Note: Bayesian consistent belief selection

Chambers, Christopher P., and Takashi Hayashi. "Bayesian consistent belief selection."Journal of Economic Theory 145.1 (2010): 432-439.

這篇

mark一下，講了subjective情況下貝葉斯更新的不可能性

「This may have implications for the common prior doctrine for example (which maintains that agents who have the same objective information should hold the same prior belief).」

: finite set of all states.

event

Belief Selection Problem:

all BSPs

$forall E,forall Fsubset E, forall Pin K(E),forall P^Fin K(F):P^F:={frac{P}{p(F)}|_{2^F}:pin P }$

is simply that set of probabilities that results from updating P prior-by-prior, restricted so that their support lies in F.

Belief Selection Rule: a function $g:X o cup_{E} Delta(E)$ such that

We say a BSR g is Bayesian consistent if $forall (E,P)in X,forall Fsubset E:g(E,P)(F) eq emptyset,g(F,P^F)=frac{g(E,P)}{g(E,P)(F)}$

Assumption 1: The totality of objective information available to the decision maker in forming a belief is summarized by a set of probability measures

Under A1, the objective informational content of learning an event has obtained must be contained solely in some set of probability measures.

Assumption 2: The only informational content of learning an event has obtained is contained in the set of updated probability measures

Assumption 3.Upon the revelation of information, the decision maker must update her subjective probability according to Bayes rule.

Bayesian consistency thus postulates the conjunction of A1-3. It requires that a decision maker using Bayes rule does not come into conflict with her pre-specified selection rule upon learning an event has obtained, when the only informational content of learning is the set of updated probability measures.

THM: if then there exists no Bayesian consistent BSR g.

The THM above demonstrates that any Bayesian decision maker who forms a belief when given imprecise information must use something more than just the set of probability measures available at hand.

A consequence of this result is the fact that the timing of the resolution of subjective uncertainty is relevant for understanding the beliefs of a decision maker. In forming beliefs, it is not only relevant what 「objective」 information is available to them, but also how they arrived at that information.