Bridging extrinsic and intrinsic variable importance

Variable importance may describe either intrinsic predictive information in a population or extrinsic importance for a fitted prediction rule. Quantifying the uncertainty in variable importance estimates is critical for interpretation. Methods for estimating intrinsic variable importance (we will refer to these as VIMP) and the minipatch leave-one-covariate-out procedure (MPLOCO) target intrinsic and extrinsic importance, respectively, and provide methods for computing standard errors. These two approaches have a shared structure, comparing prediction performance with and without features, but the relationship between them has not been formally characterized. We establish conditions under which the two perspectives align. Under squared-error loss, if the fitted full and reduced learners converge to their oracle counterparts sufficiently fast, then MPLOCO is asymptotically equivalent to VIMP. We provide further conditions extending this result to general loss functions and formalize grouped MPLOCO for potentially overlapping feature groups. Through simulations, we show that VIMP and MPLOCO agree most closely when the fitted learner is well aligned with the data-generating mechanism. In a high-dimensional grouped simulation, both procedures identified the signal-containing groups. In an analysis of HIV-1 VRC01 neutralization sensitivity, both methods placed the same three biologically relevant feature groups among their highest-ranked groups. These results clarify when intrinsic and extrinsic importance can be interpreted similarly and when they provide complementary information.