Independence preservation in expert judgment synthesis

We prove, with a few minor exceptions, that if P1 and P2 are probability distributions on the countable set S for which the fixed events E and F are independent, then, both for the standard Euclidean metric and for any metric inducing a topology coarser than the Euclidean topology, there exists a third probability distribution P3 on S that preserves this independence and is equidistant from P1 and P2. We contrast this result with an impossibility theorem from the probability pooling literature, and note its connection with the vigorously debated “epistemic peer problem” in philosophical decision theory.