Combining Experts’ Probabilities via the Product of Odds

We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert’s probability assessment for an event into his/her own posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show that: a sufficient statistic for the all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information that is common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to that of probabilities that honour the principles of uniformity and compromise.