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gwern 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.
I think this is a form of double-counting the same evidence. You can only perform Bayesian updating on information that is new; if you try to update on information that you've already incorporated, your probability estimate shouldn't move. But if you take information you've already incorporated, shuffle the terms around, and pretend it's new, then you're introducing fake evidence and get an incorrect result. You can add a term for "Bayesian updating might not work" to any model, except to a model that already accounts for that, as models of the probability that Bayesian updating works surely do. That's what's happening here; you're adding "there is an epsilon probability that Bayesian updating doesn't work" as evidence to a model that already uses and contains that information, and counting it twice (and then counting it n times).
You can also fashion a similar problem regarding priors.
Determine what method you should use to assign a prior in a certain situation.
Then determine what method you should use to assign a prior to "I picked the wrong method to assign a prior in that situation".
Then determine what method you should to assign a prior to "I picked the wrong method to assign a prior to "I picked the wrong method to assign a prior in that situation" ".
This doesn't seem like double-counting of anything to me; at no point can you assume you have picked the right method for any prior-assigning with probability 1.
This one is different, in that the evidence you're introducing is new. However, the magnitude of the effect of each new piece of evidence on your original probability falls off exponentially, such that the original probability converges.