
0.13 ± 0.00
Accuracy (LR)
100.00 ± 0.00
78.75 ± 0.05
78.75 ± 0.05
Precision (LR)
100.00 ± 0.00
68.20 ± 0.07
67.16 ± 0.10
Recall (LR)
100.00 ± 0.00
69.48 ± 0.12
65.82 ± 0.10
Multi-IMV (XGB)
0.52 ± 0.00
0.49 ± 0.00
0.45 ± 0.00
Table 4: Multi-IMV results for Nursery using logistic regression, XGBoost, and LightGBM. Means and sample standard deviations are calculated across ten seeds, each evaluated with five stratified folds. Metric definitions and units follow Table 3 . Global metrics (LR): F1 Score: 78.74 ± 0.05; AUC-ROC: 91.77 ± 0.01; Log Loss: 0.39 ± 0.00; Hamming Loss: 0.21 ± 0.00; Cohen’s Kappa: 68.11 ± 0.08; Jaccard: 67.87 ± 0.06. Global metrics (XGB): F1 Score: 99.40 ± 0.05; AUC-ROC: 99.99 ± 0.00; Log Loss: 0.08 ± 0.00; Hamming Loss: 0.01 ± 0.00; Cohen’s Kappa: 99.10 ± 0.08; Jaccard: 98.82 ± 0.10. Global metrics (LGBM): F1 Score: 99.76 ± 0.06; AUC-ROC: 100.00 ± 0.00; Log Loss: 0.06 ± 0.00; Hamming Loss: 0.00 ± 0.00; Cohen’s Kappa: 99.65 ± 0.09; Jaccard: 99.53 ± 0.12.
Methods
Seker
Barbunya
Bombay
Cali
Dermason
Horoz
Sira
Multi-IMV (LR)
0.17 ± 0.00
0.10 ± 0.00
0.04 ± 0.00
0.13 ± 0.00
0.32 ± 0.00
0.16 ± 0.00
0.21 ± 0.00
Brier Loss (LR)
0.01 ± 0.00
0.01 ± 0.00
0.00 ± 0.00
0.01 ± 0.00
0.03 ± 0.00
0.01 ± 0.00
0.04 ± 0.00
Accuracy (LR)
98.45 ± 0.02
98.55 ± 0.03
100.00 ± 0.00
98.53 ± 0.01
95.83 ± 0.03
98.71 ± 0.01
94.76 ± 0.04
Precision (LR)
94.93 ± 0.10
94.22 ± 0.13
100.00 ± 0.00
93.41 ± 0.10
92.28 ± 0.06
95.47 ± 0.10
85.61 ± 0.12
Recall (LR)
94.68 ± 0.08
90.63 ± 0.22
99.96 ± 0.08
94.36 ± 0.06
91.66 ± 0.06
95.41 ± 0.06
87.69 ± 0.12
Multi-IMV (XGB)
0.17 ± 0.00
0.10 ± 0.00
0.04 ± 0.00
0.13 ± 0.00
0.33 ± 0.00
0.16 ± 0.00
0.21 ± 0.00
Table 5: Multi-IMV results for Dry Bean using logistic regression, XGBoost, and LightGBM. Means and sample standard deviations are calculated across ten seeds, each evaluated with five stratified folds. Metric definitions and units follow Table 3 . Global metrics (LR): F1 Score: 92.43 ± 0.04; AUC-ROC: 99.34 ± 0.00; Log Loss: 0.21 ± 0.00; Hamming Loss: 0.08 ± 0.00; Cohen’s Kappa: 90.83 ± 0.05; Jaccard: 86.10 ± 0.06. Global metrics (XGB): F1 Score: 92.75 ± 0.11; AUC-ROC: 99.42 ± 0.01; Log Loss: 0.19 ± 0.00; Hamming Loss: 0.07 ± 0.00; Cohen’s Kappa: 91.23 ± 0.14; Jaccard: 86.65 ± 0.19. Global metrics (LGBM): F1 Score: 92.75 ± 0.11; AUC-ROC: 99.41 ± 0.01; Log Loss: 0.20 ± 0.00; Hamming Loss: 0.07 ± 0.00; Cohen’s Kappa: 91.23 ± 0.13; Jaccard: 86.66 ± 0.18.
Figure 4: Ablate-IMV for MNIST odd/even digits (CNN; a. , d. ), UCI HAR upstairs/downstairs activity (bidirectional GRU; b. , e. ), and MAGIC Gamma gamma/hadron events (MLP; c. , f. ). Heatmaps compare enhanced rows with basic columns; positive values favor rows, and reverse comparisons need not be negatives. Diagonals are zero; heatmaps share a zero-centred colour scale. Bars compare models with constant 0.5 predictions, not the full model shown first. Cells, bars, and labels show ten-seed means (42–51); whiskers show one sample standard deviation, not confidence intervals. MNIST/HAR ablate architecture; MAGIC removes feature groups with hidden architecture fixed. Training lasts one, 15, and 12 epochs, respectively. Official MNIST/HAR test sets and stratified 75/25 MAGIC splits align observations across variants within seeds.
Figure 5: Test performance during training of a three-layer GCN on a. PROTEINS (1,113 graphs), b. NCI1 (4,110), c. NCI109 (4,127), d. Mutagenicity (4,337), e. AIDS (2,000), and f. DD (1,178). Counts refer to full datasets. Lines show means across ten seeds (42–51): solid for IMV, dashed for accuracy, dash-dotted for precision, and dotted for recall. Shading spans the minimum and maximum across seeds at each epoch, not confidence intervals. Each seed defines a stratified 70/15/15 training/validation/test split; the same test graphs are evaluated from untrained epoch 0 through epoch 200. IMV compares probabilities with the training-prevalence null; classification uses a 0.5 threshold with class 1 positive. Curves are not smoothed, and negative IMV values are retained.
Dataset
Graphs
Mean nodes
Mean edges
PROTEINS
1,113
39.06
72.82
NCI1
4,110
29.87
32.30
NCI109
4,127
29.68
32.13
Mutagenicity
4,337
30.32
30.77
AIDS
2,000
15.69
16.20
DD
1,178
284.32
715.66
Table 6: Datasets used in the GNN experiments. Mean edge counts refer to undirected edges before GCN self-loops are added.
n
IMV
Median (ms)
IQR (ms)
Time/obs. ( μ s)
300
0.282
0.078
0.002
0.260
3,000
0.279
0.124
0.000
0.041
30,000
0.303
0.719
0.010
0.024
300,000
0.303
6.987
0.319
0.023
Table 7: Runtime of vanilla_imv for the simulation in Box 3.3 . The median and interquartile range (IQR) summarize 9 batches of repeated calls; n is the number of test observations. Timing excludes fitting and prediction.
Feature attribution analysis is critical for interpreting machine learning models and supporting reliable data-driven decisions. However, feature attribution measures often exhibit stochastic variation: different train--test splits, random seeds, or model-fitting procedures can produce substantially different attribution values and feature rankings. This paper proposes a framework for incorporating stochastic nature of feature attribution and a robust attribution metric, RoSHAP, for stable feature ranking based on the SHAP metric. The proposed framework models the distribution of feature attribution scores and estimates it through bootstrap resampling and kernel density estimation. We show that, under mild regularity conditions, the aggregated feature attribution score is asymptotically Gaussian, which greatly reduces the computational cost of distribution estimation. The RoSHAP summarizes the distribution of SHAP into a robust feature-ranking criterion that simultaneously rewards features that are active, strong, and stable. Through simulations and real-data experiments, the proposed framework and RoSHAP outperform standard single-run attribution measures in identifying signal features. In addition, models built using RoSHAP-selected features achieve predictive performance comparable to full-feature models while using substantially fewer predictors. The proposed RoSHAP approach improves the stability and interpretability of machine learning models, enabling reliable and consistent insights for analysis.
Lanxin Xiang, Liang Shi, Youhui Ye +3
Department of Statistics Virginia Tech · Department of Statistics Virginia Tech Transportation Institute · Department of Computer Science Virginia Tech
Changes in input distribution can induce shifts in the average predictions of machine learning models. Such prediction shifts may impact downstream business outcomes (e.g. a bank's loan approval rate), so understanding their causes can be crucial. We propose \ours{}: a Shapley value method for attributing prediction shifts to changes in the conditional probabilities of interpretable subgroups of data, where these subgroups are defined by the structure of decision trees. We initially apply this method to single decision trees, providing exact explanations based on conditional probability changes at split nodes. Next, we extend it to tree ensembles by selecting the most explanatory tree and accounting for residual effects. Finally, we propose a model-agnostic variant using surrogate trees grown with a novel objective function, allowing application to models like neural networks. While exact computation can be intensive, approximation techniques enable practical application. We show that \ours{} provides simple, faithful, and near-complete explanations of prediction shifts across model classes, aiding model monitoring in dynamic environments.
Interpretability is central for scientific machine learning, as understanding why models make predictions enables hypothesis generation and validation. While tabular foundation models show strong performance, existing explanation methods like SHAP are computationally expensive, limiting interactive exploration. We introduce ShapPFN, a foundation model that integrates Shapley value regression directly into its architecture, producing both predictions and explanations in a single forward pass. On standard benchmarks, ShapPFN achieves competitive performance while producing high-fidelity explanations (R2=0.96, cosine=0.99) over 1000\times faster than KernelSHAP (0.06s vs 610s). Our code is available at https://github.com/kunumi/ShapPFN
Luan Borges Teodoro Reis Sena, Francisco Galuppo Azevedo
Kunumi Institute, Belo Horizonte, Minas Gerais, Brazil · Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil