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Warrigal comments on Open Thread: February 2010, part 2 - Less Wrong
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So it sounds like even though there are 2,000 separate prophecies, the probability of every prophecy coming true is much greater than 2^(-2000).
Maybe you just need to explain more but I don't see that.
Let P(2,000) be the probability that all 2,000 prophecies come true, and P(500) be the probability that the initial 500 all come true. Suppose P(2,000) = 2^(-2000) and P(500) = 2^(-500). We know that P(500|2,000) = 1, so P(2,000|500) = P(2,000)*P(500|2,000)/P(500) = 2^(-2000)*1/2^(-500) = 2^(-1500). A probability of 2^(-1500) is not pretty darned high, so either P(2,000) is much greater than we supposed, or P(500) is much lower than we supposed. The latter is counterintuitive; one wouldn't expect the Believable Bible's existence to be strong evidence against the first 500 prophecies.
And this doesn't depend on prophecies in particular. Any claims made by the religion will do. For example, the same sort of argument would show that according to our subjective probabilities, all the various claims of a religion should be tightly intertwined. Suppose (admittedly an extremely difficult supposition) we discovered it to be a fact that 75 million years ago, an alien named Xeno brought billions of his fellow aliens to earth and killed them with hydrogen bombs. Our subjective probability that Scientology is a true religion would immediately jump (relatively) high. So one's prior for the truth of Scientology can't be anywhere near as low as one would think if one simply assigned an exponentially low probability based on the complexity of the religion. Likewise, for very similar reasons, komponisto's claim elsewhere that Christianity is less likely to be true than that a statue would move its hand by quantum mechanical chance events, is simply ridiculous.
If nobody had ever proposed Scientology, though, learning Xenu existed wouldn't increase our probabilities for most other claims that happen to be Scientological. So it seems to me that our prior can be that low (to the extent that Scientological claims are naturally independent of each other), but our posterior conditioning on Scientology having been proposed can't.
Right, because that "Scientology is proposed" has itself an extremely low prior, namely in proportion to the complexity of the claim.
In proportion to the complexity of the claim given that humans exist, which is much lower (=> higher prior) than its complexity in a simple encoding, since Scientology is the sort of thing that a human would be likely to propose.
Use \* to get stars * instead of italics.
Oops! It seems I assumed everything would come out right instead of checking after I posted.
Edit: Yeah, I was being dumb.
Where A = "events occur" and B = "events are predicted", you're saying P(A and B) < P(A). Warrigal is saying it would be counterintuitive if P(A|B) < P(B).