I'm saying that the conditional probability of A given B, P(A|B), is the probability of A under the assumption that B is true.
we're both saying exactly the same thing. you're assuming that B is true. B, in this case, is that God exists and can lead people to water, but the point I'm making is that you're assuming that's true in the data. I'm ok with that in the prior. but assuming that it's true in the data has been the issue, and the only issue, I've had with your approach from pages back. recall, you wrote:
(I can't include your equation, so i'll write it out)Our data consist of 2 explorers, both of whom got lost, but found water and lived. Thus n=2 and y=2. Plugging those values into the above expression and carrying out the integration, we find that the (marginal) likelihood under H1, L(Y|H1) = .34013.
so the marginal likelihood under the God assumption uses a binomial:
p^y(1-p)^(n-y)
and this is where you include the data you're using, i.e., the two explorers. and the way you include them is to say that y=2 and n=2. n being the sample size of 2 which is fine. but p is the probability of God directing them to water and surviving, and you're squaring it, which means you're treating both explorers' anecdotes as evidence that the God hypothesis is true. the (1-p)^(n-y) is the probability that they survived without God. and since
n=y=2
(1-p)^(2-2) = 1
so this term drops out. you're saying there is no evidence that anyone may have survived outside the God hypothesis as you are accepting the explorers anecdote that God saved them. continuing on...
so including the prior, your p(live|God)=34% (that's correct, I checked). you then divide this by the chance that these two found water by random chance, or p^2=.02^2=.0004 and conclude that God is 850 times more helpful! what if we had 3 explorers? well then p(live|God)=0.255 and the p(live|not God)=0.02^3=.000008. so now we can conclude that God is 32,875 times more helpful than chance alone!
given that your assuming all your data confirms the God hypothesis, the more data you have the more it's confirmed. this is the reason why I keep saying that you're assuming the thing you're trying to show.
and since they're all anecdotes, there are reasons other than God as to why these people survived. many of them. and I can't see any way you can confirm that they survived due to God. you're taking their word for it and you're not including contrary evidence (to say the least).
this is all a very tedious way of asking the same question I've asked before:
why are you assuming that the data you collected of the explorers anecdotes about surviving due to God are true? how do you justify this given it's an anecdote?
*done in haste so please excuse typos, etc.