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：源于博弈论的Shapley值通过量化每个特征值对预测的贡献，被广泛用于解释机器学习模型的预测结果。这要求预测值为标量（如二分类问题），但多分类概率预测是定义在多维单纯形上的离散概率分布。在此类多分类场景中，Shapley值通常以“一对其余”的方式针对每个类别单独计算，忽视了输出分布的组合本质。本文引入Shapley组合作为合理解释多分类概率预测的严谨方法，采用成分数据分析中的艾奇逊几何。我们证明Shapley组合是艾奇逊单纯形上唯一满足线性性、对称性和效率性的量，将标准Shapley值的对应公理性质扩展至多分类场景。我们在多种场景中展示这种合理的多分类处理方法。