A Further Look at the Bayes Blind Spot

Gyenis and Rédei (G&R) have shown that any prior p on a finite algebra A, however chosen, significantly restricts the set of posteriors derivable from p by Jeffrey conditioning (JC) on a nontrivial measurable partition (i.e., a partition consisting of members of A, at least one of which is not an atom of A). They support this claim by proving that the set of potential posteriors not derivable from p in this way, which they call the Bayes blind spot of p, is large, having cardinality c and normalized Lebesgue measure 1, as well as being of second Baire category for a natural metrizable topology. In the present paper, we establish results analogous to those of G&R for probability measures on any infinite sigma algebra of subsets of a countably infinite set (which requires distinctly different treatments of the topological and measure-theoretic cases). We also show, in both the finite and infinite cases, that all of the limitative results for a single prior p continue to hold for the intersection of the Bayes blind spots of countably many priors. This leads us to reject the claim of G&R that the large size of blind spots in the single prior case is attributable to the limitations imposed by priors. We argue instead that it is the so-called rigidity property of JC that accounts for the large size of Bayes blind spots. G&R also prove that any potential posterior q can be derived from a prior p by at most two applications of JC on nontrivial partitions. But they remark that their particular two-stage derivation of an \(r \in BS(p)\) $r\in BS(p)$ from p is barely distinguishable from a single, complete reassessment on a trivial partition. We show, however, that there are two-stage derivations of r from p that require no direct assessment of the probability of any atom at either stage. Finally, in order to situate the aforementioned limitative results in the proper context, we demonstrate that a probability revision that would amount to complete reassessment within a Jeffrey evidentiary framework may arise, in a different evidentiary framework, in a way that involves no direct assessment of the probability of any atom, as, for example, in a generalization of Jeffrey’s solution to the problem of old evidence and new explanation.