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benelliott comments on Confidence levels inside and outside an argument - Less Wrong
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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?
Assign a probability 1-epsilon to your belief that Bayesian updating works. Your belief in "Bayesian updating works" is determined by Bayesian updating; you therefore believe with 1-epsilon probability that "Bayesian updating works with probability 1-epsilon". The base level belief is then held with probability less than 1-epsilon.
As the recursive nature of holding Bayesian beliefs about believing Bayesianly allows chains to tend toward large numbers, the probability of the base level belief tends towards zero.
There is a flaw with Bayesian updating.
I think this is just a semi-formal version of the problem of induction in Bayesian terms, though. Unfortunately the answer to the problem of induction was "pretend it doesn't exist and things work better", or something like that.
Thank-you for expressing my worry in much better terms than I managed to. If you like, I'll link to your comment in my top-level comment.
I still don't know why everyone thinks this is the problem of induction. You can certainly have an agent which is Bayesian but doesn't use induction (the prior which assigns equal probability to all possible sequences of observation is non-inductive). I'm not sure if you can have a non-Bayesian that uses induction, because I'm very confused about the whole subject of ideal non-Bayesian agents, but it seems like you probably could.
Interesting that Bayesian updating seems to be flawed if an only if you assign non-zero probability to the claim that is flawed. If I was feeling mischievous I would compare it to a religion, it works so long as you have absolute faith, but if you doubt even for a moment it doesn't.
It's similar to Hume's philosophical problem of induction (here and here specifically). Induction in this sense is contrasted with deduction - you could certainly have a Bayesian agent which doesn't use induction (never draws a generalisation from specific observations) but I think it would necessarily be less efficient and less effective than a Bayesian agent that did.
Feel free! I am all for increasing the number of minds churning away at this problem - the more Bayesians that are trying to find a way to justify Bayesian methods, the higher the probability that a correct justification will occur. Assuming we can weed out the motivated or biased justifications.