Suppose that you’re thinking of adopting radical probabilism, or some more moderate form of Bayesianism, as your epistemology, but you’re hesitant because you think there are beliefs which are properly basic (including, perhaps, the belief in God), and you think that Bayesianism won’t make room for such beliefs. Reformed Epistemology, after all, is an externalist epistemology, while Bayesianism appears to be an internalist epistemology. Here’s an obvious way to put these commitments together coherently.
As a preliminary note, I want to call the reader’s attention to the fact, of which I only just became aware while trying to articulate this idea, that Pruss has suggested that (objective) Bayesianism should be regarded as a hybrid epistemology, where belief-updating is an internalist epistemic procedure, but the particular calibration of prior probability assignments is potentially (un/)warranted given an externalist story. I think this account sounds right. I have also been thinking, lately, about a way to fill out an epistemology such that it tells both an internalist and an externalist story (in other words, I’ve been thinking about how to articulate an epistemological commitment which bridges the internalist-externalist divide) and thus, if Pruss is right, it appears that Natural Law Bayesianism can commend itself to us (or to me) in light of this philosophical virtue.
I am increasingly convinced that Natural Law Bayesianism is correct, but what I am about to suggest will hold for any version of objective Bayesianism (and, as we will see, perhaps even for subjective Bayesianism as well). Perhaps we should regard a belief as properly basic if and only if its prior probability is (and ought to be) set at higher than 0.5 (on a scale from 0 to 1). We can then say that we have a defeater D for some properly basic belief H iff:
- P(H) >0.5
- P(H|D) ≤ 0.5
Recall that a properly basic belief is a belief which one is rational to maintain even in the absence of inferential evidence or rational argument, so long as genuine defeaters are not forthcoming. Take inferential evidence and rational argument to be species of evidence in the Bayesian sense; something E is evidence in the Bayesian sense for hypothesis/proposition H if and only if the probability of H given E is higher than the prior probability of H (prior, that is, relative to the condition ‘E’). Thus, we might have some beliefs the appropriate assignment of whose prior probabilities is in the range 0.5 < x ≤ 1, and these beliefs we will be able to rationally maintain even in the absence of inferential evidence or rational argument.
In fact, if one has coherentist leanings (as opposed to foundationalist leanings), one could even talk about a form of proper-basicality for a coherentist using precisely this language, but just adopt a subjective Bayesianist account of the priors in place of the objective Bayesianist account (mutatis mutandis).
 See: Richard Jeffrey, “Radical Probabilism (Prospectus for a User’s manual),” in Philosophical Issues 2 (1992): 193-204.
 See: Peter Forrest, “The Epistemology of Religion” in The Stanford Encyclopedia of Philosophy (Summer 2017 Edition), Edited by Edward N. Zalta, (June 15th, 2019): https://plato.stanford.edu/entries/religion-epistemology/#RefoEpis
 See: Alexander R. Pruss, “Internalism, Externalism and Bayesianism,” Alexander Pruss’ Blog, (June 15th, 2019): http://alexanderpruss.blogspot.com/2017/03/internalism-externalism-and-bayesianism.html
 One might think to suggest that this is more a vice than a virtue, for perhaps an account which is both internalist and externalist inherits all the problems peculiar to either side of that divide. I think this is mistaken; the only way to have an epistemology which is adequate for answering both internalist and externalist concerns is going to be an epistemology whose framework allows us to address puzzles peculiar to each front, and no argument for the indispensability of internalism or externalism will stand as an objection to such an epistemology.