Post date: Jan 30, 2014 9:10:18 PM
Vladik asked Scott:
Do you have more info re c-boxes? I am very curious about your email a few months ago that this is the latest hot thing and not p-boxes anymore
Scott responded to Vladik:
Did I send you Michael's paper? If not, it is attached.
I wouldn't say that p-boxes aren't hot anymore. Instead, I'd say that c-boxes are the solution to my quest over many years for the sampling theory to accompany probability bounds analysis (PBA).
C-boxes are a way to go from random sample data (either points or intervals) to structures that estimate unobservable parameters. These structures often look like p-boxes, but they have a slightly different interpretation. The most important thing, however, is that they can be used directly within the PBA technology, and they yield p-boxes as outputs. For instance, if you have c-box for the binomial rate parameter p, you can create a p-box for the implied binomial random variable by projecting the c-box for p through the formula for the binomial distribution. This is quite easy to do, whether or not convenient formulas exist for this composition.
C-boxes are analogous to the posteriors that Bayesians produce, but c-boxes make a performance statement that posteriors don't have. From a c-box, you automatically get (traditional) confidence intervals for the parameter, at any confidence level you wish. The thing that is amazing is that this performance propagates through calculations. So you can make confidence statements about any quantities you want. It means that the resulting p-boxes inherit the performance interpretation.
I'll try tomorrow or Tuesday to send you a slide show we're working up that tries to spell out some of the significant parts of the story. I will be trying to put all the material for a paper on this subject on another Google Site at https://sites.google.com/site/computingwithconfidence/, but I've not had any time to do this yet, and there's hardly anything there right now.