Pooling operators with the marginalization property

In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P 1 , …, P n , over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σ n i = 1 W i P i + (1 ‐ Σ n i = 1 W i ) Q , where Q is an arbitrary measure and W 1 , …, W n ϵ [—1,1] are weights such that| Σ J Σ j w j | ≤ 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 P i , then either T = P i for some 1 ≤ i ≤ n or T = Q , in which case Q takes values in {0, l}.