Another look at statistical inference with machine learning-imputed data

From structural biology to epidemiology, predictions from machine learning (ML) models increasingly complement costly gold-standard data, enabling faster, more affordable, and scalable scientific inquiry. In response, prediction-based (PB) inference has emerged to support statistical analysis that combines a large volume of predicted data with a small amount of gold-standard data. The goals of PB inference are twofold: (i) to mitigate bias arising from prediction error and (ii) to improve efficiency relative to classical inference based solely on gold-standard data. While early PB inference methods primarily focused on bias mitigation, improving efficiency remains an active area of research. Motivated by connections between PB inference and longstanding problems in statistics and related fields, we draw on the two-phase sampling literature to introduce an approach for Z-estimation with ML-imputed outcomes that is guaranteed to match or exceed the efficiency of classical inference, regardless of prediction quality. We demonstrate the utility of our approach through theoretical and numerical analyses as well as an application to UK Biobank data. We further establish new connections between existing PB inference approaches and foundational and contemporary statistical methods.