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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 a personal 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 all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information 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 the class of combining functions that honour the principles of uniformity and compromise.

This paper displays the productive role of the judgment of exchangeability and even conditional exchangeability that should replace the misleading assertion of independence of various experts' opinions regarding uncertain situations. The application is technically rather complicated, and a basic understanding of exchangeability in its simple applications is presumed. One component of the analysis is novel. It suggests how we might specify a likelihood function that allows us to learn about one exchangeable sequence of events from the outcomes of another exchangeable sequence. The substantive content of the the paper concerns the use of personal probabilities by two experts in assessing the sex of human skulls found in anthropological investigations. Although we initially value the two experts' assertions exchangeably, we learn to value one of the expert's assertions more than the other's. Moreover we identify precisely how much to value the elicitation from a second expert after we have already learned the assertion of the first. Valuation is based on a decision theoretic procedure assessing reduction in risk.

It is known that the problem of combining a number of expert probability evaluations is frequently solved with additive or multiplicative rules. In this paper we try to show, with the help of a behavioural model, that the additive rule (or linear pooling) derives from the application of Bayesian reasoning. In another occasion we will discuss the multiplicative rule.