Ontic randomness, which may be better called physical indeterminism, is given as the best explanation for epistemic randomness for which no conditional variable exists (in the best theories of physics, etc.) to remove the epistemic randomness.

So, for a given epistemic-random Y, "0 < P(Y) < 1" => Y is ontic-random iff there is no such X st. P(Y|X) = 1 or P(Y|-X) = 1 where dim(X) is abitarily large

The existence of X is not epistemic, and is decided by the best interpretation of the best available science.

Bell's theorem limits the conditions on `X` so that either (X does not exist) or (X is non-local).

If you take the former branch then ontic-randomness falls out "for free" from highly specific cases of epistemic; if you take the latter, then there is no case in all of physics where one implies the other.

Personally, I lean more towards saying there is no case of ontic randomness, only "ontic vagueness" or measurement-indeterminacy -- which gives rise to a necessary kind of epistemic randomness due to measurement.

So that P(Y|X) = 1 if X were known, but X isn't in principle knowable. This is a bit of a hybrid position which allows you to have the benefits of both: reality isn't random, but it necessarily must appear so because P(X|measure(X)) is necessarily not 1. (However this does require X to be non-local still).

This arises, imv, because I think there are computability constraints on the epistemic P(Y|X, measure(X)), ie., there has to be some f: X -> measure(X) which is computable -- but reality isn't computable. ie., functions of the form f : Nat -> Nat do not describe reality.

This is not an issue for most macroscopic systems because they have part-whole reductions that make "effectively computable" descriptions fine. But in systems whether these part-whole reductions dont work, including QM, the non-computability of reality creates a necessary epistemic randomness to any possible description of it.

The word ontic makes it sound more esoteric than it is. In quantum mechanics, you could have a model that predicts everything you can practically know and calls the rest "random," or you could have a model with far more complexity and a lot of unusual internal mechanisms, that predicts everything you can practically know and explains the rest as not knowing the initial conditions. If you simulate quantum measurement on a computer, you're doing the latter - computers have nothing to do with nondeterminism, and would be choosing the results with a PRNG. "A PRNG decides everything between each step of the universal process," is an example of the kind of unusual internal mechanism that deterministic QM must have.

In classical mechanics these are the same model. So what is presented as physical evidence for metaphysics is actually quantum mechanics splitting apart two ways of looking at randomness, which classically are equally complex and hard to tell apart (thinking of the future as a probability distribution vs. believing that the future is a definite point about which your knowledge is described by a probability distribution), but in quantum mechanics are not.

Physicists have thought long and hard about this. This is very far outside my area, but here is a ten year old review paper that discusses some of these issues [1].

[1] https://arxiv.org/pdf/1409.1570