From Log-Odds to Shapley Values: An Explanatory Geometry for the Weighted Naive Bayes Classifier

Vincent Lemaire; Fabrice Clérot

stat.MLOct 7, 2026

From Log-Odds to Shapley Values: An Explanatory Geometry for the Weighted Naive Bayes Classifier

Authors: Vincent Lemaire, Fabrice Clérot

Organizations: Orange Research, France

Abstract

This paper studies the construction of an explanatory space for a weighted naive Bayes classifier from the supervised representation induced by the model. We start from the classical supervised distance based on conditional log-likelihoods and introduce a discriminative reformulation based on log-odds, which is more directly related to the classification decision. We then show that this representation induces a distance that exactly coincides with the $\ell_1$ distance between vectors of analytical Shapley values, thereby providing a formal explanatory interpretation of the geometry induced by the model. Finally, we empirically compare several supervised distances derived from these representations using a $k$ -nearest neighbors classifier. This work highlights a close link between supervised distance, local explanation, and predictive behavior, from a primarily methodological perspective.

Figures & tables

Dataset	# examples	$V_{n}$	$V_{c}$	$C$	Dataset	# examples	$V_{n}$	$V_{c}$	$C$
Adult	48 842	7	8	2	Pima	768	8	0	2
Heart	270	10	3	2	Segmentation	2 310	19	0	7
Letter	20 000	16	0	26	Shuttle	58 000	9	0	7
Mushroom	8 416	0	22	2	SickEuthyroid	3 163	7	18	2
Pendigits	10 992	16	0	10	Vehicle	846	18	0	4
Phoneme	5 404	5	0	2	Waveform	5 000	40	0	3

Table 1 : Datasets used in the experiments.

V_{n}

denotes the number of numerical variables,

V_{c}

the number of categorical variables, and

C

the number of classes.

Dataset	$d_{global}^{1}(D,D^{\prime})$		$d_{shap}^{1}(D,D^{\prime})$		$d_{SNB}^{1}(D,D^{\prime})$
	ACC	AUC	ACC	AUC	ACC	AUC
Adult	0,8239	0,7541	0,8241	0,7547	0,8256	0,7557
Heart	0,7889	0,7867	0,8148	0,8117	0,8148	0,8117
Letter	0,9348	0,9659	0,9398	0,9686	0,9431	0,9703
Mushroom	1,0000	1,0000	1,0000	1,0000	1,0000	1,0000
Pendigits	0,9731	0,9852	0,9773	0,9874	0,9758	0,9867

Table 2 : Performance moyenne en Test sur 10 folds stratifiés par dataset

Figure 1 : Critical Diagram between the 3 distances (AUC OVR).

Panel of Figure 2 — Figure 2 : Evolution of $\mathrm{LocAgree}_{k}$ as a function of $k$ . A high value indicates that the neighbors selected based on the given distance more frequently share the same class predicted by the SNB than the test observation. Subfigure ( 2(a) ) shows the average results across the twelve datasets, while subfigure ( 2(b) ) details this behavior for the Adult dataset over a wide range of $k$ values.

Figure 3 : Dispersion values as a function of k (average across the 12 datasets).

dataset	$d_{global}^{1}(D,D^{\prime})$	$d_{shap}^{1}(D,D^{\prime})$	$d_{SNB}^{1}(D,D^{\prime})$
Adult	0.4838	0.5264	0.5015
Heart	0.4626	0.4645	0.4698
Letter	0.5931	0.6304	0.6210
Mushroom	0.9999	1.0000	1.0000
Pendigits	0.5276	0.5576	0.5443
Phoneme	0.3073	0.3004	0.3017

Table 3 : Average performance on test across 10 folds stratified by dataset

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Originating in game theory, Shapley values are widely used for explaining a machine learning model's prediction by quantifying the contribution of each feature's value to the prediction. This requires a scalar prediction as in binary classification, whereas a multiclass probabilistic prediction is a discrete probability distribution, living on a multidimensional simplex. In such a multiclass setting the Shapley values are typically computed separately on each class in a one-vs-rest manner, ignoring the compositional nature of the output distribution. In this paper, we introduce Shapley compositions as a well-founded way to properly explain a multiclass probabilistic prediction, using the Aitchison geometry from compositional data analysis. We prove that the Shapley composition is the unique quantity satisfying linearity, symmetry and efficiency on the Aitchison simplex, extending the corresponding axiomatic properties of the standard Shapley value. We demonstrate this proper multiclass treatment in a range of scenarios.

Abstract

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