A statistical approach to the representation of uncertainty in beliefs using spread of opinions

Reasoning with uncertainty is a field with many different approaches and viewpoints, with important applications to sensor design and autonomous system development. It is important to have calculi for propagating measures of "probability" or "likelihood" even in cases of subjective information, and it is just as important to be able to propagate the "certitude" of this information. By choosing the semantics properly, this information can be handled by keeping track of certain statistics on a different probability space, (which we will call the opinion space). The semantics assume that the "likelihood" or "probability numbers" are in fact averages over many (perhaps subjective) opinions and that uncertainty is represented by the spread in these opinions, which can be technically maintained by a covariance matrix. Different calculi result from different design choices consistent with this choice of semantics. While there are several new calculi that can be developed this way, it also turns out that certain mechanisms that are frequently considered "non-Bayesian", such as the Dempster combination formula or Kalman filtering, result from specific choices for representing the statistics and dependency assumptions.