From Shapley back to Pearson: Hypothesis Testing via the Shapley Value

The complex nature of artificial neural networks raises concerns on their reliability, trustworthiness, and fairness in real-world scenarios. The Shapley value -- a solution concept from game theory -- is one of the most popular explanation methods for machine learning models. More traditionally, from the perspective of statistical learning, feature importance is defined in terms of conditional independence. So far, these two approaches to interpretability and feature importance have been considered separate and distinct. In this work, we show that Shapley-based explanation methods and conditional independence testing are closely related. We introduce the $\textbf{SHAP}$ley $\textbf{L}$ocal $\textbf{I}$ndependence $\textbf{T}$est ($\textbf{SHAPLIT}$), a novel testing procedure inspired by the Conditional Randomization Test (CRT) for a specific notion of local (i.e., on a sample) conditional independence. With it, we prove that for binary classification problems, each marginal contribution in the Shapley value is an upper bound to the $p$-value of this conditional independence test. Furthermore, we show that the Shapley value itself provides an upper bound to the $p$-value of a global SHAPLIT null hypothesis. As a result, we grant the Shapley value with a precise statistical sense of importance with false positive rate control