Post date: Feb 21, 2014 4:37:27 PM
Gert de Cooman wrote on 22 January 2014:
Dear Scott,
I was looking at a (toy) problem of fairness of representation of political parties in political discussion programs on Flemish TV using imprecise probabilities, and was led to consider this as a binomial or multinomial parameter estimation problem with an imprecise Beta prior. This reminded me of Walley’s 2002 paper in JSPI, "Reconciling frequentist properties with the likelihood principle", where he discusses how to reconcile credible interval approaches with confidence interval approaches by allowing for imprecision. His discussion of the Imprecise Beta Model (with hyperparameter 1) includes the case for equitailed credible intervals, and he finds that they coincide with the frequentist Clopper-Pearson confidence intervals.
This reminded me of our discussions during walks on Chinese soil, and so I went and looked at your web site, and found formulas there that bear more than a passing resemblance to some of the formulas in the above-mentioned paper.
I wasn’t sure whether you were aware of this, so I thought I’d let you know, just in case.
Every good wish,
Scott replied on 22 January 2014:
Gert:
Fairness in political representation. Ha ha ha. Ha ha ha ha ha. Ha ha ha. That's funny. ...But imprecise probabilities are on Flemish TV?!
Yes, certainly we're aware of Peter's paper. We should have cited it, but I hope that we did a good job in the ISIPTA paper on c-boxes (accessible on the main c-box page via the tab labeled 'Compare with Imprecise Beta Model') of acknowledging that Peter (his two 1996 papers) and Art Dempster, and in a sense Clopper and Pearson, came up with this answer for the binomial rate first. See the section that includes our Figure 4.
As our paper notes, the c-box is identical in form to the IBM, but it is defined in a different way, which permits it to be easily generalized to other statistical inference problems such as the normal and lognormal problems, and nonparametric problems. Does Peter address those problems? The IBM depends on the class of priors being all beta distributions, and it wants you to pick a value for s. C-boxes don't assume anything about priors and they don't depend on any choice about s.
I'm actually not sure whether our way of handling the extension to predictive inference is compatible with Peter's way discussed in the 2002 paper or not. They may be different. His discussion certainly sounds different from our simplistic why-not approach. Peter's numerical example in section 7.4 in his web-accessible draft gives the interval [0.121, 0.356] for the problem of one success out of six trials. When I compute nextvalue.bernoulli(c(0,0,0,0,0,1)) using the R script on the c-box software page, I get an imprecise Bernoulli distribution with mean of [0.142,0.286]. I can't tell as I sit here whether the difference is just because he is using s=2, or if it's something deeper than that.
In any case, I think Peter's suggestion in this paper is surely essentially true, that frequentist and Bayesian approaches can be reconciled in the imprecise domain. That's the most exciting thing perhaps.
I have some questions about Peter's paper(s) though. For instance, he seems to say in one of them that the IBM with s=2 is required for confidence intervals with proper coverage, but I think it is clear that you only need s=1 to get them.
Best regards,
Scott