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Explaining a probabilistic prediction on the simplex with Shapley compositions

This paper introduces "Shapley compositions," a novel framework grounded in Aitchison geometry that provides a mathematically rigorous and unique method for explaining multiclass probabilistic predictions by properly accounting for their compositional nature, overcoming the limitations of traditional one-vs-rest Shapley value approaches.

Original authors: Paul-Gauthier Noé, Miquel Perelló-Nieto, Jean-François Bonastre, Peter Flach

Published 2026-06-04

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Paul-Gauthier Noé, Miquel Perelló-Nieto, Jean-François Bonastre, Peter Flach

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why a machine learning model made a specific guess. Let's say the model is a weather forecaster that predicts the chance of three things happening: Sunny, Rainy, or Snowy.

The Problem: The "One-vs-Rest" Trap

Traditionally, tools used to explain these guesses (called Shapley values) work like a single-lane road. They are great for binary choices (Yes/No or Sunny/Rainy). But when you have three or more options, the old tools try to explain each option separately.

They might say:

"Feature A made 'Sunny' 10% more likely."
"Feature A made 'Rainy' 5% less likely."
"Feature A made 'Snowy' 2% less likely."

The paper argues this is like trying to understand a dance by watching each dancer move in isolation. It misses the big picture: the probabilities are connected. If the chance of "Sunny" goes up, the chances of "Rainy" and "Snowy" must go down because the total probability always has to equal 100%. The old methods ignore this "tug-of-war" relationship between the classes.

The Solution: Shapley Compositions

The authors introduce a new method called Shapley Compositions. They treat the prediction not as separate numbers, but as a single, unified "recipe" or "mixture."

To make this work, they use a special mathematical toolkit called Aitchison geometry. Think of this as a new way of measuring space that respects the rules of the mixture.

The Analogy: The Flavor Blender
Imagine the model's prediction is a smoothie made of three flavors: Strawberry, Blueberry, and Banana.

The Base Prediction is the average smoothie the machine makes for everyone (maybe 33% of each).
The Actual Prediction is the specific smoothie for your order (maybe 80% Strawberry, 10% Blueberry, 10% Banana).

In this new method, every feature (like "sugar level" or "ice amount") is a blender push.

Instead of saying "Sugar increased Strawberry," the new method says: "The 'Sugar' feature pushed the smoothie mixture from the average spot toward the Strawberry corner of the triangle."
Because the flavors are linked, pushing toward Strawberry automatically pulls away from Blueberry and Banana. The new method captures this movement as a single, unified vector (an arrow) rather than three separate numbers.

How It Works (The "Geometry" Part)

The paper proves that this new method is the only way to fairly split the credit among features while respecting three golden rules:

Linearity: If you combine two models, the explanation is just the combination of their explanations.
Symmetry: If two features do the exact same thing, they get the exact same credit.
Efficiency: The sum of all the "pushes" (features) must perfectly move the smoothie from the average spot to your specific order. No credit is lost, and no extra credit is invented.

Visualizing the Result

The paper shows how to draw these explanations:

The Triangle Map: They map the three flavors onto a triangle. The center is the average. The corners are pure flavors. The "push" from a feature is drawn as an arrow moving the smoothie from the center toward a specific corner.
The Histogram: They also show the "push" as a bar chart. If a feature strongly pushes toward "Sunny," the bar for Sunny will be high, while the bars for the others drop.

Why This Matters

The paper demonstrates this with examples like identifying flowers (Iris dataset) and recognizing handwritten digits.

Old Way: You get a confusing list of numbers for each flower type, and you have to guess how they fit together.
New Way: You get a clear picture of how the features (like petal length) physically "moved" the prediction from the average flower to the specific type you have.

In short: This paper gives us a new, mathematically sound way to explain complex, multi-option predictions by treating them as a single, shifting mixture rather than a list of separate numbers. It ensures that the explanation respects the fact that increasing one option's probability must decrease the others.

Technical Summary: Explaining a Probabilistic Prediction on the Simplex with Shapley Compositions

Problem Statement
Machine learning models, particularly those performing multiclass classification, often output discrete probability distributions rather than scalar values. These outputs reside on a multidimensional simplex. While Shapley values are a standard tool for explaining scalar predictions by quantifying feature contributions, their direct application to multiclass settings is problematic. Current approaches typically compute Shapley values separately for each class (one-vs-rest or one-vs-one). This method ignores the compositional nature of the output, where the relative values between probabilities are more significant than absolute values, and fails to respect the geometric structure of the probability simplex. Furthermore, computing separate values for each class can lead to incoherent explanations where the cumulative path from a base prediction to the final prediction leaves the simplex space.

Methodology
The authors propose Shapley compositions, a framework that extends Shapley values to the space of discrete probability distributions using Aitchison geometry from compositional data analysis.

Aitchison Geometry: The paper treats the probability simplex ( $S_D$ ) not as a standard Euclidean space but as a space with specific operations:
- Perturbation ( $\oplus$ ): Acts as addition between compositions.
- Powering ( $\odot$ ): Acts as scalar multiplication.
- Inner Product: Defined via log-ratios of parts.
- Isometric Log-Ratio (ILR) Transformation: Maps the simplex to a $(D-1)$ -dimensional Euclidean space ( $\mathbb{R}^{D-1}$ ), preserving the metric and allowing standard vector operations (addition, subtraction) to represent perturbations and powerings.
Definition of Shapley Composition:
- The characteristic function is redefined to map subsets of features to the simplex using Aitchison subtraction ( $\ominus$ ), representing the change in the expected prediction when features are observed.
- The Shapley composition for a feature is defined as the average of these perturbations over all possible coalitions, calculated using Aitchison operations (perturbation and powering) rather than standard summation and scaling.
- Mathematically, the Shapley composition $\phi_i$ is the unique quantity satisfying Linearity, Symmetry, and Efficiency on the Aitchison simplex.
- Efficiency in this context ensures that the perturbation of the base distribution (expected prediction) by the sum of all Shapley compositions equals the final predicted distribution: $\bigoplus_{i=1}^d \phi_i = f(x) \ominus E_A[f(X)]$ .
Visualization and Analysis:
- ILR Space: Shapley compositions are visualized as vectors in the $(D-1)$ -dimensional ILR space. This allows for the analysis of angles (orthogonality/redundancy), norms (strength of contribution), and projections onto class-compositions (direction of influence).
- Alternative Visualizations: For higher dimensions, the paper suggests visualizing compositions as histograms (showing the distributional shift per feature) or using parallel coordinates to track the successive perturbation of the probability distribution from the base to the prediction.

Key Contributions

Formal Definition: The paper defines Shapley compositions as a principled, multidimensional extension of Shapley values to the probability simplex.
Axiomatic Proof: The authors prove that the Shapley composition is the unique quantity satisfying linearity, symmetry, and efficiency on the Aitchison simplex, thereby extending the theoretical foundation of Shapley values to multiclass settings.
Geometric Framework: The work establishes a coherent geometric interpretation where feature contributions are additive perturbations on the simplex, avoiding the incoherence of one-vs-rest approaches that may exit the probability space.
Visualization Tools: The paper introduces specific visualization techniques (ILR plots, histograms, parallel coordinates) tailored to interpreting these multidimensional contributions.

Results and Demonstrations
The paper provides proof-of-concept demonstrations rather than large-scale empirical benchmarks:

Iris Dataset (3 Classes): Visualized in a 2D ILR space, showing how features like petal length and width perturb the base distribution toward the "versicolor" region. The analysis highlights how Shapley compositions capture the trade-off between classes (e.g., a feature increasing one class probability necessarily decreases others) more coherently than separate SHAP values.
Digit Recognition (4 and 10 Classes):
- A 4-class task (digits 0–3) is visualized in 3D ILR space, illustrating the tetrahedral structure of the simplex.
- A 10-class task demonstrates the use of balances (log-ratios of groups of classes) to interpret specific ILR components. This allows users to focus on subspaces comparing specific groups of classes (e.g., odd vs. even, or specific digit pairs) without visualizing the full high-dimensional space.
Estimation: The authors adapt existing sampling algorithms (from the SHAP toolkit) to estimate Shapley compositions, noting that adjustments are required to ensure the efficiency property holds for approximations.

Significance and Claims
The paper claims to be the first to propose a formal extension of the Shapley value framework to the multidimensional simplex for explaining multiclass probabilistic predictions.

Theoretical Foundation: The primary contribution is theoretical and methodological, providing a "well-founded" way to handle the compositional nature of multiclass outputs.
Coherence: It resolves the issue of one-vs-rest explanations by ensuring the explanation path remains within the probability simplex and treats the distribution as a single entity.
Future Work: The authors modestly note that their experimental results are limited to illustrative examples. They identify the computational complexity (scaling with the number of features) and the assumption of feature independence as current limitations, leaving the development of estimation algorithms for dependent features and large-scale applications for future research.