When we refer to a concrete probability space the events to which probabilities are assigned should have a specific well defined meaning. Now when we refer to a prior distribution of some parameter it should therefore be clear what is being said about reality if the parameter takes a certain value.
Now if the different values of the parameter aren't refering to different members of a population (for example urns with a different amount of red balls) then the parameter is only refering to different distributions. For this to have any meaning it seems to me that these distributions must be seen as "real" in the sense of the frequentist interpretation of a "physical probability". For the degree of belief interpretation of probability it would be nice not to have to refer to the concept of a physical probability, since it is not without it's own issues. My question is how/if it would be possible to assign a meaning to different values of a prior parameter, which does not stem from a population, without refering to physical probabilities.
Using probabilities of distributions is only irreducible if we’re dealing with repeatable experiments, and in such situations an evidential interpretation of probabilities has no problem with frequentist probabilities.
Consider a paradigmatic non-frequentist scenario, say, a one-off, non-repeatable life choice where I want to decide whether I’ll be better off becoming a mathematician or a coin manufacturer. As part of the decision-making process, I might try to estimate the probability that I’ll make more money as a mathematician, and as part of that, I might model the income for both options with normal distributions and assign some probabilities to their parameters. None of this means that there are any “physical” probabilities involved – this is merely a device to assign some reasonable set of probabilities, and we can entirely do away with the normal distribution, its parameters and their distribution by integrating over the parameters to obtain individual probabilities for individual income values instead.
By contrast, talk of probabilities of distributions is irreducible when we’re dealing with a repeatable experiment. But an evidential interpretation of probabilities doesn’t deny that there are (to a very good approximation) such things as repeatable experiments, and that in a repeatable experiment the probability to be assigned to an outcome is equal to the relative frequency of the outcome when the experiment is repeated. What’s characteristic of an evidential interpretation of probabilities is not that probablities are never frequencies, but that they don’t have to be and that that’s not how they’re defined and conceptualized. For a repeatable experiment, the distribution of its outcomes is a specific, well-defined feature of the experiment to which we can assign probabilities just like to any other feature.
(Here sampling from a population is a special case of a repeatable experiment.)