A Characterization Theorem for Externally Bayesian Groups

A contribution is made to the problem of combining the subjective probability density functions f₁, ⋯, fₙ of n individuals for some parameter θ. More precisely, the situation is addressed which occurs when the members of a group share a common likelihood for some data and want to ensure that combining their posterior distributions for θ will yield the same result obtained by applying Bayes' rule to the aggregated prior distribution. Under certain regularity conditions to be discussed below, the logarithmic opinion pool ∏ⁿᵢ₌₁ fwᵢᵢ / ∫ ∏ⁿᵢ₌₁ fwᵢᵢ dμ with wᵢ ≥ 0 and ∑ⁿᵢ₌₁ wᵢ = 1 is shown to be the only pooling formula which satisfies this criterion of group rationality.