So abductive inference (IBE) is distinguished from enumerative induction. The former basically involves inverting deductive conditionals to generate hypotheses. So when asked why the grass is wet we immediately start coming up with deductive stories for how the grass might have gotten wet "It rained therefore the grass got wet", "The sprinkler was on therefor the grass got wet" (note that the idea that explanans deductively entail explanandum is almost certainly false but this was the dominant position in philosophy of science when abductive inference was first discussed and a lot of times people don't update their versions of IBE to take into account the, much better, causal model of explanation). Then the available hypotheses are compared and the best one chosen according to some set of ideal criteria. In contrast, enumerative induction involves looking at lots of particular pieces of evidence and generalizing "I've seen the grass be wet 100 times and 90 of those times it had rained, therefore it rained (with a 10% chance I'm wrong)".

Now, the "ideal criteria" for an IBE differ from philosopher to philosopher but everyone worth their salt will include degree of consistency with past observation so IBE essentially subsumes enumerative induction. Usually the additional criteria are vaguer things like parsimony and generality. Now, since enumerative induction is about the frequency of observations it is more conducive to mathematical analysis and the Bayesian method. But the things IBE adds to the picture of inference aren't things the Bayesian method has to ignore, you just have to incorporate the complexity of a hypothesis into its prior. But since objective Occam priors are the most confusing, controversial and least rigorous aspect of Bayesian epistemology there is room to claim that somehow our incorporation of economy into our inferences requires a vaguer, more subjective approach.

But that's stupid. The fact that we're bad Bayesian reasoners isn't a rebuttal to the foundational arguments of Bayesian epistemology (though, those aren't fantastic either). Your inferential method still has to correspond to Bayes' rule or your beliefs are vulnerable to being dutch booked and your behavior can be demonstrated to be suboptimal according to your own notion of utility (assuming certain plausible axioms about agency).

That the authors say things like "If one wants to evaluate the probability that this world exists and there are infinitely many possibilities, n, then no matter how small a probability one assigns to each one, the sum will be infinite" suggests they are either unfamiliar with or reject an approach identifying the ideal prior with the computational complexity of the hypothesis (note that a strict enumerative inductive approach can be redeemed if facts about computational complexity are nothing more than meta-inductive facts about our universe(s)).

Whether one accepts that approach or not it plainly can't be worse than relying on evolved, instinctual or aesthetic preference when picking hypotheses-- which I assume is where they're going with this. One needn't apply an explicitly Bayesian method at all to reject God on IBE grounds. Theism plainly fails any economy criteria one would want- and I could go on about why but this comment needs to end.

Is this just funny games?

Eh, yes.

Is there something to the proposed limitations of Bayes' theorem they present?

Pretty much, no. The questions they are trying to answer, of "how does a system deal with a lack of exact numbers" and such, are approached with tools like replacing discrete probabilities with probability distributions and using maximum ignorance or Kolgomorov complexity-approximating priors.

More broadly, be skeptical of any weakenings. Going from quantitative posterior probabilities to qualitative "probably or approximately yes" judgments is suspicious (especially because posterior probabilities are about "probably or appoximately" yes or no answers). Similarly, moving from cardinal probabilities (posteriors that are a decimal between 0 and 1 representing how likely a thing is to happen) to ordinal probabilities as Richard Carrier (see quote below) does is likely bad.

you can at least say that something wouldn't be less likely than X or more likely than Y

I googled "Inference to the Best Explanation", and this paper of Gilbert Harman appears to be where the phrase was first coined, although the general idea goes back further. More recently, there's a whole book on the subject, and lukeprog's web site has an introductory article by the author of that book.

There isn't a single piece of mathematics in either of the two papers, which leads me to expect little of them. The book (of which I've only seen a few pages on Amazon) does contain a chapter on Bayesian reasoning, arguing that it and IBE are "broadly compatible". This appears to come down to the usual small-world/large-world issue: Bayes is sound mathematics (say the small-worlders) when you already have a hypothesis that gives you an explicit prior, but it must yield to something else when it comes to finding and judging hypotheses.

That something else always seems to come down to magic. It may be called IBE, or model validation, or human judgement, but however many words are expended, no method of doing it is found. It's the élan vital of statistics.

ETA: I found the book in my university library, but only the first edition of 1991, which is two chapters shorter and doesn't include the Bayes chapter (or any other mathematics). In the introduction (which is readable on Amazon) he remarks that IBE has been "more a slogan than an articulated philosophical theory", and that by describing inference in terms of explanation, it explains the obscure by the obscure. From a brief scan I was not sufficiently convinced that he fixes these problems to check the book out.

Bayesian probability works well when the specific values are known (they use the example of predicting the color of a ball to be drawn out of a container). In theology, the values are not known.

If you don't allow exact probabilities for things, there are decisions it becomes impossible to make, such as whether or not to take a given bet. If you try to come up with a different method of choosing, you either end up with paradoxes, or you end up behaving exactly as if you were using Bayesian statistics.

If one wants to evaluate the probability that this world exists and there are infinitely many possibilities, n, then no matter how small a probability one assigns to each one, the sum will be infinite.

This is only true if we assign them all the same probability. We tend to weight them by their complexity. Also, if you didn't, the more complex possibilities would tend to contain the simpler ones, which may approach a limit as the number of possibilities considered increases.

... establish that Christianity is the true formulation of what [God] wants ... by examining revealed theistic truths (aka the Bible)

This traditional line of apologetics is all very weird to me, a committed Christian of nearly 30 years. Seriously studying the scriptures is just what convinced me the Bible is not inspired by an intelligent or loving God.

I don't have the knowledge (yet) to answer your questions, but I'm very interested in what others will say.

Obviously the key lies in their definition of "ideal explanation."

That seems non-obvious to me. It's highly problematic, sure -- but not "key". "Key" is "adequate range of data". That cannot be an objective measure. It occurs to me that Bayes' theorem has no such problem; it simply takes additional input and revises its conclusions as they come -- it makes no presumption of its conclusions necessarily being representative of absolute truth.

I also, personally, take objection to:

(2) It is not highly unlikely that a world-creator exists.

I find it is highly unlikely that "a world-creator" exists. For two reasons. 1) Our universe necessarily possesses an infinite history (Big Bang + Special Relativity says this.) 2) Any ruleset which allows for spontaneous manifestation of an agentless system is by definition less unlikely than the rulesets which allow for the spontaneous manifestation of an agent that can itself manifest rulesets. (The latter being a subset of the former, and possessed of greater 'complexity' -- an ugly term but there just isn't a better one I am familiar with; in this case I use it to mean "more pieces that could go wrong if not assembled 'precisely so'.)

I can't say, as a person who is still neutral on this whole "Bayesian theory" thing (i.e.; I feel no special attachment to the idea and can't say I entirely agree with the notion that our universe in no ways truly behaves probabilistically) -- I can't say that this topic as related is at all convincing.