A Further Look at the Bayes Blind Spot

TL;DR

This paper extends the analysis of the Bayes blind spot from finite to infinite sigma algebras, revealing that large blind spots persist and may impact Bayesian learning.

Contribution

It generalizes previous finite results to infinite settings using new approaches, and shows that large blind spots remain for countable intersections of priors.

Findings

Large Bayes blind spots exist in infinite sigma algebras.

Results hold for intersections of multiple priors.

Bayesian learning may be inherently limited by these blind spots.

Abstract

Gyenis and Redei have demonstrated that any prior p on a finite algebra, however chosen, severely restricts the set of posteriors accessible from p by Jeffrey conditioning on a nontrivial partition. Their demonstration involves showing that the set of posteriors not accessible from p in this way (which they call the Bayes blind spot of p) is large with respect to three common measures of size, namely, having cardinality c, (normalized) Lebesgue measure 1, and Baire second category with respect to a natural topology. In the present paper, we establish analogous results for probability measures defined on any infinite sigma algebra of subsets of a denumerably infinite set. However, we have needed to employ distinctly different approaches to determine the cardinality, and especially, the topological and measure-theoretic sizes of the Bayes blind spot in the infinite case. Interestingly, all of the results that we establish for a single prior p continue to hold for the intersection of the Bayes blind spots of countably many priors. This leads us to conjecture that Bayesian learning itself might be just as culpable as the limitations imposed by priors in enabling the existence of large Bayes blind spots.