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Peter_de_Blanc comments on Demands for Particular Proof: Appendices - Less Wrong
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There is some extremely small probability that the theory of evolution is false, and the evidence of this has been withheld from us by some kind of plot. This hypothesis is supported by the absence of time cameras (since time cameras would resolve the matter), and so the absence of time cameras must increase the probability that evolution is false... even if only by 1/3^^^^^3, or something like that.
Silly Unknowns. 0, 1, and 1/3^^^^^3 are not probabilities.
Why not 1/3^^^^^3?
You can't imagine anything that improbable. Unless we adopt Robin's anthropic penalty, in which case "I am in a unique position to affect 3^^^^^3 other people" is that improbable.
Actually, the beauty of mathematics is that it enables us to imagine such things -- just as surely as it tells us that there ain't nothin' we're talkin' about that's anywhere near that.
I can't imagine quarks either.
1/3^^^^^3 is a probability. A stupid probability, but a probability nonetheless. And if you declare 1/3^^^^^3 to be not a probability because of it's unimaginable uselessness then by the same standard I expect you to consider 3^^^^^3 'Not a Number'. I know you routinely use arbitrarily large numbers like 3^^^3 for decision theoretic purposes (on Halloween costumes!) and that is a number that is more or less chosen because it is already unimaginable.
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.