Using Large Language Models as Low-Cost Statistical Estimators for Human-Response Data

Abstract

Quantitative research across the social and behavioral sciences depends on human subject experiments that are expensive, slow, and subject to sampling bias. Here we show that pretrained large language models induce risk-equivalent estimators of conditional expectations under squared loss, establishing restricted functional risk equivalence: under squared loss, the LLM induces an estimator whose risk matches the Bayes optimal risk for squared-loss prediction of conditional expectations for any inference that depends on the data only through the conditional mean. We formalize the LLM as a misspecified functional estimator $T(\hat{P}_n)$ trained on i.i.d.\ data, decompose the estimation error into representation bias $\epsilon_{\mathrm{rep}}$ and optimization error, and prove that under mild regularity conditions the LLM's expected error converges to the irreducible population variance plus the squared representation bias, with the representation bias bounded by the Pinsker inequality. The identifiability error $\delta$ propagates into the effective bias, inflating the asymptotic risk floor. We establish restricted functional risk equivalence via a bidirectional Le Cam deficiency analysis: the forward deficiency vanishes asymptotically while the reverse deficiency is exactly zero. We provide finite-sample concentration bounds and a calibration protocol with explicit decision rules. The result is a precise, provable statement: a well-calibrated LLM achieves the Bayes-optimal risk for conditional-mean-dependent inference, bounded by explicit scope conditions. In practical applications, this means that under satisfied conditions and well-calibrated models, large language models can be used in many prediction and decision-making tasks that originally relied on human experiments, approximating near-optimal statistical inference at lower cost.