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Perplexed comments on Confidence levels inside and outside an argument - Less Wrong
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Comments (174)
The problem I was specifically asking to solve is "what if Bayesian updating is flawed", which I thought was an appropriate discussion on an article about not putting all your trust in any one system.
Bayes theorem looks solid, but I've been wrong about theorems before. So has the mathematical community (although not very often and not for this long, but it could happen and should not be assigned 0 probability). I'm slightly sceptical of the uniqueness claim, given I've often seen similar proofs which are mathematically sound, but make certain assumptions about what it allowed, and are thus vulnerable to out-of-the-box solutions (Arrow's impossibility theorem is a good example of this). In fact, given that a significant proportion of statisticians are not Bayesians, I really don't think this is a good time for absolute faith.
To give another example, suppose tomorrow's main page article on LW is about an interesting theorem in Bayesian probability, and one which would affect the way you update in certain situations. You can't quite understand the proof yourself, but the article's writer is someone whose mathematical ability you respect. In the comments, some other people express concern with certain parts of the proof, but you still can't quite see for yourself whether its right or wrong. Do you apply it?
You have piqued my curiosity. A trick to get around Arrow's theorem? Do you have a link?
Regarding your main point: Sure, If you want some members of your army of mutant rational agents to be so mutated that they are no longer even Bayesians, well ... go ahead. I suppose I have more faith in the rough validity of trial-and-error empiricism than I do in Bayes's theorem. But not much more faith.
I'm afraid I don't know how to post links.
I think there is already a main-page article on this subject, but the general idea is that Arrow's theorem assumes the voting system is preferential (you vote by ranking voters) and so you can get around it with a non-preferential system.
Range voting (each voter gives each candidate as score out of ten, and the candidate with the highest total wins) is the one that springs most easily to mind, but it has problems of its own, so somebody who knows more about the subject can probably give you a better example.
As for the main point, I doubt you actually put 100% confidence in either idea. In the unlikely event that either approach led you to a contradiction, would you just curl up in a ball and go insane, or abandon it.
Ah. You mean this posting. It is a good article, and it supports your point about not trusting proofs until you read all of the fine print (with the warning that there is always some fine print that you miss reading).
But it doesn't really overthrow Arrow. The "workaround" can be "gamed" by the players if they exaggerate the differences between their choices so as to skew the final solution in their own favor.
All deterministic non-dictatorial systems can be gamed to some extent (Gibbard Satterthwaite theorem, I'm reasonably confident that this one doesn't have a work-around) although range voting is worse than most. That doesn't change the fact that it is a counter-example to Arrow.
A better one might be approval voting, where you have as many votes as you want but you can't vote for the same candidate more than once (equivalent to a the degenerate case of ranging where there are only two rankings you can give.
Thanks for the help with the links.
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