**A** few days ago, I noticed the paper Estimation under matrix quadratic loss and matrix superharmonicity by Takeru Matsuda and my friend Bill Strawderman had appeared in Biometrika. *(Disclaimer: I was not involved in handling the submission!)* This is a “classical” shrinkage estimation problem in that covariance matrix estimators are compared under under a quadratic loss, using Charles Stein’s technique of unbiased estimation of the risk is derived. The authors show that the Efron–Morris estimator is minimax. They also introduce superharmonicity for matrix-variate functions towards showing that generalized Bayes estimator with respect to a matrix superharmonic priors are minimax., including a generalization of Stein’s prior. Superharmonicity that relates to (much) earlier results by Ed George (1986), Mary-Ellen Bock (1988), Dominique Fourdrinier, Bill Strawderman, and Marty Wells (1998). (All of whom I worked with in the 1980’s and 1990’s! in Rouen, Purdue, and Cornell). This paper also made me realise Dominique, Bill, and Marty had published a Springer book on Shrinkage estimators a few years ago and that I had missed it..!

## Archive for Rutgers University

## estimation of a normal mean matrix

Posted in Statistics with tags Biometrika, Charles Stein, Cornell University, James-Stein estimator, Purdue University, Rutgers University, shrinkage estimation, Springer-Verlag, superharmonicity, Université de Rouen on May 13, 2021 by xi'an## Fisher, Bayes, and predictive Bayesian inference [seminar]

Posted in Statistics with tags fiducial inference, Foundations of Probability, inverse probability, Jerzy Neyman, Karl Pearson, R.A. Fisher, Rutgers University, seminar, Thomas Bayes, webinar on April 4, 2021 by xi'an**A**n interesting Foundations of Probability seminar at Rutgers University this Monday, at 4:30ET, 8:30GMT, by Sandy Zabell (the password is Angelina’s birthdate):

R. A. Fisher is usually perceived to have been a staunch critic of the Bayesian approach to statistics, yet his last book (Statistical Methods and Scientific Inference, 1956) is much closer in spirit to the Bayesian approach than the frequentist theories of Neyman and Pearson. This mismatch between perception and reality is best understood as an evolution in Fisher’s views over the course of his life. In my talk I will discuss Fisher’s initial and harsh criticism of “inverse probability”, his subsequent advocacy of fiducial inference starting in 1930, and his admiration for Bayes expressed in his 1956 book. Several of the examples Fisher discusses there are best understood when viewed against the backdrop of earlier controversies and antagonisms.

## Korean trip

Posted in Mountains, Running, Statistics, Travel, University life with tags ABC, Bayesian asymptotics, Bukhasan, campus, CREST, jatp, Korea, Korean Statistical Society, Rutgers University, Salzburg, Seoul, Seoul National University, tutorial, University of Seoul on November 24, 2019 by xi'an**A** fairly short but exciting trip to Seoul and to the Fall meeting of the Korean Statistical Society there. Plus giving a seminar at Seoul National University, where I stayed and enjoyed its beautiful campus surrounded by hills painted in the flamboyant reds and yellows of trees. Running to the top of Gwanaksan in the early morning, with some scrambling moments, was a fantastic beginning for the day! Although it was quite unintentional Sacha Tsybakov from CREST happened to be another invited speaker at the meeting (along with Regina Liu from Rutgers, whom I was also met in Salzburg two months ago) and we had a nice stroll together on the University of Seoul campus during a break in the sessions, gaining another view of the city from the top of the Bukhasan mountain. The talk I gave there on the asymptotics of ABC happened to be more attended than my tutorial lecture delivered at the beginning of JSM in Denver this summer. I am thus quite grateful to the organisers for their invitation and this opportunity to meet Korean statisticians and to get a glimpse of Korean culture and cuisine!

## Confidence distributions

Posted in Books, Statistics, Travel, University life with tags confidence distribution, confidence region, discussion, International Statistical Review, Kemar Singh, marginalisation paradoxes, Min-ge Xie, p-values, Rutgers University 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.

**O**ne 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 (p_{0},p_{1}), and a binomial likelihood, the posterior distribution on p_{1}-p_{0} has its main mass in the tails of both the prior and the likelihood (“the marginal posterior of d = p_{1}-p_{0} 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 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 parameteras long as the center of a posterior distribution is not on the line joining the two centers of the joint prior and likelihood function.

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. p_{1}+p_{0}. This indeed provides the marginal prior of p_{1}-p_{0}, the marginal posterior of p_{1}-p_{0}, but…not the marginal likelihood of p_{1}-p_{0}! 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.)

**M**ost 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 ABC, ABC model choice, Harvest Moon Brewery, Princeton University, Rutgers University, William Strawderman on April 7, 2012 by xi'an**A**fter 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.

**T**he 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!)*