Less Wrong: Open Thread, September 2010

matt

This thread is for the discussion of Less Wrong topics that have not appeared in recent posts. If a discussion gets unwieldy, celebrate by turning it into a top-level post.

I haven't read your post and my understanding is still hazy, but surely at least the theorems don't depend on the agents being able to fully reconstruct each other's evidence? If they do, then I don't see how it could be true that the probability the agents end up agreeing on is sometimes different from the one they would have had if they were able to share information. In this sort of setting I think I'm comfortable calling it "updating on each other's opinions".

Regardless of Aumann-like results, I don't see how:

one can learn from knowing other people's opinions without knowing their arguments

could possibly be controversial here, as long as people's opinions probabilistically depend on the truth.

You have to also be able to deduce how much of the other agent's information is shared with you. If you and them got your posteriors by reading the same blogs and watching the same TV shows, then this is very different from the case when you reached the same conclusion from completely different channels.

3Wei Dai16y

You're right, sometimes the agreement protocol terminates before the agents fully reconstruct each other's evidence, and they end up with a different agreed probability than if they just shared evidence. But my point was mainly that exchanging information like this by repeatedly updating on each other's posterior probabilities is not any easier than just sharing evidence/arguments. You have to go through these convoluted logical deductions to try to infer what evidence the other guy might have seen or what argument he might be thinking of, given the probability he's telling you. Why not just tell each other what you saw or what your arguments are? Some of these protocols might be useful for artificial agents in situations where computation is cheap and bandwidth is expensive, but I don't think humans can benefit from them because it's too hard to do these logical deductions in our heads. Also, it seems pretty obvious that you can't offload the computational complexity of these protocols onto a third party. The problem is that the third party does not have full information of either of the original parties, so he can't compute the posterior probability of either of them, given an announcement from the other. It might be that a specialized "disagreement arbitrator" can still play some useful role, but I don't see any existing theory on how it might do so. Somebody would have to invent that theory first, I think.

3Perplexed16y

They don't necessarily reconstruct all of each other's evidence, just the parts that are relevant to their common knowledge. For example, two agents have common priors regarding the contents of an urn. Independently, they sample from the urn with replacement. They then exchange updated probabilities for P(Urn has Freq(red)<Freq(black)) and P(Urn has Freq(red)<0.9*Freq(black)). At this point, each can reconstruct the sizes and frequencies of the other agent's evidence samples ("4 reds and 4 blacks"), but they cannot reconstruct the exact sequences ("RRBRBBRB"). And they can update again to perfect agreement regarding the urn contents. Edit: minor cleanup for clarity. At least that is my understanding of Aumann's theorem.