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In practice, this difference is obvious, but how to put it in words?

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I would describe hyper parameters as "indirectly learned/trained". Think of the inference chain.

Data --> Parameter(s) --> Hyper parameter(s)

The hyper parameters are not directly connected to the data observed.

Another way is to consider a scenario where you know the parameter(s). Is there any point using the data? If the answer is no, then it's a hyper parameter.

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