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Pooling operators with the marginalization property. (English) Zbl 0553.62003

This paper is concerned with the finding of a consensus distribution for a group of Bayesians. The author characterizes those pooling procedures which have the marginalization property (i.e. a procedure which yields the same consensus distribution whether it is applied before or after the probability measures have been marginalized), and shows that such a procedure T must be of the form \[ T=\sum^{n}_{1}w_ iP_ i+(1- \sum^{n}_{1}w_ i)Q. \] Here \(P_ 1,...,P_ n\) are the probability measures representing the initial opinions of the Bayesians, Q is an arbitrary measure, \(\forall w_ i\in [\)-1,1] and \(| \sum_{j\in J}w_ j| \leq 1\) for every subset J of \(\{\) 1,2,...,n\(\}\). If, moreover, T is required to preserve independence exhibited by each \(P_ i\), then it is shown that either \(T\equiv P_ i\) for some i, or \(T\equiv Q\), in which case Q takes on values in \(\{\) 0,1\(\}\). The paper contains a good discussion of related work.
Reviewer: A.Dale

MSC:

62A01 Foundations and philosophical topics in statistics
39B99 Functional equations and inequalities
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