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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 represen","title":"Using Large Language Models as Low-Cost Statistical Estimators for Human-Response Data","url":"https://arxiv.org/abs/2606.30372","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.30372v1 Announce Type: new \nAbstract: 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. 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