  We consider the forecast aggregation problem in repeated settings, where the
forecasts are done on a binary event. At each period multiple experts provide
forecasts about an event. The goal of the aggregator is to aggregate those
forecasts into a subjective accurate forecast. We assume that experts are
Bayesian; namely they share a common prior, each expert is exposed to some
evidence, and each expert applies Bayes rule to deduce his forecast. The
aggregator is ignorant with respect to the information structure (i.e.,
distribution over evidence) according to which experts make their prediction.
The aggregator observes the experts' forecasts only. At the end of each period
the actual state is realized. We focus on the question whether the aggregator
can learn to aggregate optimally the forecasts of the experts, where the
optimal aggregation is the Bayesian aggregation that takes into account all the
information (evidence) in the system.
  We consider the class of partial evidence information structures, where each
expert is exposed to a different subset of conditionally independent signals.
Our main results are positive; We show that optimal aggregation can be learned
in polynomial time in a quite wide range of instances of the partial evidence
environments. We provide a tight characterization of the instances where
learning is possible and impossible.
