I'm pretty happy with just assuming under H1 that the God we're talking about isn't a slacker, and so the probability that the explorer lives under H1 is 1.
and that's where we differ! for p(live|God)=1, you not only have to assume God exists, but that He told that person where to find the water. Even assuming God exists, I'm pretty sure anyone lost in the desert about to die will most likely call on God to help (atheist or not) like in the old saying "there're are no atheists before a calculus exam." God obviously doesn't help all of them, and there's no way to distinguish the ones He helps from random chance. I'm looking for empirical evidence. nonetheless...
That said, the problem you've brought up is a standard problem in Bayesian inference, namely, that we may have one or more parameters in our models whose values we do not know. The correct way (both theoretically and practically) to deal with this situation is to represent our state of knowledge about the parameter by using a probability distribution.
agreed
In the current problem, we "know" that the probability that an explorer finds water randomly on his own is 0.02.
I'm fine with that. it would obviously depend on which desert we're talking about (some are more harsh than others), but I'm sure we could find one where a 2% probability of randomly finding water is reasonable (keep in mind the explorers survival experience would offset some of the 'randomness' of his finding water. think Bear Grylls, etc. still I'm fine with your approach so far).
Even if we don't know how much help God would be, it seems reasonable to assume that with God's help the explorer doesn't do any worse than random chance. So under the God hypothesis (H1), it seems reasonable to say that the probability p of finding water is in the interval [.2, 1].
here we disagree. I would posit that there's no difference between having God's help and finding water randomly as there is no evidence that would distinguish God's help from randomly finding water.
you're saying that p ~ uniform[.02, 1] which implies a mean of 1.02/2 = 0.51, or pretty much a coin flip. Heads God helped you tails He didn't. I find this way too high for something that has never been shown to happen. think of it this way, if heads is God and tails random chance and you flipped this coin an infinite number of times and it always landed on it's edge would you think this coin had close to a 51% prob of landing on heads? I would think this coin was weighted to only land on it's edge (i.e., not 50/50). since there is no proof for the God hypothesis, this is the scenario that makes most sense (at least to me). but to counter my
opinion, we would need empirical evidence. otherwise, why do we run experiments at all? we could just conclude whether a drug works base on our own subjective judgment, and I'm sure ours will differ from, say, that of a drug company's.
I've argued with frequentists about this sort of thing for years. their major criticism of Bayes is it's subjectivity, so it's always been important to me that model assumptions have empirical evidence to back them up. otherwise, we just have opinion. so although it seems reasonable to you to assign a probability of finding water assuming God is in the interval [.2,1] (on average 51%), this seems
extremely unreasonable to me, and assuming that it's this high 'games' the result in favor of the God hypothesis, as the minimum of your interval is equal to random chance. no where to go but up. my frequentist friends would flip (no pun intended).
I'm perfectly willing to say this is an unanswerable question. but to make it answerable, we would need to have objective empirical evidence, like the way I calculated randomly finding water as the (#live/Total number lost). I can't use my (or your) subjective judgment. all the reviewer has to say is 'how did you come up with that?" and we're stuck.
*note that I am totally fine with the rest of your formulation, although I may use a dirichilet prior to keep things conjugate, you know, avoid nasty looking posterior distributions requiring MCMC methods or having to integrate. but still, I won't quibble about this. reasonable and nicely done.
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