In my first article on the Leibnizian Cosmological Argument (LCA) I gave a bit of credit to the LCA proponent. If the principle of sufficient reason (PSR) were true with respect to the universe, if there were an external cause to physical reality, the cause would have to be nonphysical, and a nonphysical cause would imply a supernatural cause. Yet I also said this:
Typically we accept a theory explaining something (when there is no better explanation) as evidence for a theory, e.g. the big bang theory explaining the cosmic microwave background radiation. But there’s a weakness to exploit here that I think is all too often overlooked with theistic arguments. One way an argument can fail to be convincing is if it provides no rational support for its conclusion, but we should not make the mistake of thinking that’s the only way an argument can fail to be convincing. Another way is if the argument provided nonzero but nonetheless too little support for its conclusion. So we can accept that ceteris paribus a worldview that explains e.g. why there is something rather than nothing is better than one that doesn’t, but given the high plausibility of physical reality existing eternally without an external cause, the degree of evidential support this provides is quite small. This, I submit, is the real weakness of the LCA and the real reason atheists should embrace for thinking the Leibnizian Cosmological Argument to be not a particularly good argument. The alternative hypothesis that the universe exists eternally and uncaused is conceivable and too plausible to ignore, with only very weak grounds for thinking it does in fact have an external cause.
A blogger named Maverick Christian responded in an article called, Bayes’ Theorem and the LCA. Maverick Christian’s response is a bit convoluted (it’s five pages long) so I’ll do what I can to give a brief version of it. Basically, he uses a version of Bayes’ theorem to show that the existence of the universe gives nonnegligible evidence for the hypothesis that the universe has an explanation of its existence. Where H stands for the hypothesis and E stands for the evidence, the version of Bayes’ theorem Maverick Christian uses is the following, where Pr(X) is a shorthand for “the probability of X:”
Pr(HE) = 

Using the following symbolization key:
 E = The universe exists.
 H = The hypothesis that the universe has an explanation (a sufficient reason) for its existence,.
 ¬H = The universe does not have a sufficient reason (explanation) for its existence.
Maverick Christian imagines a fictional character he calls “Agnostic Al” for whom Pr(H) and Pr(¬H) are both 0.5. He imagines the values for Bayes’ theorem to be as follows:
 Pr(H) = 0.5
 Pr(EH) = 1
 Pr(¬H) = 0.5
 Pr(E¬H) = 0.5
Maverick Christian estimates the likelihood of E given ¬H (i.e. the probability that the universe would exist in the absence of a sufficient reason for its existence) is 0.5, so that Pr(E⫬H) is 0.5. But how do we calculate the probability of the universe existing in the absence of a sufficient reason? What are our grounds for thinking it’s something like Pr(E¬H) is 0.5 instead of 0.6 or 0.4 or some other number? Arguably, Pr(E¬H) is too inscrutable for the Bayesian approach to be a successful argument.
But should we be like agnostic Al in the first place? I will argue the answer is No. The best way to attack this Bayes’ theorem approach is to attack the prior probability of H, or what we can call HE (the hypothesis that the universe has an explanation of its existence) as opposed to HN (the hypothesis that the universe exists but has no explanation for its existence. Note that the following equation is also true:

= 

× 

Note that both Pr(EHE) and Pr(EHN) are the same (they both equal 1, since both HE and HN entail the universe’s existence). What the above equation means then is if HN and HE are of equal prior probability (i.e. if Pr(HE) equals Pr(HN)), then E (the evidence of the universe’s existence) doesn’t permit us to favor HE over HN. That is, while it make HE more likely than it was before (i.e. Pr(HEE) > Pr(HE)), it won’t be enough for Pr(HEE) to be higher than 0.5.
But why think Pr(HN) ≥ Pr(HE)? Because HN implies a magical force creating the universe, and magical forces creating stuff is the sort of thing that substantially deviates from the sorts of things we know exist (I used similar reasoning when arguing for the presumption of atheism), whereas physical reality existing eternally without any external cause is very plausible and doesn’t require positing anything so extravagant as a magical being or force creating something. Thus, Pr(HN) ≥ Pr(HE), and while Pr(HEE) > Pr(HE), the fact remains that Pr(HEE) ≤ 0.5, and thus Pr(HE) ≤ 0.5 even though Pr(HE) > Pr(H).