Tim Hanson sends along this paper (coauthored with Adam Branscum and Wesley Johnson):
Eliciting information from experts for use in constructing prior distributions for logistic regression coefficients can be challenging. The task is especially difficult when the model contains many predictor variables, because the expert is asked to provide summary information about the probability of “success” for many subgroups of the population. Often, however, experts are confident only in their assessment of the population as a whole. This paper is about incorporating such overall, marginal or averaged, information easily into a logistic regression data analysis by using g-priors. We present a version of the g-prior such that the prior distribution on the probability of success can be set to closely match a beta dis- tribution, when averaged over the set of predictors in a logistic regression. A simple data augmentation formulation that can be implemented in standard statistical software packages shows how population-averaged prior information can be used in non-Bayesian contexts.
The g-prior is a class of models defined on transformed coefficients. Strictly speaking, the g-prior is improper because it depends on the data (in a regression model, the g-prior is scaled based on the observed matrix of predictors). But it’s an interesting case because, although it’s improper, it can be informative and have a finite integral.
Our “noninformative” version of the “informative” g-prior did not do as well as the Gelman et al. prior, but the informative version did quite well. A little real prior information goes a long way!
I agree with this message. Prior information can make a big difference. This is a point that we downplayed in BDA, but over the past few years I’ve moved toward respecting prior information.