PPI is the Difference Estimator: Recognizing the Survey Sampling Roots of Prediction-Powered Inference

TL;DR

This paper reveals that prediction-powered inference (PPI) estimators are equivalent to classical survey sampling estimators, bridging machine learning inference with established statistical survey methods.

Contribution

It establishes the algebraic equivalence between PPI estimators and traditional survey sampling estimators, and discusses implications for theory and practice.

Findings

PPI estimator for population mean equals the difference estimator from survey sampling.

PPI plus corresponds to the GREG estimator in survey sampling.

Recognizing the equivalence enables cross-fertilization of methods and theory.

Abstract

Prediction-powered inference (PPI) is a rapidly growing framework for combining machine learning predictions with a small set of gold-standard labels to conduct valid statistical inference. In this article, I argue that the core estimators underlying PPI are equivalent to well-established estimators from the survey sampling literature dating back to the 1970s. Specifically, the PPI estimator for a population mean is algebraically equivalent to the difference estimator of Cassel et al. (1976), and PPI plus corresponds to the generalized regression (GREG) estimator of Sarndal et al. (2003). Recognizing this equivalence, I consider what part of PPI is inherited from a long-standing literature in statistics, what part is genuinely new, and where inferential claims require care. After introducing the two frameworks and establishing their equivalence, I break down where PPI diverges from model-assisted estimation, including differences in the mode of inference, the role of the unlabeled data pool, and the consequences of differential prediction error for subgroup estimands such as the average treatment effect. I then identify what each framework offers the other: PPI researchers can draw on the survey sampling literature's well-developed theory of calibration, optimal allocation, and design-based diagnostics, while survey sampling researchers can benefit from PPI's extensions to non-standard estimands and its accessible software ecosystem. The article closes with a call for integration between these two communities, motivated by the growing use of large language models as measurement instruments in applied research.