Posted: Dec 18, 2011 6:12 am
jlowder wrote:Hmmm... I may have oversimplified in an earlier post. In this example, I would say that the ratio of Pr(2 heads in a row | heliocentrism) to Pr(not 2 heads in a row | heliocentrism) is 1, which is just a fancy way of saying that heliocentrism is irrelevant.
Assuming that this is a typo (or not a typo, but a miswriting or whatever the technical word is) for
Pr(2 heads in a row | heliocentrism) to Pr(2 heads in a row | ¬heliocentrism) is 1
then I surely agree with that. My problem remains though, that according to this definition your class of "predictions" overlaps with your class of "irrelevancies" which to me seems like the definition of prediction doesn't capture what we would intend it to.
"H predicts E just in case H predicts that E is more likely than not, i.e., Pr(E|H&B) > 0.5."
It would seem more natural to define it in terms more like those you use at the end here, i.e. in terms of the ratio of the probability of the evidence given the hypothesis divided by the probability of the evidence given the falsity of the hypothesis, or informally that:
(1) E is evidence for H
is equivalent to
(2) H predicts E
is equivalent to
(3) Pr(E|H)/Pr(E) > 1
Although there may be the odd issue here with the non-finite sample space cases.
Edit: Removed a mis-pasted text fragment.
