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In this page, it's written (emphasis mine)

If probabilities are thought to describe orderly opinions, Bayes theorem describes how the opinions should be updated in the light of new information

What is your understanding/definition of "orderly" opinion?

Maybe something like: a probability that is not arbitrarily chosen but well-founded and explainable?

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That term exactly refers to the difference between two main paradigms in probability and statistics: Frequentism vs Bayesianism. You can find many texts for explaining the difference, for example [1] and [2].

By the way, we can briefly say it means that we assume that there exists a fixed opinion out there (like a parameter of a distribution function) and we are going to estimate it based on the observation. Despite the Baysian probability, in frequentist probability, we assume that the opinion is not fixed and changing randomly based on observations.

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