We consider a Bayesian forecast aggregation model where $n$ experts, after observing private signals about an unknown binary event, report their posterior beliefs about the event to a principal, who then aggregates the reports into a single prediction for the event. The signals of the experts and the outcome of the event follow a joint distribution that is unknown to the principal, but the principal has access to i.i.d. "samples" from the distribution, where each sample is a tuple of the experts' reports (not signals) and the realization of the event. Using these samples, the principal aims to find an $\varepsilon$-approximately optimal aggregator, where optimality is measured in terms of the expected squared distance between the aggregated prediction and the realization of the event. We show that the sample complexity of this problem is at least $\tilde \Omega(m^{n-2} / \varepsilon)$ for arbitrary discrete distributions, where $m$ is the size of each expert's signal space. This sample complexity grows exponentially in the number of experts $n$. But, if the experts' signals are independent conditioned on the realization of the event, then the sample complexity is significantly reduced, to $\tilde O(1 / \varepsilon^2)$, which does not depend on $n$. Our results can be generalized to non-binary events. The proof of our results uses a reduction from the distribution learning problem and reveals the fact that forecast aggregation is almost as difficult as distribution learning.

翻译：我们考虑一个贝叶斯预测聚合模型，其中 $n$ 位专家在观察到关于未知二元事件的私有信号后，向委托人报告其对该事件的后验信念，委托人随后将这些报告聚合成一个关于该事件的单一预测。专家的信号及事件结果遵循一个联合分布，该分布对委托人而言是未知的，但委托人可以获取该分布中独立同分布的"样本"，每个样本包含专家报告（而非信号）及事件实现的元组。利用这些样本，委托人旨在找到一个 $\varepsilon$-近似最优聚合器，其中最优性通过聚合预测与事件实现之间的期望平方距离来衡量。我们证明，对于任意离散分布，该问题的样本复杂度至少为 $\tilde \Omega(m^{n-2} / \varepsilon)$，其中 $m$ 是每位专家信号空间的大小。该样本复杂度随专家数量 $n$ 呈指数增长。然而，若专家信号在事件实现条件下相互独立，则样本复杂度显著降低至 $\tilde O(1 / \varepsilon^2)$，且与 $n$ 无关。我们的结果可推广至非二元事件。证明过程采用了从分布学习问题出发的归约方法，揭示了预测聚合几乎与分布学习同等困难这一事实。