# Population distributions/data generating functions in Bayesian Statistics

In many frequentist stats courses, random variables come from some distribution at the population level and as such we could say that $$y=X \beta + \epsilon$$ is the true function for something like regression.

Does Bayesian statistics still believe in that and use the Bayesian approach to (hopefully) more faithfully model the population or is everything probabilistic in Bayesian statistics?

If everything is probabilistic and there is no "fixed" population, then how do we hope to ever model given that the distributions for the parameters have parameters which themselves are also probabilistic? It seems somewhat endlessly recursive.