Empirical Likelihood Meets Prediction-Powered Inference

2025-12-18

AI Summary

This paper addresses statistical inference under a setting featuring scarce labeled data, abundant unlabeled data, and high-quality external predictions. We propose a prediction-augmented empirical likelihood framework. Methodologically, we unify supervised estimating equations with auxiliary moment conditions derived from external predictions within an empirical likelihood optimization—learning the optimal auxiliary function via cross-fitting and basis expansion to yield a calibrated empirical distribution. Theoretically, our estimator achieves the semiparametric efficiency bound and enables valid estimation and uncertainty quantification for arbitrary differentiable functionals. Simulation and empirical studies demonstrate that the proposed method substantially reduces mean squared error, shortens confidence interval length, and strictly maintains nominal coverage—achieving both efficiency and robustness.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

Abstract

We study inference with a small labeled sample, a large unlabeled sample, and high-quality predictions from an external model. We link prediction-powered inference with empirical likelihood by stacking supervised estimating equations based on labeled outcomes with auxiliary moment conditions built from predictions, and then optimizing empirical likelihood under these joint constraints. The resulting empirical likelihood-based prediction-powered inference (EPI) estimator is asymptotically normal, has asymptotic variance no larger than the fully supervised estimator, and attains the semiparametric efficiency bound when the auxiliary functions span the predictable component of the supervised score. For hypothesis testing and confidence sets, empirical likelihood ratio statistics admit chi-squared-type limiting distributions. As a by-product, the empirical likelihood weights induce a calibrated empirical distribution that integrates supervised and prediction-based information, enabling estimation and uncertainty quantification for general functionals beyond parameters defined by estimating equations. We present two practical implementations: one based on basis expansions in the predictions and covariates, and one that learns an approximately optimal auxiliary function by cross-fitting. In simulations and applications, EPI reduces mean squared error and shortens confidence intervals while maintaining nominal coverage.

Problem

• Inference with small labeled and large unlabeled data using external model predictions
• Linking prediction-powered inference with empirical likelihood via joint constraints
• Developing efficient estimators and confidence sets for general functionals

Innovation

• Empirical likelihood integrates labeled and prediction data
• Asymptotically efficient estimator with reduced variance
• Calibrated weights enable general functional estimation