Post date: Feb 23, 2014 9:44:32 PM
The question from Thursday of how c-boxes can account for prior knowledge might be very easily answered. As Mohamed suggested, it seems to work if we just follow traditional Bayesian methods.
I'd never until tonight gotten around to checking the math to see whether getting data in two batches and following the c-box approach with the first batch, but using robust Bayesian conventions to handle the second batch would give the same result as when we get both batches at the same time and pool them before applying the c-box approach.
It seems to me that this would be the minimum behavior for any reasonable scheme for accounting for prior information. I am happy to report that at least the c-boxes for the Bernoulli, binomial, Poisson, and exponential parameters obey this "pool rule".
I still think we'd need to check that the confidence interpretation would be sustained in the result if we do updating in the traditional way. (Not that I quite know how one might go about doing this.)
Michael Balch never wants to talk about this issue of handling prior information, although I'm not really sure why. Of course, allowing Bayesians to swoop in with their possibly unjustified, precise priors that they pulled out of midair would violate the spirit and intent of c-boxes.
Scott