Characterization of externally Bayesian pooling operators. (English) Zbl 0602.62005

Genest, Christian; McConway, Kevin J.; Schervish, Mark J.

Ann. Stat. 14, 487-501 (1986).

A pooling operator is a function \(T: \Delta\) \({}^ n\to \Delta\), \(\Delta\) is the class of \(\mu\)-measurable functions \(f: \Theta\to [0,\infty)\) such that \(f>0\mu\)-a.e. and \(\int f d\mu =1\), (\(\Theta\),\(\mu)\) is a measurable space, which may be used to extract a ”consensus” \(T(f_ 1,...,f_ n)\) from the different subjective opinions of the n members of a group of experts. In the case in which this group is asked to give a suggestion in form of a group probability distribution, it may be asked whether they should synthesize their opinions before or after they learn the outcome of an experiment. If the group posterior distribution is the same whatever the order in which the pooling and the updating are done, the pooling is said to be externally Bayesian.
In the paper the authors characterize all externally Bayesian pooling formulae and give conditions under which the opinion of the group will be proportional to the geometric average of the individual densities.

Reviewer: D.Costantini

MSC:

62A01	Foundations and philosophical topics in statistics
62C10	Bayesian problems; characterization of Bayes procedures

Keywords:

characterization; group decision problem; logarithmic pool; expert opinions; pooling operator; consensus; group probability distribution; group posterior distribution; externally Bayesian; geometric average of the individual densities