A Tribute to Charles Stein

Statistical Science just ran a special issue (Feb. 2012) as a tribute to Charles Stein that focused on shrinkage estimation. Shrinkage and the Stein effect have been my entries to the Bayesian (wonderful) world, so I read through this series of papers edited by Ed George and Bill Strawderman with fond remembrance. The more because most of the authors are good friends! Jim Berger, Bill Jefferys, and Peter Müller consider shrinkage estimation for wavelet coefficients and applies it to Cepheid variable stars. The paper by Ann Brandwein and Bill Strawderman is a survey of shrinkage estimation and the Stein effect for spherically elliptical distributions, precisely my PhD thesis topic and main result! Larry Brown and Linda Shao give a geometric interpretation of the original Stein (1956) paper. Tony Cai discusses the concepts of minimaxity and shrinkage estimators in functional spaces. George Casella and Juinn Gene Hwang recall the impact of shrinkage estimation on confidence sets. Dominique Fourdrinier and Marty Wells give an expository development of loss estimation using shrinkage estimators. Ed George, Feng Liang and Xinyi Xu recall how shrinkage estimation was recently extended to prediction using Kullback-Leibler losses. Carl Morris and Martin Lysy detail the reversed shrinkage defect and Model-II minimaxity in the normal case. Gauri Datta and Malay Ghosh explain how shrinkage estimators are paramount in small area estimation, providing a synthesis between both the Bayesian and the frequentist points of view. At last, Michael Perlman and Sanjay Chaudhuri reflect on the reversed shrinkage effect, providing us with several pages of Star Trek dialogues on this issue, and more seriously voicing a valid Bayesian reservation!

Improving convergence properties of the Data Augmentation algorithm

Our paper Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling, written with Jim Hobert and Vivek Roy during their latest visit to Paris a few years ago (!), has now appeared in arXiv, Statistical Science and project Euclid. I already mentioned in a previous post why this is an important paper for me. (There is nothing new  here, compared with this earlier post, except that the paper was re-posted on arXiv!)

Improving convergence of Data Augmentation [published]

Our paper with Jim Hobert and Vivek Roy, Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling, has now appeared in Statistical Science and is available on Project Euclid. (For IMS members, at least.) Personally, this is an important paper, not only for providing an exact convergence evaluation for mixtures,  not only for sharing exciting research days with my friends Jim and Vivek, but also for finalising a line of research somehow started in 1993 when Richard Tweedie visited me in Paris and when I visited him in Fort Collins… Coincidentally, my discussion of Don Fraser’s provocative Is Bayes Posterior just Quick and Dirty Confidence? also appeared in this issue of Statistical Science.

Don Fraser’s rejoinder

“How can a discipline, central to science and to critical thinking, have two methodologies, two logics, two approaches that frequently give substantially different answers to the same problems. Any astute person from outside would say “Why don’t they put their house in order?”” Don Fraser

Following the discussions of his Statistical Science paper Is Bayes posterior just quick and dirty confidence?, by Kesar Singh and Minge Xie, Larry Wasserman (who coined the neologism Frasian for the occasion), Tong Zhang, and myself, Don Fraser has written his rejoinder to the discussion (although in Biometrika style it is for Statistical Science!). His conclusion that “no one argued that the use of the conditional probability lemma with an imaginary input had powers beyond confidence, supernatural powers” is difficult to escape, as I would not dream of promoting a super-Bayes jumping to the rescue of bystanders misled by evil frequentists!!! More seriously, this rejoinder makes me reflect on lectures from the past years, from those on the diverse notions of probability (Jeffreys, Keynes, von Mises, and Burdzy) to those on scientific discovery (mostly Seber‘s, and the promising Error and Inference by Mayo and Spanos I just received).

Bayes factors and martingales

A surprising paper came out in the last issue of Statistical Science, linking martingales and Bayes factors. In the historical part, the authors (Shafer, Shen, Vereshchagin and Vovk) recall that martingales were popularised by Martin-Löf, who is also influential in the theory of algorithmic randomness. A property of test martingales (i.e., martingales that are non negative with expectation one) is that

which makes their sequential maxima p-values of sorts. I had never thought about likelihood ratios this way, but it is true that a (reciprocal) likelihood ratio

is a martingale when the observations are distributed from p.  The authors define a Bayes factor (for P) as satisfying (Section 3.2)

which I find hard to relate to my understanding of Bayes factors because there is no prior nor parameter involved. I first thought there was a restriction to simple null hypotheses. However, there is a composite versus composite example (Section 8.5, Binomial probability being less than or large than 1/2). So P would then be the marginal likelihood. In this case the test martingale is

Simulating the martingale is straightforward, however I do not recover the picture they obtain (Fig. 6):

x=sample(0:1,10^4,rep=TRUE,prob=c(1-theta,theta))
s=cumsum(x)
ma=pbinom(s,1:10^4,.5,log.p=TRUE)-pbinom(s-1,1:10^4,.5,log.p=TRUE,lower.tail=FALSE)
plot(ma,type="l")
lines(cummin(ma),lty=2) #OR lines(cummin(ma),lty=2)
lines(log(0.1)+0.9*cummin(ma),lty=2,col="steelblue") #OR cummax

When theta is not 1/2, the sequence goes down almost linearly to -infinity.

but when theta is 1/2, I more often get a picture where max and min are obtained in the first steps:

Obviously, I have not read the paper with the attention it deserved, so there may be features I missed that could be relevant for the Bayesian analysis of the behaviour of Bayes factors. However, at this stage, I fail to see the point of the “Puzzle for Bayesians” (Section 8.6) since the conclusion that “it is legitimate to collect data until a point has been disproven but not legitimate to interpret this data as proof of an alternative hypothesis within the model” is not at odds with a Bayesian interpretation of the test outcome: when the Bayes factor favours a model, it means this model is the most likely of the two given the data, not this model is true.