Hi.
A while ago, I apparently had a wrong idea about the meaning of realism. I stood corrected:

I'm still struggling with those subtleties. Would following formulation be more appropriate:
"Realism means, that every observable ##x## is attributed a probability distribution ##p_P(x)## that might depend on a set of parameters ##P## such as the settings of the measurement device (e.g. polarizer angle). Local realism means, that only parameter set in the measurement's past light cone can affect this probability distribution."

Also, I don't quite see why "predetermined" is necessarily wrong. Doesn't the Bayesian interpretation of probability basically say that probability emerges as a lack of knowledge? What's the difference between a predetermined measurement outcome that's just unknown and an observable that exists as a probability distribution until a measurement projects it onto one specific outcome?

Or do we need "not predetermined" here to allow for the experimentator to have free will when setting the measurement parameters?

I think there is a difference in the predicted distribution of results. If one could toss a coin with absolutely identical initial conditions and stimulus time after time, one predicts the exact same result for every toss, or at least I predict that. Its an impossible experiment to carry out as far as I know, but maybe its possible to build a machine that tosses a coin, makes it flip 100 times and always land heads, even if the stimulus and starting conditions are not exactly identical toss to toss. One can imagine such a device even if we can't make one. That is a pre-determined measurement outcome that is just unknown (until the first toss has completed). If there is some inherent randomness in the coin dynamics that cannot be accounted for physically except to say 'there is randomness' then even under the hypothetical identical starting conditions and stiumulus (tossing, I mean) the one expects some spread in the results due to the randomness.

It's a matter of opinion as to whether realistic in the sense of Bell (and Einstein) implies determinism. One could imagine a local stochastic process: Brownian motion, for instance, where at every moment, particles randomly choose among several options for their next state. I would consider such a model realistic, even though things are not predetermined.

But there is a double connection between realism and determinism, as applied to EPR:

There is no way to reproduce the perfect correlations predicted by quantum mechanics for the EPR experiment using a locally realistic theory unless that theory is deterministic.

Given any locally realistic theory (deterministic or not) there is an equivalent locally realistic theory that is deterministic that makes the same predictions. So if you rule out all possible deterministic locally realistic theories, then you also rule out all possible nondeterministic locally realistic theories. So it's good enough to consider deterministic theories in your analysis.