Great Expectations under Language Models with Potentials

Many applications of language models hinge not on individual samples but on the expectation of a test functional under the model. Estimating such expectations reliably can be computationally expensive. In this paper, we show how to make estimation more efficient by exploiting the next-token conditional probabilities which are available as a by-product of sampling. We do so through potentials: real-valued functions on prefixes that decompose the test functional additively. We construct an estimator whose variance depends on the chosen potential, and derive conditions under which a potential reduces this variance. We then develop practical potentials for several estimands and applications, and show that for every estimand we study our estimator achieves lower variance than existing baselines at comparable computational cost.