When a group confronts a decision involving uncertainty, different individuals may have different (probabilistic) beliefs about the world. In order for the group to make rational decisions (i.e. to maximize subjective expected utility), we must construct probabilistic beliefs for the group as a whole. Probabilistic belief aggregation rules (also called opinion pooling rules) are functions that "aggregate" the individuals' disparate beliefs into a single collective belief. An important class of such rules are the geometric (or logarithmic) aggregation rules, in which the collective belief is a (renormalized) geometric average of the individuals' beliefs. These rules have many nice properties, but they have three major shortcomings:

1. Existing definitions of geometric aggregation for uncountably infinite spaces involve a major domain restriction: they assume that all probabilities are absolutely continuous with respect to a common "reference measure".

2. Geometric aggregation is only well-defined for profiles that are coherent, meaning that the beliefs of the agents have overlapping support.

3. In many decision problems, agents do not have a clear idea of the space of possible "states of nature"; they only have in mind some set of statements (each of which could be either true or false). Formally, their beliefs take the form of a probability measure defined on an abstract Boolean algebra. But geometric aggregation is not well-defined for these kinds of beliefs.

In this presentation, I will discuss an extended version of geometric aggregation that solves all three problems.