Normative Bayesianism and Disagreement
Normative Bayesianism says that you ought to believe as you would if you were an ideal Bayesian believer and so believing is what it is to believe rationally. An ideal Bayesian believer has (1) beliefs by having credences, where a credence is a degree of belief in a proposition; (2) has a Prior = a complete consistent set of credences (capitalized to avoid confusing priors = a person’s credences with Priors = a plurality of complete consistent sets of credences), that is to say, has a credence function from the sigma algebra of propositions into the reals such that the credence function is a measure that is a probability function; (3) changes his beliefs on the basis of the evidence he has acquired by updating his credence function by the use of Bayes’ theorem.
Much of the earlier discussion about the rationality of disagreement and the requirement of modesty was advanced on the basis of the claim that Bayesian believers cannot rationally disagree. But there are different versions of what precisely that claim might be.
Strong Bayesian Agreement: Ideal Bayesian believers who have common knowledge of each others opinion of a proposition agree on that proposition.
Moderate Bayesian Agreement: Ideal Bayesian believers who have rational Priors and common knowledge of each others opinion of a proposition agree on that proposition.
Weak Bayesian Agreement: Ideal Bayesian believers who have a common Prior and common knowledge of each others opinion of a proposition agree on that proposition.
I think it is clear that WBA does not supply much support for the claim that disagreement cannot be rational or that modesty is a requirement, simply because it does not rule out the possibility of rational differences in Priors. However, and provided we take them to be universally quantified, that is to say, that they apply to any propositions whatsoever of which we have common knowledge, MBA and SBA each suffice for Normative Bayesianism to imply that there can be no reasonable disagreement. .
Aumann’s theorem in Agreeing to Disagree says nothing about the rationality of Priors. His theorem says only that if two people have the same Prior and their posteriors for an event are common knowledge, then those posteriors are the same. So Aumann’s theorem proves only WBA. If we can supplement Aumann’s theorem with a proof that there is only one Bayesianly rational Prior then together they will give us SBA. Since there is only one Bayesianly rational Prior, ideal Bayesian believers will all have that Prior, and hence by Aumann’s theorem they will agree on any propositions of which they have common knowledge. This will also give us MBA, because uniqueness of rational Prior collapses it into SBA.
What might work is if we define the rationality of Priors in terms of satisfying MBA, and that would certainly fit with the general coherentism of Bayesianism. For example, a rational Prior is one from a set of Priors for which, anyone who has a Prior from that set will agree with anyone else with a Prior from that set on a proposition of which they have common knowledge, and there is no other set of Priors for which this is true. The final clause is needed since without it every singleton set satisfies the definition, and what we need is a unique set of Priors to be the set of Rational Priors. On this definition, the set of Rational Priors could not be a singleton, so MBA would not collapse into SBA.