Aggregating opinions through logarithmic pooling (Q794889)

This paper deals with the problem of finding a reasonable formula for synthesizing or typifying the opinions of a group of n individuals. The pooling formula, which is of the type called a logarithmic pool, is characterized by a single axiom on the pooling operator called ''Relative Propensity Consistency''. If presented with n prior density functions, \(f_ 1,f_ 2,...,f_ n>0\), it would return a combined opinion of the form \(\Pi f_ i^{a_ i}\) where \(a_ i>0\), \(\sum a_ i=1\) are unspecified constants. However, the domain of the operator is not confined to densities and so would admit vague priors, for example. Another characterization of the pooling operator is also given using two axioms called ''prior-to-posterior coherence'' and ''Pareto criterion''. The first author, K. J. McConway and Schervish (in an unpublished technical report) have subsequently derived a general form of the pooling operator under the axiom of ''external Bayesianity''. And preliminary results of the second and the third author indicate that the logarithmic pool is implied by two weak axioms called the ''dominance principle'' and ''independence of irrelevent alternatives''. Although both yield logarithmic opinion pools these two unpublished analyses are different and each applies in situations where the other does not.