Previously, we saw how the Bayesian notion of “degrees of belief” offers a possible solution to the preface paradox. Here, we shall consider some philosophical or epistemic objections to this idea of “degrees of belief.” In his thought-provoking and beautiful 2015 Nous paper, for example, Kenny Easwaran poses a number of open questions regarding the nature of degrees of belief, all of which (we think) boil down to the two following queries:

Question 1: What is the difference between an old-fashioned and plain and simple “belief” and a highfalutin Bayesian “degree of belief”? In particular, is there some threshold or cut-off point (say, .9 or .95 or .99) above which a degree of belief acts like a full-fledged belief? (A related question we have is this: do we even need the notion of degrees of belief? After all, isn’t a regular or ordinary belief just as subjective and susceptible to Bayesian updating as a degree of belief is?)

Question 2: In actual human reasoning and daily practice, are degrees of belief “infinitely precise real numbers” (e.g., exact numerical values ranging from 0 to 1) or “something less precise” (e.g., high, medium, and low)? In other words, can a degree of belief really be expressed in precise numerical terms, and if so, how? Aren’t we just plucking numbers out of thin air?

This second question is especially delicate. If it turns out that for whatever reason we cannot assign a precise numerical value to a degree of belief, then we won’t be able to transpose the axioms of probability into the Bayesian framework, and the Bayesian view of probability collapses like a house of cards. In any case, thus far it has taken us three separate blog posts to summarize the first three pages of Easwaran’s 38-page paper. We hope to discuss the rest of his paper in future posts (most likely after the Labor Day holiday).

Credit: Zoubin Ghahramani, via SlideShare