 At first sight, one might think that probabilistic opinion pooling is free from any troubling impossibility results. However, linear pooling, and thus any opinion pooling rule satisfying the above-stated conditions, has some defects. The first is this. Suppose we require that whenever all individuals consider two propositions \(p\) and \(q\) probabilistically independent (perhaps conditional on a third proposition \(r)\), this probabilistic independence judgment be preserved in the collective probability assignments. A rationale for requiring this is that judgments of probabilistic independence may encode insights about which propositions (or events described by them) are causally relevant or irrelevant to which others. Unfortunately, however, linear pooling (except in its degenerate dictatorial form) does not generally preserve unanimously held conditional independence judgments (Genest and Wagner 1987), and so we would again get an impossibility result if we added this preservation requirement to our list of earlier conditions, together with non-dictatorship.  