The Sample Complexity of Forecast Aggregation
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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 experts' reports (not signals) and the realization of the event. Using these samples, the principal aims to find an ε-approximately optimal (Bayesian) aggregator. We study the sample complexity of this problem. We show that, for arbitrary discrete distributions, the number of samples must be at least Ω̃(m^n-2 / ε), where m is the size of each expert's signal space. This sample complexity grows exponentially in the number of experts n. But if experts' signals are independent conditioned on the realization of the event, then the sample complexity is significantly reduced, to Õ(1 / ε^2), which does not depend on n.
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