Fortunately, we have Jim Berger interpreting himself this evening (see December 11)
Jim Berger writes:
A few comments:
1. Objective Bayesian priors are often improper (i.e., have infinite total mass), but this is not a problem when they are developed correctly. But not every improper prior is satisfactory. For instance, the constant prior is known to be unsatisfactory in many situations. The ‘solution’ pseudo-Bayesians often use is to choose a constant prior over a large but bounded set (a ‘weakly informative’ prior), saying it is now proper and so all is well. This is not true; if the constant prior on the whole parameter space is bad, so will be the constant prior over the bounded set. The problem is, in part, that some people confuse proper priors with subjective priors and, having learned that true subjective priors are fine, incorrectly presume that weakly informative proper priors are fine.
2. My more provocative comment was based on the fact that objective Bayesians worry a lot about the prior, and work hard to get a prior that is good in situations where one does not have much prior information or is obligated to use impartial priors (e.g., by regulation). True subjective Bayesians also worry a lot about the prior, attempting to model their prior information carefully, doing sensitivity studies, etc. But, in part because Bayesian analysis has become so popular and is being used by many without training in either objective Bayesian or true subjective Bayesian methods, there are many quite adhoc choices of priors being made that have no inherent justification and would, I claim, be much less ‘Bayesian’ than the objective Bayesian priors. All that is less exciting than my provocative comment, but it is what I had in mind when writing the comment.
UPDATE: Dec. 29, 2011: Andrew Gelman today fired back with a lengthy post on his own blog. I hope others will join in the discussion in the commentary here.