AI 中文总结
该研究刻画了同伴预测中互信息无偏估计量的样本复杂度，确定了不同样本量下唯一的互信息，提出了DMI的改进估计量，还发现提前停止估计量可降低方差及存在可在少于三个样本内估计的互信息。
AI 中文摘要
同伴预测旨在通过奖励联合报告集来激励智能体真实报告观测到的信号，而无需观测真实值。遵循Kong和Schoenebeck（2019）引入的信息论互信息的推广，我们将信号联合分布上的函数称为互信息，当它是非负的，并且对于所有信息结构都能抑制报告的混淆。互信息的无偏估计量从分布中抽取一定数量的样本，并为两个智能体返回奖励，使得期望奖励等于互信息。我们旨在为给定的样本数量刻画具有无偏估计量的互信息集合。我们表明，对于三个或更少的采样报告对，唯一具有无偏估计量的互信息是平凡零；对于二元报告空间的四个或五个样本，Kong（2024）的行列式互信息（Determinant Mutual Information，DMI）是唯一的互信息（相差一个标量倍数）。我们进一步表明，DMI在六个样本时不再唯一。我们为任意给定的样本数量提供了DMI的改进估计量，并刻画了其收敛速率。我们还研究接受随机样本数量的互信息估计量：首先，我们表明，事前有界样本数量上的互信息估计量（称为“提前停止估计量”）可以比等效的固定样本估计量（针对DMI）实现更低的方差；其次，我们引入基于评分规则的互信息类别，并在该类别中识别出一种互信息，其期望可在三个样本以内估计。
英文摘要
Peer prediction seeks to incentivize agents to truthfully report an observed signal by rewarding joint sets of reports without observing a ground truth. Following the generalization of information-theoretic mutual information introduced in Kong and Schoenebeck (2019), we call a function of a joint distribution over signals a mutual information when it is non-negative and disincentivizes garbling reports for all information structures. An unbiased estimator for a mutual information takes some number of samples from the distribution and returns rewards for both agents, such that the expected reward is equal to the mutual information. We seek to characterize the set of mutual informations with unbiased estimators for a given number of samples. We show that for three or fewer sampled report pairs, the only mutual information with an unbiased estimator is trivially zero, and for four or five samples with a binary report space, the Determinant Mutual Information (DMI) of Kong (2024) is the unique mutual information (up to a scalar multiple). We further show that DMI ceases to be unique at six samples. We provide an improved estimator of DMI for any given number of samples and characterize its convergence rate. We also examine mutual information estimators that accept a randomized number of samples. First, we show that mutual information estimators on an ex-ante bounded number of samples (termed "stop-short estimators") can achieve a lower variance than an equivalent fixed-sample estimator (for DMI). Second, we introduce the class of scoring-rule-based mutual informations and identify in this family a mutual information that can be estimated with under three samples in expectation.
