3.5 - Characterization of Externally Bayesian Pooling Operators

Summary

ABSTRACT

When a panel of experts is asked to provide some advice in the form of a group probability distribution, the question arises as to 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 mechanism is said to be externally Bayesian by Madansky (1964). In this chapter, we characterize all externally Bayesian pooling formulas and we give conditions under which the opinion of the group will be proportional to the geometric average of the individual densities.

INTRODUCTION

Let (Θ,µ) be a measure space and let Δ be the class of µ-measurable functions f:Θ→ (0,∞) such that f > 0 µ-a.e. and ∫fdµ = 1. In the language of multiagent statistical decision theory (cf. Weerahandi and Zidek, 1981), a pooling operator is any function T: AⁿΔ → Δ which may be used to extract a “consensus” T (f₁,…,f_n) from the different subjective opinions f₁,…,f_n ∈ Δ of the n members of a group. The current interest for pooling operators seems to stem from a theorem due to Wald (1939) concerning the optimality of Bayesian decision rules.