MEC: Machine-Learning-Assisted Generalized Entropy Calibration for Semi-Supervised Mean Estimation

Obtaining high-quality labels is costly, whereas unlabeled covariates are often abundant, motivating semi-supervised inference methods with reliable uncertainty quantification. Prediction-powered inference (PPI) leverages a machine-learning predictor trained on a small labeled sample to improve efficiency, but it can lose efficiency under model misspecification and suffer from coverage distortions due to label reuse. We introduce Machine‑Learning‑Assisted Generalized Entropy Calibration (MEC), a cross‑fitted, calibration‑weighted variant of PPI. MEC improves efficiency by reweighting labeled samples to better align with the target population, using a principled calibration framework based on Bregman projections. This yields robustness to affine transformations of the predictor and relaxes requirements for validity by replacing conditions on raw prediction error with weaker projection‑error conditions. As a result, MEC attains the semiparametric efficiency bound under weaker assumptions than existing PPI variants. Across simulations and a real‑data application, MEC achieves near‑nominal coverage and tighter confidence intervals than CF‑PPI and vanilla PPI.

Many scientific studies have a small amount of high-quality labeled data, such as carefully measured outcomes, but a much larger amount of unlabeled data, such as patient characteristics, images, or other features. Machine learning can help make use of these unlabeled data, but naive use of predictions can lead to unreliable uncertainty estimates. This paper proposes a new method called Machine-Learning-Assisted Generalized Entropy Calibration, or MEC. The method uses machine-learning predictions to guide how the labeled data are reweighted, so that the labeled sample better represents the larger target population. This improves statistical efficiency while maintaining valid confidence intervals. Compared with existing prediction-powered inference methods, MEC is more robust when the machine-learning predictor is imperfect. It also avoids the double use of the same labeled data for both training and evaluation by using cross-fitting. In simulations and a real-data example, MEC provides confidence intervals with close-to-nominal coverage and often shorter lengths than existing methods. Overall, MEC provides a principled way to combine machine learning with statistical inference when labels are expensive but unlabeled covariates are plentiful.