Having a look at the Bayes Blind Spot

Abstract

The Bayes Blind Spot of a Bayesian Agent is, by definition, the set of probability measures on a Boolean sigma -algebra that are absolutely continuous with respect to the background probability measure (prior) of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by a single conditionalization no matter what (possibly uncertain) evidence he has about the elements in the Boolean sigma -algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has the same cardinality as the set of all probability measures (continuum); it has the same measure as the measure of the set of all probability measures (in the natural measure on the set of all probability measures); and is a “fat” (second Baire category) set in topological sense in the set of all probability measures taken with its natural topology. Features of the Bayes Blind Spot are determined from the perspective of repeated Bayesian learning when the Boolean algebra is finite. Open problems about the Bayes Blind Spot are formulated in probability spaces with infinite Boolean sigma -algebras. The results are discussed from the perspective of Bayesianism.