Alternative Frameworks for the Reconciliation of Probability Assessments

Abstract
The problem of a probability assessor who has, directly or indirectly, given two different numbers for the probability of one event is addressed. In order to use these assessments in, for example, a decision analysis, these two numbers must be reconciled to give one value as the probability. A detailed mathematical exposition is given concerning the calculation of the least-squares reconciliation proposed by Lindley, Tversky and Brown (1979), in both probability and log-odds metrics. It is shown that the method of reconciliation they have proposed is formally equivalent to that of taking a weighted average of log-odds, with weights proportional to the independent information content of each assessment. This method has the advantage of being simple in application. It is further argued that the motivation for taking multiple probability assessments is an attempt to obtain more information from the subject, and that the proposed method of reconciliation captures the essence of this motivation. The relationship between this research and the well-known 'expert-use' problem is explored. Finally, a discussion of alternative potential approaches to the reconciliation problem is included.

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