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 Priors 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.
Originally at http://www.overcomingbias.com/2006/12/normative_bayes.html

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