No. I mean you can construct prior sets that will result in them moving radically in one direction or another. Exercise: For any epsilon>0, there is a set of priors and hypothesis space containing a hypothesis H such that one can construct two Bayesians who start off with P(H)< eps, and after updating have P(H)>1-eps.
If and only if their priors do not match one another. That’s the whole point of Aumann’s Agreement Theorem. For which there are proofs.
I’m not sure what you mean here. For any epsilon >0 I can start with two Bayesians who share the same priors and have different estimates for different statements (before their probabilities become common knowledge) and a hypothesis H such that that before the updating P(H) < eps, for both and after the updating P(H) > 1- eps for both. I’m not at all sure why you think I need two sets of priors to pull this off. Nothing in this contradicts Aumann’s theorem.
Also, you are wrong about Aumann’s theorem. It isn’t an iff, the implication goes only one way. You can start off with different Bayesians who have different priors and who after updating get the same posteriors. Aumann simply is talking about the case where they have the same priors. It says nothing about what happens if they have different priors. In fact, there are weaker theorems about limiting behavior in certain contexts where the priors disagree but as long as they aren’t too pathological you can get as the number of observations increases they start to agree.
A toy example that may help here:
Assume that there is a coin. Alice and Bob have different priors about this coin. Alice assigns a 25% chance that the coin is fair, a 20% that the coin always turns up heads, a 25% chance that the coin always turns up tailss and a 30% chance that the coin turns up heads 2/3rds of the time. Bob has a 20% chance that the coin is fair, a 25% chance that the coin always turns up heads, a 20% chance that the coin always turns up tails, and a 30% chance that the coin turns up heads 2/3rds of the time. Now, first consider what happens if on the first flip the coin turns up heads. Bob and Alice will now assign zero probability to the possibility that the coin always turns up tails. They now agree on that possibility. Furthermore, assume they keep flipping the coin and observing the results. Then it isn’t that hard to see that as long as the coin actually is one of the four options in the limiting situation Alice and Bob will agree. And you can explicitly state with what probability you should expect any given degree of disagreement between them.
I honestly do not recall having seen it
This suggests to me that you may not be paying that much attention what others (especially I) have written in reply to you. It may therefore may make sense to go back and reread the thread when you have time.
That being said, the fusion power problem is actually a very good example of this. The overwhelming majority of the endeavor in that field has gone into Tokamak-style fusion power generation
This seems to be a valid point. The method of approach doesn’t to some extent look like the approach to anti-aging research in so far as most of the research is focusing on a single method. But I’m not convinced that this argument is that strong. There’s also been a fair bit of research into laser confinement fusion for example. And before it became apparent that they could not be efficient enough to provide power, Farnsworth style fusors were also researched heavily. Industry still researches scaling and making Farnsworth fusors more efficient because they make very nice portable neutron generators that one can turn on and off. So while the majority of the research funding has been going to toroidal magnetic confinement there’s been a very large amount of money put into other types. It is only in the context of percentage of the total that the amount put in looks small.
We’re in a binary state here. I assert P(X)=~1 is true, you assert this is false. These positions are diametrical / binary opposites.
By this definition any two people who disagree about a probability estimate our diametric. This seems like not a good definition if one wants to capture common intuition of the terms. Certainly in contrast for example I don’t think that if you told someone in the general public that “this person thinks that life extension is likely in the next fifty years and this other person considers it to be a near certainity” that they would describe this as diametric opposition.
See also lessdazed’s remark about the difficulty humans have of exchanging all relevant information.
Exchanging relevant information is exceedingly difficult when both parties can relate the same statements of fact and cite the same materials as necessarily resulting in opposing conclusions.
Well, yes. That’s part of the problem. Humans have massive amounts of information that they’ve moved into their background processing. I have pretty decent intuition for certain classes of mathematical problems. But that’s from accumulated experience. I can pretty reliably make conjectures about those classes of problems. But I can’t point explicitly to what is causing me to do so. It is possible that we have differing background sets of data that are impacting our processing at a base level.
I’m not sure what you mean here. For any epsilon >0 I can start with two Bayesians who share the same priors and have different estimates for different statements (before their probabilities become common knowledge) and a hypothesis H such that that before the updating P(H) < eps, for both and after the updating P(H) > 1- eps for both. I’m not at all sure why you think I need two sets of priors to pull this off. Nothing in this contradicts Aumann’s theorem.
Also, you are wrong about Aumann’s theorem. It isn’t an iff, the implication goes only one way. You can start off with different Bayesians who have different priors and who after updating get the same posteriors. Aumann simply is talking about the case where they have the same priors. It says nothing about what happens if they have different priors. In fact, there are weaker theorems about limiting behavior in certain contexts where the priors disagree but as long as they aren’t too pathological you can get as the number of observations increases they start to agree.
A toy example that may help here:
Assume that there is a coin. Alice and Bob have different priors about this coin. Alice assigns a 25% chance that the coin is fair, a 20% that the coin always turns up heads, a 25% chance that the coin always turns up tailss and a 30% chance that the coin turns up heads 2/3rds of the time. Bob has a 20% chance that the coin is fair, a 25% chance that the coin always turns up heads, a 20% chance that the coin always turns up tails, and a 30% chance that the coin turns up heads 2/3rds of the time. Now, first consider what happens if on the first flip the coin turns up heads. Bob and Alice will now assign zero probability to the possibility that the coin always turns up tails. They now agree on that possibility. Furthermore, assume they keep flipping the coin and observing the results. Then it isn’t that hard to see that as long as the coin actually is one of the four options in the limiting situation Alice and Bob will agree. And you can explicitly state with what probability you should expect any given degree of disagreement between them.
This suggests to me that you may not be paying that much attention what others (especially I) have written in reply to you. It may therefore may make sense to go back and reread the thread when you have time.
This seems to be a valid point. The method of approach doesn’t to some extent look like the approach to anti-aging research in so far as most of the research is focusing on a single method. But I’m not convinced that this argument is that strong. There’s also been a fair bit of research into laser confinement fusion for example. And before it became apparent that they could not be efficient enough to provide power, Farnsworth style fusors were also researched heavily. Industry still researches scaling and making Farnsworth fusors more efficient because they make very nice portable neutron generators that one can turn on and off. So while the majority of the research funding has been going to toroidal magnetic confinement there’s been a very large amount of money put into other types. It is only in the context of percentage of the total that the amount put in looks small.
By this definition any two people who disagree about a probability estimate our diametric. This seems like not a good definition if one wants to capture common intuition of the terms. Certainly in contrast for example I don’t think that if you told someone in the general public that “this person thinks that life extension is likely in the next fifty years and this other person considers it to be a near certainity” that they would describe this as diametric opposition.
Well, yes. That’s part of the problem. Humans have massive amounts of information that they’ve moved into their background processing. I have pretty decent intuition for certain classes of mathematical problems. But that’s from accumulated experience. I can pretty reliably make conjectures about those classes of problems. But I can’t point explicitly to what is causing me to do so. It is possible that we have differing background sets of data that are impacting our processing at a base level.