Multi-source Stable Variable Importance Measure via Adversarial Machine Learning

The quantification and inference of predictive importance for exposure covariates have recently gained significant attention in the context of interpretable machine learning. Contemporary scientific investigations often involve data originating from multiple sources with distributional heterogeneity. It is imperative to introduce a new notation of the variable importance measure that is stable across diverse environments. In this paper, we introduce MIMAL (Multi-source Importance Measure via Adversarial Learning), a novel statistical framework designed to quantify the importance of exposure variables by maximizing the worst-case predictive reward across source mixtures. The proposed framework is adaptable to a broad spectrum of machine learning methodologies for both confounding adjustment and exposure effect characterization. We establish the asymptotic normality of the data-dependent estimator of the multi-source variable importance measure under a general machine learning framework. Our framework requires the similar learning accuracy conditions compared to those required for single-source variable importance analysis. The finite-sample performance of MIMAL is demonstrated through extensive numerical studies encompassing diverse data generation scenarios and machine learning implementations. Furthermore, we illustrate the practical utility of our approach in a real-world case study of air pollution in Beijing, analyzing data collected from multiple locations.