[Q & A]Prior or hyperprior, Which should I Choose?

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Question:

I have a linear regression (say) model p(t|x;w) = N(t ; m , D); Being Bayesian, I can put a Gaussian prior on parameter w. However, I've realized for some models we can put Gaussian-Wishart hyperprior on the Gaussian to be 'more' Bayesian. Is this correct ? Are both of these two models valid Bayesian models ? It seems to me that we can always put hyperprior, hyperhyperprior ......because it will still be a valid probabilistic model.I am wondering what's the difference between putting a prior and putting the hyperprior on the prior. Are they both Bayesian ?

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关键词：Should choose PRIOR Which hyper difference because between correct putting

Using hyperpriors only makes sense in a hierarchical Bayesian model. In that case you would be looking at multiple groups and estimate a group specific coefficient w_group based on group specific priors, with coefficients drawn from a global hyperprior.

If your prior and hyperprior reside on the same hierarchical level, which seems to be the case you are think about, then the effect on the results is the same as using a simple prior with a wider standard deviation. Since it still requires additional computational costs, such stacking should be avoided.

There is a lot of statistical literature on how to pick non-informative priors, often theoretically best solutions are improper distributions (their total integral is infinite) and there is a large risk of getting improper posterior solutions without well defined means or even medians. So for practical purposes picking wide normal distributions usually works best.