Fisher’s lost information

After a post on X validated and a good discussion at work, I came to the conclusion [after many years of sweeping the puzzle under the carpet] that the (a?) Fisher information obtained for the Uniform distribution U(0,θ) as θ⁻¹ is meaningless. Indeed, there are many arguments:

1. The lack of derivability of the indicator function for x=θ is a non-issue since the derivative is defined almost everywhere.
2. In many textbooks, the Fisher information θ⁻² is derived from the Fréchet-Darmois-Cramèr-Rao inequality, which does not apply for the Uniform U(0,θ) distribution.
3. One connected argument for the expression of the Fisher information as the expectation of the squared score is that it is the variance of the score, since its expectation is zero. Except that it is not zero for the Uniform U(0,θ) distribution.
4. For the same reason, the opposite of the second derivative of the log-likelihood is not equal to the expectation of the squared score. It is actually -θ⁻²!
5. Looking at the Taylor expansion justification of the (observed) Fisher information, expanding the log-likelihood around the maximum likelihood estimator does not work since the maximum likelihood estimator does not cancel the score.
6. When computing the Fisher information for an n-sample rather than a 1-sample, the information is n²θ⁻², rather than nθ⁻².
7. Since the speed of convergence of the maximum likelihood estimator is of order n⁻², the central limit theorem does not apply and the limiting variance of the maximum likelihood estimator is not the Fisher information.

Jubilee at the University of Calcutta

The main reason for my trip to India was taking part in the celebrations of the 75th anniversary of the Department of Statistics at the University of Calcutta and of the 100th anniversary of the birth of P.K. Bose (whom I did not know before visiting Kolkata). The Department of Statistics was created in 1941 by Mahalanobis, the very first statistics department in Asia. (Mahalanobis was also instrumental in creating the ISI in 1932. And Sankhyā in 1933.)  Fisher visited Calcutta very often and was very supportive of Mahalanobis’ efforts: in the corridor, the above picture of Fisher is displayed, with him surrounded by faculties and graduates from the Department when he came in 1941.

Although I missed the first two days of the conference (!), I enjoyed very much the exchanges I had with graduate students there, about my talk on folded MCMC and other MCMC and Bayesian issues. (With The Bayesian Choice being an easy conversational bridge-way between us as it is their Bayesian textbook.) The setting reminded me of the ISBA conference in Varanasi four years ago, with the graduate students being strongly involved and providing heavy support in the organisation, as well as eager to discuss academic and non-academic issue. (Plus offering us one evening an amazing cultural show of songs and dances.)

Varanasi unholy air

One of my lasting memories of my trip to Varanasi four years ago is the poor quality of the air, with an almost constant fog over the city, fed by open air fires everywhere and aggressive vehicle exhaust, rather than by the few cremation pyres by the Ganges… I read today in The Guardian that the city actually ranks worst in India for its air quality. (I also read in that article that Gwalior had similar issues, although I remember a pleasant walk around the fort there, in the sun. Presumably on one of the few “good” air quality days.) Not that Paris is doing great in the past days, with a whole week of ineffective driving restrictions that left wood heating operating at full blast. I did not feel the air difference while biking, but I presume the impact of the micro-particles central to those pollution alerts is more long-term!

Current trends in Bayesian methodology with applications

When putting this volume together with Umesh Singh, Dipak Dey, and Appaia Loganathan, my friend Satyanshu Upadhyay from Varanasi, India, asked me for a foreword. The book is now out, with chapters written by a wide variety of Bayesians. And here is my foreword, for what it’s worth:

It is a great pleasure to see a new book published on current aspects of Bayesian Analysis and coming out of India. This wide scope volume reflects very accurately on the present role of Bayesian Analysis in scientific inference, be it by statisticians, computer scientists or data analysts. Indeed, we have witnessed in the past decade a massive adoption of Bayesian techniques by users in need of statistical analyses, partly because it became easier to implement such techniques, partly because both the inclusion of prior beliefs and the production of a posterior distribution that provides a single filter for all inferential questions is a natural and intuitive way to process the latter. As reflected so nicely by the subtitle of Sharon McGrayne’s The Theory that Would not Die, the Bayesian approach to inference “cracked the Enigma code, hunted down Russian submarines” and more generally contributed to solve many real life or cognitive problems that did not seem to fit within the traditional patterns of a statistical model.
Two hundred and fifty years after Bayes published his note, the field is more diverse than ever, as reflected by the range of topics covered by this new book, from the foundations (with objective Bayes developments) to the implementation by filters and simulation devices, to the new Bayesian methodology (regression and small areas, non-ignorable response and factor analysis), to a fantastic array of applications. This display reflects very very well on the vitality and appeal of Bayesian Analysis. Furthermore, I note with great pleasure that the new book is edited by distinguished Indian Bayesians, India having always been a provider of fine and dedicated Bayesians. I thus warmly congratulate the editors for putting this exciting volume together and I offer my best wishes to readers about to appreciate the appeal and diversity of Bayesian Analysis.

Bangalore workshop [ಬೆಂಗಳೂರು ಕಾರ್ಯಾಗಾರ]

As I am now back home after a rather lengthy and somewhat eventful trip [getting too early to Bangalore airport with 3 hours to spend in the nice and very quiet lounge, followed by another 5 hour wait in the very nice but no so quiet Bombay airport lounge, no visit to the cockpit this time!, and then the usual sick passenger blocking all trains from Paris-Charles de Gaulle airport for one hour, reaching home to find my 97-year old neighbour fallen in her kitchen and calling for help!], I cannot but reflect on the difference between my two trips to India, from the chaos of Varanasi to the orderly peace of the campus of the Indian Institute of Science of Bangalore and even to some extent of the whole city of Bangalore, all proportions guarded. Even managing to get a [new] pair of [new] prescription glasses (or rather spectacles) within three days!

I thus found this trip much less stressful and much profitable, from enjoying the local food to discussing with Indian statisticians. The purpose of the IFCAM workshop was to bring both groups together for potential joint projects funded by IFCAM (at the travel level). While I found most talks were driven by specific applications, esp. in genomics, there are directions where we could indeed collaborate, from capture-recapture to astrostatistics. So it may be that I’ll be back in India in a near future!