Do you (or anyone else reading this) know of any attempts to give a precise non-frequentist interpretation of the exact numerical values of Bayesian probabilities? What I mean is someone trying to give a precise meaning to the claim that the "degree of plausibility" of a hypothesis (or prediction or whatever) is, say, 0.98, which wouldn't boil down to the frequentist observation that relative to some reference class, it would be right 98/100 of the time, as in the above quoted example.

Or to put it in a way that might perhaps be clearer, suppose we're dealing with the claim that the "degree of plausibility" of a hypothesis is 0.2. Not 0.19, or 0.21, or even 0.1999 or 0.2001, but exactly that specific value. Now, I have no intuition whatsoever for what it might mean that the "degree of plausibility" I assign to some proposition is equal to one of these numbers and not any of the other mentioned ones -- except if I can conceive of an experiment or observation (or at least a thought-experiment) that would yield that particular exact number via a frequentist ratio.

I'm not trying to open the whole Bayesian vs. frequentist can of worms at this moment; I'd just like to find out if I've missed any significant references that discuss this particular question.