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Eliezer_Yudkowsky comments on Demands for Particular Proof: Appendices - Less Wrong
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log_2(3^^^^^3) heads in a row?
Coin's fixed.
Ah, so you meant: No physically possible series of Bayesian updates can promote a hypothesis to prominence if its prior probability is that low. And Peter meant: It is decision-theoretically useless to include a subroutine for tracking probability increments of 1/3^^^^^3 in your algorithm.
But the non-Bayesian source of your Bayesian prior might output 1/3^^^^^3 as the prior probability of an event -- as surely for the coin flip example as for Robin Hanson's anthropic one.
To be precise, it's impossible to describe any sense event with a prior probability that low. You can describe hypotheses conditional on which a macro-event has a probability that low. For example, conditional on the hypothesis that a coin is fixed to have a 1/3^^^3 probability of coming up heads, the probability of seeing heads is 1/3^^^3. But barring the specific and single case of Hanson's hypothesized anthropic penalty being rational, I know of no way to describe, in words, any hypothesis which could justly be assigned so low a prior probability as 1/3^^^3. Including the hypothesis that purple is falling upstairs, that my socks are white and not white, or that 2 + 2 = 5 is a consistent theorem of Peano arithmetic.
How many dustspecks in the eye are you willing to bet on that?
The log_2(3^^^^^3) consecutive binary digits of pi starting from number 3^^^^^3 are 0?
Then our minds are "fixed" too, just like the coin.