Post date: Jan 30, 2014 9:38:34 PM
Guys, the reviews for the ICVRAM paper came back. They were very minor (see below), so I'll respond this weekend with a revision. No need for you to do anything, unless you want to make an amendment or change of some kind yourself (in which case, let me know by Sunday night or so).
Here in France, there are apparently a lot of people who use p-boxes, and I just attended a lecture today that introduced and talked about c-boxes. I restrained an urge to squeal like a little girl.
Sébastien Destercke (pronounced "de-STACK") asked from the audience how you could account for prior knowledge. There was some discussion, and the tentative consensus was that maybe we could do what Bayesians do at that point. During the discussion, I dutifully pointed out that we'd need to check that the confidence interpretation would be sustained in the result if we did that. (Not that I quite know how one might go about doing this.)
I'd never gotten around to checking the math until tonight to see whether getting data in two batches and following Michael's 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 Michael's approach. As I've said many times to Michael, 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-box for the Bernoulli parameter obeys this "pool rule". I'll try to check normal and exponential cases too. If they obey the pool rule too, I'm prepared to declare victory over this critical element of the puzzle. If something weird happens in other cases, that can be someone else's paper.
Was there a reason you didn't wanna talk about this issue of handling prior information, Michael? Of course, allowing Bayesians to swoop in with their possibly ill-conceived precise priors that they pulled out of midair would violate the spirit and intent of c-boxes, but having a tenable solution to the updating problem--and even better, one that fundamentally agrees with current practice--is going to make the difference between ultimate broad acceptability of the approach and potentially bitter rivalry. Don't you think?
I'll put this email on the Say page of the c-boxes site.
Sc
Reviewer 1 – Report on Computing with Confidence: Imprecise Posteriors and Predictive Distributions. By S. Ferson et al., for ICVRAM 2014
Confidence structures (c-boxes) have recently been introduced by one of the authors. They transfer the concept of confidence distributions (introduced by Cox in 1958) to the setting of imprecise probability. Confidence structures are much more flexible than confidence distributions (e.g., one can construct a c-box for the binomial distribution) and can be propagated in computational models. The full paper definitely delivers what the abstract promised. It contains a well-written introduction to the concept and its history, a wealth of connections with competing notions, and a very illuminating selection of examples.
The exposition is concise and clear. In my view, the concept of c-boxes will become an important one both in imprecise probability and in classical probability, extending the range of applicability and promoting the notion of confidence distributions.
I recommend that the paper be accepted for publication at the ICVRAM conference. I have only a few minor remarks:
p. 2, line 18: The word “interval” is missing after “confidence”.
pp. 3-4: A bit more explanations of Table 1 would be welcome. At least, explain what is the meaning of “next.binomialnp”, “cbox.binomialnp.n” etc.
p. 4, line 4 after table: Shouldn’t it be “online”?
p. 6, second paragraph: I was a bit confused at first reading by the interjection about the ratio k/n, followed by the estimate p = 0 and the right graph in Figure 1, before I remembered that the example was about k = 0. May be the sentence about general k/n can be entangled from the example, in which k = 0.
Reviewer 2 – Report on Computing with Confidence: Imprecise Posteriors and Predictive Distributions. By S. Ferson et al., for ICVRAM 2014
By raising again attention to confidence distributions it makes a valuable contribution to the conference. The paper clearly is interesting, and it will initiate interesting discussions about foundations of statistics that are also of great importance for risk modelling.