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.

翻译：起源于博弈论的Shapley值被广泛用于解释机器学习模型的预测，其通过量化每个特征值对预测的贡献来实现。这要求预测结果为标量（如二分类问题），而多类概率预测作为离散概率分布存在于多维单纯形上。在此类多分类场景中，Shapley值通常以一对多的方式在每个类别上单独计算，忽略了输出分布的组合特性。本文引入Shapley组合作为解释多类概率预测的严谨方法，该方法基于组合数据分析中的Aitchison几何结构。我们证明Shapley组合是在Aitchison单纯形上满足线性性、对称性与效率性的唯一度量，从而扩展了标准Shapley值对应的公理性质。我们通过一系列场景验证了这种多分类处理的合理性。