I think that I understand this proof now. Does the following dialogue capture it?
AGENT 1: My observations establish that our world is in the world-set S. However, as far as I can tell, any world in S could be our world.
AGENT 2: My observations establish that our world is in the world-set T. However, as far as I can tell, any world in T could be our world.
TOGETHER: So now we both know that our world is in the world-set S ∩ T—though, as far as we can tell, any world in S ∩ T could be our world. Therefore, since we share the same priors, we both arrive at the same value when we compute P(E | S ∩ T), the probability that a given event E occurred in our world.
ETA: janos’s comment indicates that I’m missing something, but I don’t have the time this second to think it through. Sounds like the set that they ultimately condition on isn’t S ∩ T but rather a subset of it.
ETA2: Well, I couldn’t resist thinking about it, even though I couldn’t spare the time :). The upshot is that I don’t understand janos’s comment, and I agree with Psy-Kosh. As stated, for example, in this paper:
The meet [of partitions π1 and π2] has as blocks [i.e., elements] all nonempty intersections of a block from π1 with a block from π2.
From this it follows that the element of I∧J containing w is precisely I(w) ∩ J(w). So, unless I’m missing something, my dialogue above completely captures the proof in the OP.
ETA3: It turns out that both possible ways of orienting the partial order relation are in common use. Everything that I’ve seen discussing the theory of set partitions puts refinements lower in the lattice. This was the convention that I was using above. But, as Vladimir Nesov points out, it’s natural to use the opposite convention when talking about epistemic agents, and this is the usage in Wei Dai’s post. The clash between these conventions was a large part of the cause of my confusion. At any rate, under the convention that Wei Dai is using, the element of I∧J containing w is not in general I(w) ∩ J(w).
Your dialog is one way to achieve agreement, and what I meant when I said “simply tell each other I(w) and J(w)” however it is not what Aumann’s proof is about. The dialog shows that two Bayesians with the same prior would always agree if they exchange enough information.
Aumann’s proof is not really about how to reach agreement, but why disagreements can’t be “common knowledge”. The proof follows a completely different structure from your dialog.
From this it follows that the element of I∧J containing w is precisely I(w) ∩ J(w).
No, this is wrong. Please edit or delete it to avoid confusing others.
From this it follows that the element of I∧J containing w is precisely I(w) ∩ J(w).
No, this is wrong. Please edit or delete it to avoid confusing others.
The implication that I asserted is correct. The confusion arises because both possible ways of orienting the partial order on partitions are common in the literature. But I’ll note that in the comment.
The problem is not in conventions and the literature, but in whether your interpretation captures the statement of the theorem discussed in the post. Ambiguity of the term is no excuse. By the way, “meet” is Aumann’s usage as well, as can be seen from the first page of the original paper.
Indeed. I plead guilty to reading hastily. I saw the term “meet” being used in a context where I already knew its definition (the only definition it had, so far as I knew), so I only briefly skimmed Wei Dai’s own definition. Obviously I was too careless.
However, it really bears emphasizing how strange it is to put refinements higher in the partial order of partitions, at least from the perspective of the general theory of partial orders. Under the category theoretic definition of partial orders, P ≤ Q means that there is a map P → Q. Now, to say that a partition Q is a coarsening of a partition P is to say that Q is a quotient P/~ of P. But such a quotient corresponds canonically to a map P → Q sending each element p of P to the equivalence class in Q containing p. Indeed, Wei Dai is invoking just such maps when he writes “I(w)”. In this case, Ω is construed as the discrete partition of itself (where each element is in its own equivalence class) and I is used (as an abuse of notation) for the canonical map of partitions I: Ω → I. The upshot is that one of these canonical partition maps P → Q exists if and only if Q is a coarsening of P. Therefore, that is what P ≤ Q should mean. In the context of the general theory of partial orders, coarser partitions should be greater than finer ones.
I think that I understand this proof now. Does the following dialogue capture it?
AGENT 1: My observations establish that our world is in the world-set S. However, as far as I can tell, any world in S could be our world.
AGENT 2: My observations establish that our world is in the world-set T. However, as far as I can tell, any world in T could be our world.
TOGETHER: So now we both know that our world is in the world-set S ∩ T—though, as far as we can tell, any world in S ∩ T could be our world. Therefore, since we share the same priors, we both arrive at the same value when we compute P(E | S ∩ T), the probability that a given event E occurred in our world.
ETA: janos’s comment indicates that I’m missing something, but I don’t have the time this second to think it through. Sounds like the set that they ultimately condition on isn’t S ∩ T but rather a subset of it.
ETA2: Well, I couldn’t resist thinking about it, even though I couldn’t spare the time :). The upshot is that I don’t understand janos’s comment, and I agree with Psy-Kosh. As stated, for example, in this paper:
From this it follows that the element of I∧J containing w is precisely I(w) ∩ J(w). So, unless I’m missing something, my dialogue above completely captures the proof in the OP.
ETA3: It turns out that both possible ways of orienting the partial order relation are in common use. Everything that I’ve seen discussing the theory of set partitions puts refinements lower in the lattice. This was the convention that I was using above. But, as Vladimir Nesov points out, it’s natural to use the opposite convention when talking about epistemic agents, and this is the usage in Wei Dai’s post. The clash between these conventions was a large part of the cause of my confusion. At any rate, under the convention that Wei Dai is using, the element of I∧J containing w is not in general I(w) ∩ J(w).
Your dialog is one way to achieve agreement, and what I meant when I said “simply tell each other I(w) and J(w)” however it is not what Aumann’s proof is about. The dialog shows that two Bayesians with the same prior would always agree if they exchange enough information.
Aumann’s proof is not really about how to reach agreement, but why disagreements can’t be “common knowledge”. The proof follows a completely different structure from your dialog.
No, this is wrong. Please edit or delete it to avoid confusing others.
The implication that I asserted is correct. The confusion arises because both possible ways of orienting the partial order on partitions are common in the literature. But I’ll note that in the comment.
The problem is not in conventions and the literature, but in whether your interpretation captures the statement of the theorem discussed in the post. Ambiguity of the term is no excuse. By the way, “meet” is Aumann’s usage as well, as can be seen from the first page of the original paper.
Indeed. I plead guilty to reading hastily. I saw the term “meet” being used in a context where I already knew its definition (the only definition it had, so far as I knew), so I only briefly skimmed Wei Dai’s own definition. Obviously I was too careless.
However, it really bears emphasizing how strange it is to put refinements higher in the partial order of partitions, at least from the perspective of the general theory of partial orders. Under the category theoretic definition of partial orders, P ≤ Q means that there is a map P → Q. Now, to say that a partition Q is a coarsening of a partition P is to say that Q is a quotient P/~ of P. But such a quotient corresponds canonically to a map P → Q sending each element p of P to the equivalence class in Q containing p. Indeed, Wei Dai is invoking just such maps when he writes “I(w)”. In this case, Ω is construed as the discrete partition of itself (where each element is in its own equivalence class) and I is used (as an abuse of notation) for the canonical map of partitions I: Ω → I. The upshot is that one of these canonical partition maps P → Q exists if and only if Q is a coarsening of P. Therefore, that is what P ≤ Q should mean. In the context of the general theory of partial orders, coarser partitions should be greater than finer ones.