The unit information prior and its BIC approximation

Question

Just moments ago, I asked this question because I've been reading Wagenmakers 2007. I now have a better understanding of what a unit information prior is and can push my knowledge further with two (by which I mean 3) questions:

1. The unit information prior has been considered by some to be too conservative. Are there other criticisms unique to the unit information prior that are not covered by criticisms of "objective priors" as a family? And, does this prior compare favourably to similar "objective priors"?
2. It was raised in the previous question that there is some discomfort using the BIC approximation. I would appreciate an elaboration as to why this may be.

Answer 1

Looking at BIC's formula $$ BIC = -2 \log\left(\sup_\theta f(x\mid\theta)\right) + k \, \log n $$ you will see that there is no trace of the prior $\pi(\theta)$ on it. That's because the derivation of the BIC by Schwarz is based on an asymptotic result under which his prior (a formal prior which puts mass on subspaces of the parameter space) is "washed out". So, to argue that the BIC is truly Bayesian ammounts to saying that it is possible to do Bayesian inference without any priors (not even improper/default/reference/whatever priors). The question is if, in certain cases, we may see the BIC as an (useful) approximation, in some sense, of a full Bayesian criterion. Schwarz original paper is very short and worth reading.