I know Zellner's prior is using data in order to set prior information, but actually the whole model depends on the data. Is there any other reason?

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    $\begingroup$ Can you post a citation/reference for the claim that it is unacceptable? And also tell us as much of the context for this claim as you can? $\endgroup$ – Jake Westfall Mar 16 '17 at 3:32
  • $\begingroup$ I read it at some paper (can't remember the name or the author), about regression and the Bayesian approach. It didn't have much information. (i remember they used the same arguments that people use with Jeffreys'). I understand the controversy talking about non informative priors at an independent context, but using regression I cannot see the point $\endgroup$ – Red Noise Mar 16 '17 at 3:45

In our book, Bayesian Essentials with R, we state almost the same thing:

Zellner's prior somehow appears as a data-dependent prior through its dependence on $X$, but this is not really a problem since the whole model is conditional on $X$.

Zellner's prior writes down as $$ \beta|\sigma \sim \mathscr{N}_p\left(\tilde\beta,g\sigma^2(X^\text{T}X)^{-1}\right)\qquad \sigma\sim\pi(\sigma)=1/\sigma $$ and its major inconvenient is the dependence on the constant $g$, that impacts in a significant manner the resulting inference. This is illustrated in the book. A way out of this problem is to associate $g$ with a prior distribution, as detailed in Bayesian Essentials with R. A more expedite way out is settle for $g=n$.

A second issue with the Zellner prior is that this is an improper prior (because of $\sigma$) hence faces difficulties for model comparison as in variable selection. A somewhat dirty trick bypasses this difficulty: again quoting from the book:

we are compelled to denote by $\sigma^2$ and $\alpha$ the variance and intercept terms common to all models, respectively. Although this is more of a mathematical trick than a true modeling reason, the prior independence of $(\alpha; \sigma2)$ and the model index allows for the simultaneous use of Bayes factors and an improper prior on those nuisance parameters.

Therefore, it does not seem right to call Zellner's inacceptable. In my opinion, the only inacceptable priors are those that conflict with prior information. In a non-informative situation, any prior should be acceptable, at least a priori. (It may be that the data reveals a conflict between the prior and the parameter that could have been behind the data.)


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