Downvoted for misapplication of math.
Good and evil are poorly defined subjective concepts (Were the Crusades good or evil? Was 9/11? The destruction of the Soviet Union? Depends on who you ask, and when.)
Are you working with an idealized toy model where everyone agrees on what's good and what's evil, rather than with the messy world we actually live in? If yes, then state it upfront. You also have many other implicit assumptions, false dichotomies (like god vs naturalism) and statements bereft of meaning (like god's explanatory power).
Until you define your terminology, all your calculations are of the type garbage in -- garbage out.
There is an argument in the vicinity of yours that I think is interesting. One popular theodicy is skeptical theism, the claim that we have basically no access to god's reasons for action, so we should not make claims like "God would probably have created a universe with less evil."
A consequence of skeptical theism seems to be that the distribution over possible world states, given the hypothesis that god created the world, should be close to maximum entropy. This means no observation of the world state could count as much evidence for the hypothesis.
This seems like the point you're flirting with here, and I think it's a good one, but unfortunately it is obscured by your bad math.
"unfortunately it is obscured by your bad math."
That's what I'm hoping someone will explain
Are you hoping someone will explain why your point is obscured, or do you want an explanation of how your math is bad? I assume the latter, so I'll try to point out a couple of places where your math goes wrong (really just re-iterating stuff that has already been said, but hopefully in a clearer way). Apologies if I'm misinterpreting your question. Here are some of the more obvious errors in your analysis (not all mathematical):
H and ~H are not merely mutually exclusive, they are also exhaustive. Every possible state of the world must be compatible with either H or ~H. Intuitively, if a hypothesis H picks out a certain set of possible worlds (such as the ones with an all powerful creator god), then ~H must be true in all other possible worlds. You seem to choose H and ~H (and later in your article E and ~E) to be two exclusive hypotheses without bothering about them exhausting all possibilities. "Universe created by an all-powerful god" and "Naturalism" do not exhaust the space of hypotheses. There are possible worlds that are neither created by an omnipotent deity nor naturalistic.
Your rationale for assigning a high value to Pr(Our universe | Naturalism) is weak. You say that naturalism postulates a fundamentally uncaring universe. OK, but why does this entail that we should expect a naturalistic universe to contain the precise mix of good and evil we see? I would think that the hypothesis of an uncaring universe should lead us to think that no particular ratio of good and evil is more favored than any other. So, conditional on the naturalism hypothesis, we should have a uniform distribution over various ratios of good and evil. But this is precisely the sort of distribution you attribute to the theist position -- a distribution that doesn't restrict anticipation -- so I don't see the advantage for naturalism. Of course, all of this assumes that we can make precise sense of the good:evil ratio in a world in the first place (see shminux's comment).
I doubt that any moderately sophisticated defender of the fine-tuning argument would concede that the god hypothesis could equally account for any set of physical constants. The whole point of the argument is that the fine tuning of the constants makes the god hypothesis more likely. If they have even a modicum of knowledge of Bayesianism (and I'm sure some of them do), proponents of this argument would recognize that it follows that there must be certain possible ways the universe could be (not conducive to life, perhaps) that would make the god hypothesis less likely. I actually tend to agree that the fine-tuning of the constants is some evidence for the existence of god, but given the miniscule prior probability of the hypothesis, it's not nearly enough evidence to make the hypothesis worthy of serious consideration.
How does the hypothesis of naturalism restrict anticipation when it comes to the fundamental physical constants? Is there any value of those constants that is ruled out by naturalism? Is there any value of those constants that is more likely than any other value given naturalism, assuming equal priors?
At the end, you say that any hypothesis that fails to restrict anticipation over an infinite sample space must be infinitely less likely than a hypothesis that restricts anticipation over the same sample space. This is wrong. A Gaussian distribution over some continuous sample space restricts anticipation, and a uniform distribution doesn't. It is, however, not the case for any piece of evidence from this space that the likelihood of the Gaussian hypothesis is infinitely greater than that of the uniform hypothesis. Perhaps you intended to talk about an unbounded sample space rather than just an infinite one, but there is no such thing as a probability distribution over an unbounded sample space that does not restrict anticipation. You can't have a uniform probability distribution over all the real numbers, for instance, because there is no way that such a distribution could be normalized.
the probability of an all powerful god creating the universe (H) verses the probability of naturalism (~H)
the probability of an all powerful god creating the universe (H) verses the probability of not an all powerful god creating the universe, including naturalism, or that there is an all powerful god except that a devil gets a veto on one out of every 1,000 decisions, or that an all powerful god created the universe and hates people (because by 'H' you intended only a good one, yes?), or...or... (~H).
The disjunctons in ~H are why arbitrary articulable hypotheses are basically always wrong.
among an infinite number of possible evidences, effectively favoring any alternative hypothesis
Naturalism also has things approaching infinities. Since humans don't handle infinities very well, anything ruled out only because of reasoning with infinities isn't so strongly ruled out, unless hard math is involved.
Another instance of the kind of thing lessdazed is talking about:
So it seems that E and ~E can be broken up into these three scenarios. E1 being our current world, E2 being a world of all good an no evil, and E3 being a world of all evil and no good. Then we have P(E1 | H) + P(E2 | H) + P(E3 | H) = 1.00.
You don't need to posit gods to be infinitely powerful or knowledgeable to realize they don't exist.
~E would be any other ratio of good::evil that we can imagine outside of the current state of affairs
So it seems that E and ~E can be broken up into these three scenarios. E1 being our current world, E2 being a world of all good an no evil, and E3 being a world of all evil and no good.
So, does it mean that you cannot imagine any good:evil ratio save 1:0, 0:1 and the current state of affairs? Why it is so? Why couldn't be there e.g. 1547 times less evil and 243 times more good than today?
It doesn't mean anything to ask whether a god is all-knowing. There is no hypothetical scenario in which the god can use an infinite amount of knowledge, therefore it does not mean anything to say he has it. You can of course posit gods with arbitrarily large amounts of knowledge and power, just not infinite knowledge or power.
I'm posting this here because I want to see if my reasoning is incorrect.
Generally, when people talk about the problem of evil, the underlying problem is actually one of indifference. Given that God exists, he doesn't seem to care some (most) of the time bad things happen, and seems to sometimes reward bad people with good fortune. This makes sense, of course, if there indeed was no god, but we have thousands of years of theodicy that argues that an all powerful, all knowing god exists despite the problem of indifference.
So, I attempted to formulate the problem of indifference in terms of probability - the probability of an all powerful god creating the universe (H) verses the probability of naturalism (~H) - to see which one was more likely. E would be the state of the current universe which seems to have both "good" and "evil" in it. I had no idea how to determine the probability of P(E | H), but if the 2,500+ years of theodicy explaining the problem of indifference was in fact correct, then to be fair to theism I might grant that P(E | H) = .99. However, this seems to not be correct; I did know that P(E | H) + P(~E | H) = 1.00 so this would mean that the all powerful god of traditional theism wouldn't make sense if our universe were indeed ~E instead of E given that P(E | H) = .99.
~E would be any other ratio of good::evil that we can imagine outside of the current state of affairs, or at least that's my reasoning. This means that if the universe were all good and zero evil, or all evil and zero good, granting that P(~E | H) = .01 doesn't seem like something traditional theism would accept. If we woke up tomorrow, and there was absolutely no evil in the world, and that was how the world always was, would traditional theism have no theodicies that explained why this world was evidence for their god(s)? That doesn't seem likely. Similarly, but less so, for a world that was overwhelmingly evil with very little good.
So it seems that E and ~E can be broken up into these three scenarios. E1 being our current world, E2 being a world of all good an no evil, and E3 being a world of all evil and no good. Then we have P(E1 | H) + P(E2 | H) + P(E3 | H) = 1.00. This would also apply to naturalism, but it seems as though a world of all good/evil and no evil/good wouldn't make sense under naturalism which predicts a fundamentally uncaring universe; P(E | ~H) = .99 and P(~E | ~H) would make sense to be the remainder. If this is the case, then the likelihood ratio favors naturalism. P(E1 | H) / P(E1 | ~H) is less than 1 since traditional theism doesn't seem to be restricting anticipation on what type of world(s) it can explain while naturalism does.
But then I thought that this applies to more situations beyond just the problem of evil/indifference. If an all powerful all knowing god can explain any and every sort of evidence we can imagine, and if there are 100 different types of evidence in a given class of evidence, then this god's explanatory power gets stretched across all 100 types of evidence, with any other hypothesis that restricts the type of evidence it can explain being favored over the god hypothesis. For something like the fine tuning argument for god, since there are infinite combinations of physical constants that an all powerful, all knowing god could create, this stretches the god hypothesis out among an infinite number of possible evidences, effectively favoring any alternative hypothesis by infinite decibles via the likelihood ratio. Meaning that something like the fine tuning argument is ironically an argument against H.
And if this is the case, then an infinitely powerful, infinitely knowing god is infinitely less likely than any other hypothesis that restricts itself. Thoughts? Is my reasoning off somewhere? Is the the ultimate penalty for not restricting anticipation?