Aggregating Forecasts

How can we benefit from the wisdom of the crowd? How should we pool diverse expert opinions? And how can we do so in a precise, quantitative way?

In this sequence I set out to answer these questions.

The precise problem we are trying to solve is this one:

We are interested in the outcome of an event - an election, a war, the development of a technology or something else.
We consult a number of experts, with access to different information. Each of them is a calibrated forecaster, and elicits a probability distribution over possible outcomes of the event.

How should we combine these probability distribution to get an accurate estimate of the event outcomes?  

For example, they could be forecasting an event that might happen or not. Their predictions would be a single probability, reflecting how likely they think is the event. And we might aggregate these by taking an average. Can we do better?