Learning and Pooling, Pooling and Learning

We explore which types of probabilistic updating commute with convex IP pooling (Stewart and Ojea Quintana 2017). Positive results are stated for Bayesian conditionalization (and a mild generalization of it), imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of (precise) externally Bayesian pooling operators due to Wagner (Log J IGPL 18(2):336-345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.