Mutual information-based group explainers with coalition structure for machine learning model explanations

In this article, we study game-theoretical group explainers for machine learning (ML) models in a functional analytic setting as operators defined on appropriate functional spaces. Specifically, we focus on game values with coalition structure applied to random games based on the conditional and marginal expectation. In particular, we investigate the stability of the explanation operators which showcases the differences between the two games, such as showing that the marginal explanations can become unstable in the natural data-based metric. Furthermore, we formulate novel group explanation methodologies based on game values with coalition structure applied to both marginal and conditional games. They allow us to unify the two types of explanations and turn out to have lower complexity. In addition, we study the effect of predictor grouping on the stability of the corresponding explanation operators. Finally, we establish the two-step representation for a coalitional game value consisting of two game values and a family of intermediate games. We use this representation to generalize our grouping approach to the case of nested partitions represented by a parameterized partition tree. Specifically, we introduce a theoretical scheme that generates recursive coalitional game values and group explainers under a given partition tree structure and investigate the properties of the corresponding group explainers. We verify our results in a number of experiments with data where the predictors are grouped based on an information-theoretic measure of dependence.