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I disagree that for Bayesians "how you obtain prior probability" does not matter. The only thing that a Bayesian would say is that the probability theory - being a purely mathematical theory - does not provide any advice in that matter; it ensures internal consistency, not external validity of inference.

However, prior probabilities matters a lot in any real application, and the Bayesian methods employed for prior probability assignments have a "frequentist" touch. Not in the sense of repeatedly performing some random experiment, but in the sense of considering combinatorics, symmetries, and populations. See for example the priniciple of maximum entropy, which is related to the idea of multiplicities, which in turn is simply based on counting and sets.

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