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benelliott comments on Confidence levels inside and outside an argument - Less Wrong
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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.