No, under the current formalization, even if we are not in class C we have to trust A[C] over our own beliefs. Specifically, we need Eus[X∣info]=Eus[EA[C][X]∣info] for any X and information about A[C] . But then if we are given the info that EA[C][X]=Y, then we have:
Eus[X∣info]
=Eus[EA[C][X]∣info] (definition of universality)
=Eus[EA[C][X]∣EA[C][X]=Y] (plugging in the specific info we have)
=Y (If we are told that A[C] says Y, then we should expect that A[C] says Y)
Putting it together, we have Eus[X∣info]=Y, that is, given the information that A[C] says Y, we must expect that the answer to X is Y.
This happens because we don’t have an observer-independent way of defining epistemic dominance: even if we have access to the ground truth, we don’t know how to take two sets of beliefs and say “belief set A is strictly ‘better’ than this belief set B” [1]. So what we do here is say “belief set A is strictly ‘better’ if this particular observer always trusts belief set A over belief set B”, and “trust” is defined as “whatever we think belief set A believes is also what we believe”.
You could hope that in the future we have an observer-independent way of defining epistemic dominance, and then the requirement that we adopt A[C]’s beliefs would go away.
We could say that a set of beliefs is ‘strictly better’ if for every quantity X its belief is more accurate, but this is unachievable, because even full Bayesian updating on true information causes you to update in the wrong direction for some quantities, just by bad luck. ↩︎
Hmm, maybe I’m missing something basic and should just go re-read the original posts, but I’m confused by this statement:
So what we do here is say “belief set A is strictly ‘better’ if this particular observer always trusts belief set A over belief set B”, and “trust” is defined as “whatever we think belief set A believes is also what we believe”.
In this, belief set A and belief set B are analogous to A[C] and C (or some c in C), right? If so, then what’s the analogue of “trust… over”?
If we replace our beliefs with A[C]’s, then how is that us trusting it “over” c or C? It seems like it’s us trusting it, full stop (without reference to any other thing that we are trusting it more than). No?
In this, belief set A and belief set B are analogous to A[C] and C (or some c in C), right?
Yes.
If we replace our beliefs with A[C]’s, then how is that us trusting it “over” c or C? It seems like it’s us trusting it, full stop
So I only showed the case where info contains information about A[C]’s predictions, but info is allowed to contain information from A[C]andC (but not other agents). Even if it contains lots of information from C, we still need to trust A[C].
In contrast, if info contained information about A[A[C]]’s beliefs, then we would not trust A[C] over that.
No, under the current formalization, even if we are not in class C we have to trust A[C] over our own beliefs. Specifically, we need Eus[X∣info]=Eus[EA[C][X]∣info] for any X and information about A[C] . But then if we are given the info that EA[C][X]=Y, then we have:
Eus[X∣info]
=Eus[EA[C][X]∣info] (definition of universality)
=Eus[EA[C][X]∣EA[C][X]=Y] (plugging in the specific info we have)
=Y (If we are told that A[C] says Y, then we should expect that A[C] says Y)
Putting it together, we have Eus[X∣info]=Y, that is, given the information that A[C] says Y, we must expect that the answer to X is Y.
This happens because we don’t have an observer-independent way of defining epistemic dominance: even if we have access to the ground truth, we don’t know how to take two sets of beliefs and say “belief set A is strictly ‘better’ than this belief set B” [1]. So what we do here is say “belief set A is strictly ‘better’ if this particular observer always trusts belief set A over belief set B”, and “trust” is defined as “whatever we think belief set A believes is also what we believe”.
You could hope that in the future we have an observer-independent way of defining epistemic dominance, and then the requirement that we adopt A[C]’s beliefs would go away.
We could say that a set of beliefs is ‘strictly better’ if for every quantity X its belief is more accurate, but this is unachievable, because even full Bayesian updating on true information causes you to update in the wrong direction for some quantities, just by bad luck. ↩︎
Hmm, maybe I’m missing something basic and should just go re-read the original posts, but I’m confused by this statement:
In this, belief set A and belief set B are analogous to A[C] and C (or some c in C), right? If so, then what’s the analogue of “trust… over”?
If we replace our beliefs with A[C]’s, then how is that us trusting it “over” c or C? It seems like it’s us trusting it, full stop (without reference to any other thing that we are trusting it more than). No?
Yes.
So I only showed the case where info contains information about A[C]’s predictions, but info is allowed to contain information from A[C] and C (but not other agents). Even if it contains lots of information from C, we still need to trust A[C].
In contrast, if info contained information about A[A[C]]’s beliefs, then we would not trust A[C] over that.