Probabilistic opinion pooling generalized Part two: The premise-based approach

How can different individuals' probability functions on a given σ-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require `event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be `basic' and others `derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize this idea by introducing a `premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the `premises'.