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原文摘要

Repeated sampling with majority selection is an effective means of enforcing self-consistency in large language models. Recently, a variant has also been introduced that terminates early to save costs when the output distribution is favorable. But it still cannot yield a correct confidence estimate from the answer distribution. For this issue, we formalize this as majority identification under an unknown distribution with a budget and a target error rate, and propose BARS, a sequential rule certifying the top-two gap and returning an early stop, a budgeted majority vote, or a refusal when feasibility cannot be established. A refusal is a statement about the rule and its budget, not a claim that a question is unanswerable, and it tells practitioners when to widen the budget or switch models. A certified mode carries proven bounds on the joint answering error and on refusing an instance the budget could handle, while a practical mode, used in our experiments, replaces them with empirical calibration. On synthetic instances drawn from known distributions over thousands of seeds, the difficulty-driven behavior is recovered and the price of the certified mode is measured.On GSM8K the practical mode draws 22.3\% fewer samples than fixed-8 sampling over the same pool, refused questions included, at the cost of declining 18.8\% of the questions. On MATH, it greatly improves the accuracy of its output answers into 82.5\% at the cost of honestly refusing to answer half of the difficult questions that the model itself is unable to solve.