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"It is worth emphasizing that probabilities are assigned not measured." -- TJ Loredo "From Laplace to SN1987A" in "Max Entropy and Bayesian Methods" 1990.

Is there a logical way to distinguish what is assigned versus measured?

Or is this a scientific rather than a mathematical issue? If there is a logical distinction, it would be nice if it covered both frequentist and Bayesian concepts of probability.

(Maybe a starting point could be the distinction between the modes $X \to A$ versus $A \to X$ where, loosely, $A$ is better characterized than $X$)

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