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I am fairly new to Bayesian Modeling, however I am experimenting with such framework in order to produce several estimates. The part I am struggling the most with is the selection of prior distributions for the model parameters. The choice of such distributions in everything that I have been reading about Bayesian Modeling appears somewhat arbitrary to me.
Is there a formally defined procedure in order to choose prior distributions that is not directly influenced by the modeler? I do not understand on which basis a specific parameter should be distributed according to a distribution rather than another one. Are there any specific elements that needs to be taken into consideration in order t make the choice?

Thank you, Marco

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Yours is a common fear among those who were trained as frequentists, but are now entering the Bayesian world.

First off: you can have your model (likelihood) "dictate" the shape of your priors if you insist. Examples of such formal rules are the Jeffreys prior or, more generally, reference priors. With these types of priors you aim to minimize the influence of the prior on the posterior, pretending to have no prior knowledge whatsoever.

With noninformative priors, however, you deprive yourself of the main advantages of Bayesian modelling. For instance, carefully chosen informative priors expand the space of models you can estimate as compared to frequentist (or non-informative Bayesian) approaches, by providing identification for an otherwise unidentified likelihood. You probably encountered this fact already, albeit under different names, for example "regularization".

And last but not least, it stands to reason that you barely ever not know anything about the parameters of your model. Remember that a prior does not fix the parameter at specific values; it just declares certain ranges more plausible than others. And often you do know that some parameter is more likely to be positive than negative, or more likely to be between 0 and 1 than between 100 and 1000, for instance.

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