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Perplexed 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'm pretty sure there is an error in your reasoning. And I'm pretty sure the source of the error is an unwarranted assumption of independence between propositions which are actually entangled - in fact, logically equivalent.
But I can't be sure there is an error unless you make your argument more formal (i.e. symbol intensive).
I think it would take the form of X being an outcome, p(X) being the probability of the outcome as determined by Bayesian updating, "p(X) is correct" being the outcome Y, p(Y) being the probability of the outcome as determined by Bayesian updating, "p(Y) is correct" being the outcome Z, and so forth.
If you have any particular style or method of formalising you'd like me to use, mention it, and I'll see if I can rephrase it in that way.
I don't understand the phrase "p(X) is correct".
Also I need a sketch of the argument that went from the probability of one proposition being 1-epsilon to the probability of a different proposition being smaller than 1-epsilon.
p(X) is a measure of my uncertainty about outcome X - "p(X) is correct" is the outcome where I determined my uncertainty about X correctly. There are also outcomes where I incorrectly determined my uncertainty about X. I therefore need to have a measure of my uncertainty about outcome "I determined my uncertainty correctly".
The argument went from the initial probability of one proposition being 1-epsilon to the updated probability of the same proposition being less than 1-epsilon, because there was higher-order uncertainty which multiplies through.
A toy example: We are 90% certain that this object is a blegg. Then, we receive evidence that our method for determining 90% certainty gives the wrong answer one case in ten. We are 90% certain that we are 90% certain, or in other words - we are 81% certain that the object in question is a blegg.
Now that we're 81% certain, we receive evidence that our method is flawed one case in ten - we are now 90% certain that we are 81% certain. Or, we're 72.9% certain. Etc. Obviously epsilon degrades much slower, but we don't have any reason to stop applying it to itself.