Maximum entropy aggregation of individual opinions

It is shown that Bordley's result is overly restrictive when looked at in terms of maximum entropy derivations that are clearly using more information.

Abstract

This paper presents several formulations for aggregating opinions about the outcome of a future event-e.g., opinions in the form of probabilities are aggregated into a single probability. The approach used to derive these aggregation formulas is maximum entropy inference, with the assumption that the opinions and the event being predicted are joint random variables. As part of the presentation, the novel, axiomatically developed aggregation formula due to Bordley is used as a foil for contrasting and comparing the results developed here. It is shown that Bordley's result is overly restrictive when looked at in terms of maximum entropy derivations that are clearly using more information. Initially, we suppose that the opinions given on the future event are in the form of odds; then we consider the case where these opinions are in the form of probabilities; and finally, we reach a general formulation in which they could be odds, probabilities or even binary, yes/no, predictions of a future event with interactions between experts. In regard to this last point, the use of Darroch and Ratcliff's method in maximum entropy aggregation is sketched.>