Bayesian inference: are parameters fixed or random?

Yesterday I had an interesting discussion with a friend about how parameters are thought of in Bayesian inference. Coming from a predominantly frequentist statistical education, I had somewhere along the line picked up the notion that for Bayesians, like frequentists, the model parameters (their true values) are unknown but fixed quantities. The prior distribution then represents the prior belief about the location of this fixed value, before the data are seen. Thus the prior distribution represents our uncertainty about the location of the unknown, but fixed, parameter value.

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