  The recent criticisms of the robustness of post hoc model approximation
explanation methods (like LIME and SHAP) have led to the rise of model-precise
abductive explanations. For each data point, abductive explanations provide a
minimal subset of features that are sufficient to generate the outcome. While
theoretically sound and rigorous, abductive explanations suffer from a major
issue -- there can be several valid abductive explanations for the same data
point. In such cases, providing a single abductive explanation can be
insufficient; on the other hand, providing all valid abductive explanations can
be incomprehensible due to their size. In this work, we solve this issue by
aggregating the many possible abductive explanations into feature importance
scores. We propose three aggregation methods: two based on power indices from
cooperative game theory and a third based on a well-known measure of causal
strength. We characterize these three methods axiomatically, showing that each
of them uniquely satisfies a set of desirable properties. We also evaluate them
on multiple datasets and show that these explanations are robust to the attacks
that fool SHAP and LIME.
