## Confidence distributions

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , on June 11, 2012 by xi'an

I was asked by the International Statistical Review editor, Marc Hallin, for a discussion of the paper “Confidence distribution, the frequentist distribution estimator of a parameter — a review” by Min-ge Xie and Kesar Singh, both from Rutgers University. Although the paper is not available on-line, similar and recent reviews and articles can be found, in an 2007 IMS Monograph and a 2012 JASA paper both with Bill Strawderman, as well as a chapter in the recent Fetschrift for Bill Strawderman. The notion of confidence distribution is quite similar to the one of fiducial distribution, introduced by R.A. Fisher, and they both share in my opinion the same drawback, namely that they aim at a distribution over the parameter space without specifying (at least explicitly) a prior distribution. Furthermore, the way the confidence distribution is defined perpetuates the on-going confusion between confidence and credible intervals, in that the cdf on the parameter θ is derived via the inversion of a confidence upper bound (or, equivalently, of a p-value…) Even though this inversion properly defines a cdf on the parameter space, there is no particular validity in the derivation. Either the confidence distribution corresponds to a genuine posterior distribution, in which case I think the only possible interpretation is a Bayesian one. Or  the confidence distribution does not correspond to a genuine posterior distribution, because no prior can lead to this distribution, in which case there is a probabilistic impossibility in using this distribution.  Thus, as a result, my discussion (now posted on arXiv) is rather negative about the benefits of this notion of confidence distribution.

One entry in the review, albeit peripheral, attracted my attention. The authors mention a tech’ report where they exhibit a paradoxical behaviour of a Bayesian procedure: given a (skewed) prior on a pair (p0,p1), and a binomial likelihood, the posterior distribution on p1-p0 has its main mass in the tails of both the prior and the likelihood (“the marginal posterior of d = p1-p0 is more extreme than its prior and data evidence!”). The information provided in the paper is rather sparse on the genuine experiment and looking at two possible priors exhibited nothing of the kind… I went to the authors’ webpages and found a more precise explanation on Min-ge Xie’s page:

Although the contour plot of the posterior distribution sits between those of the prior distribution and the likelihood function, its projected peak is more extreme than the other two. Further examination suggests that this phenomenon is genuine in binomial clinical trials and it would not go away even if we adopt other (skewed) priors (for example, the independent beta priors used in Joseph et al. (1997)). In fact, as long as the center of a posterior distribution is not on the line joining the two centers of the joint prior and likelihood function (as it is often the case with skewed distributions), there exists a direction along which the marginal posterior fails to fall between the prior and likelihood function of the same parameter.

and a link to another paper. Reading through the paper (and in particular Section 4), it appears that the above “paradoxical” picture is the result of the projections of the joint distributions represented in this second picture. By projection, I presume the authors mean integrating out the second component, e.g. p1+p0. This indeed provides the marginal prior of p1-p0, the marginal posterior of p1-p0, but…not the marginal likelihood of p1-p0! This entity is not defined, once again because there is no reference measure on the parameter space which could justify integrating out some parameters in the likelihood. (Overall, I do not think the “paradox” is overwhelming: the joint posterior distribution does precisely the merging of prior and data information we would expect and it is not like the marginal posterior is located in zones with zero prior probability and zero (profile) likelihood. I am also always wary of arguments based on modes, since those are highly dependent on parameterisation.)

Most unfortunately, when searching for more information on the authors’ webpages, I came upon the sad news that Professor Singh had passed away three weeks ago, at the age of 56.  (Professor Xie wrote a touching eulogy of his friend and co-author.) I had only met briefly with Professor Singh during my visit to Rutgers two months ago, but he sounded like an academic who would have enjoyed the kind of debate drafted by my discussion. To the much more important loss to family, friends and faculty represented by Professor Singh demise, I thus add the loss of missing the intellectual challenge of crossing arguments with him. And I look forward discussing the issues with the first author of the paper, Professor Xie.

## …and from Rutgers

Posted in Books, pictures, Statistics, Travel, University life, Wines with tags , , , , , on April 7, 2012 by xi'an

After my seminar in Princeton, I went to Rutgers University, in New Brunwick, New Jersey, to meet my friend Bill Strawderman and my former PhD student Aude Grelaud, and spent a pleasant evening with them. The next day, after a quick tour of the historical campus (great Old Dutch buildings!), I had a series of meetings with faculty members and with students, where we discussed extensions and applications of ABC. The seminar was on a tighter schedule than in Princeton, but we also managed to discuss the selection of summary statistics, while I insisted more on the (precision) gain brought by a reduction in the dimension of those summary statistics.

The schedule was tight as I had to catch a plane to Paris in New York (JFK) the same evening but taking advantage of the fairly efficient train facilities around New York, we still managed to share a quick beer at the Harvest Moon Brewery Café (I wish I had had time to get a tee-shirt from there!)… (The rest of the trip was 100% uneventful as I managed to sleep the whole flight back home!)

## talks on the East Coast

Posted in Statistics, Travel, University life with tags , , , , , , , , on March 29, 2012 by xi'an

On Tuesday and Wednesday, next week, I will give seminars in Princeton University and Rutgers University, respectively. My talk at Princeton actually takes place in the Department of Economics, at the Oskar Morgenstern Memorial Seminar (Tuesday, April 3, 2:40 – 4:00pm 200 Fisher Hall). I must acknowledge that the prospect is a wee daunting. For addressing the manes of Morgenstern and for speaking in Nash‘s very own institution, if nothing else! And my talk at Rutgers is in the Department of Statistics and Bostatistics (Wedn, April 4, 3:20 – 4:20, Hill Center, Busch Campus), where I will meet with my friend of many years Bill Strawderman. And my former PhD student Aude Grelaud. Both talks will cover the same ground of ABC model choice and Bayesian consistency (surprise, surprise!). The format of the econometrics seminar at Princeton being a bit longer, I will give more background on ABC, mostly in connection with the econometric methods I mentioned in my ABC tutorial in Roma and at CREST. I presume I will skip this part in Rutgers as biologists and geneticists are more likely to attend than econometricians. In preparation, here is the current version of the talk, to be updated till Monday at the very least!