Pooling operators with the marginalization property

Abstract In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P1, …, Pn, over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σni=1Wi Pi + (1 ‐ Σni =1Wi)Q, where Q is an arbitrary measure and W1, …, Wn ϵ [—1,1] are weights such that| ΣJ Σ j wj | ≤ 1 for every subset J of {1, …, n}. If, in addition, T is required to preserve the independence of arbitrary events A and B whenever these events are independent under each Pi, then either T = Pi for some 1 ≤ i ≤ n or T = Q, in which case Q takes values in {0, l}.