No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference

Post-deployment monitoring of healthcare AI requires statistically valid, label-efficient methods, but gold-standard labels from clinician chart review are expensive. Prediction-powered inference (PPI) and active statistical inference (ASI) reduce label cost by combining a small labeled sample with abundant model predictions, but both are restricted to a single predictor, a poor fit for modern clinical pipelines that have multiple predictors of differing cost and accuracy available at inference time. We propose Active Multiple-Prediction-Powered Inference (AM-PPI), which routes each instance to a cost-appropriate predictor subset, samples gold-standard labels in proportion to the chosen subset's residual uncertainty, and reweights predictions to minimize estimator variance, all under a single deployment-time budget. AM-PPI generalizes ASI to leverage multiple predictors and extends Multiple-PPI from global per-predictor allocation to per-instance adaptive routing. We derive closed-form Karush-Kuhn-Tucker (KKT) conditions for all three decisions and prove, via biconvexity and strong duality, that the resulting fixed point is a global optimum despite the joint problem being non-jointly-convex. We establish asymptotic normality with valid coverage, minimum-variance unbiasedness within the linear-prediction augmented inverse propensity weighted (AIPW) class, and a closed-form criterion identifying when multiple predictors help. On synthetic data and three healthcare monitoring tasks, AM-PPI produces 10 to 40 percent narrower confidence intervals (CIs) than single-predictor ASI in the budget regime where routing matters, and matches the better baseline elsewhere.

We study semisupervised mean estimation with a small labeled sample, a large unlabeled sample, and a black-box prediction model whose output may be miscalibrated. A standard approach in this setting is augmented inverse-probability weighting (AIPW) [Robins et al., 1994], which protects against prediction-model misspecification but can be inefficient when the prediction score is poorly aligned with the outcome scale. We introduce Calibrated Prediction-Powered Inference, which post-hoc calibrates the prediction score on the labeled sample before using it for semisupervised estimation. This simple step requires no retraining and can improve the original score both as a predictor of the outcome and as a regression adjustment for semisupervised inference. We study both linear and isotonic calibration. For isotonic calibration, we establish first-order optimality guarantees: isotonic post-processing can improve predictive accuracy and estimator efficiency relative to the original score and simpler post-processing rules, while no further post-processing of the fitted isotonic score yields additional first-order gains. For linear calibration, we show first-order equivalence to PPI++. We also clarify the relationship among existing estimators, showing that the original PPI estimator is a special case of AIPW and can be inefficient when the prediction model is accurate, while PPI++ is AIPW with empirical efficiency maximization [Rubin et al., 2008]. In simulations and real-data experiments, our calibrated estimators often outperform PPI and are competitive with, or outperform, AIPW and PPI++. We provide an accompanying Python package, ppi_aipw, at https://larsvanderlaan.github.io/ppi-aipw/.

The rapidly expanding artificial intelligence (AI) industry has produced diverse yet powerful prediction tools, each with its own network architecture, training strategy, data-processing pipeline, and domain-specific strengths. These tools create new opportunities for semi-supervised inference, in which labeled data are limited and expensive to obtain, whereas unlabeled data are abundant and widely available. Given a collection of predictors, we treat them as a mixture of experts (MOE) and introduce an MOE-powered semi-supervised inference framework built upon prediction-powered inference (PPI). Motivated by the variance reduction principle underlying PPI, the proposed framework seeks the mixture of experts that achieves the smallest possible variance. Compared with standard PPI, the MOE-powered inference framework adapts to the unknown performance of individual predictors, benefits from their collective predictive power, and enjoys a best-expert guarantee. The framework is flexible and applies to mean estimation, linear regression, quantile estimation, and general M-estimation. We develop non-asymptotic theory for the MOE-powered inference framework and establish upper bounds on the coverage error of the resulting confidence intervals. Numerical experiments demonstrate the practical effectiveness of MOE-powered inference and corroborate our theoretical findings.