## Fisher’s lost information

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , on February 11, 2019 by xi'an

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.

## Varanasi unholy air

Posted in pictures, Travel with tags , , , , , , , , , , on December 15, 2016 by xi'an

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!

## India snapshot (#21)

Posted in Statistics with tags , , , , , , , on January 12, 2013 by xi'an

We went before sunset on a boat ride along the Ganges, terminating in the main ghat,Dashashwamedh Ghat (दशाश्वमेध घाट), in a pleasant temperature announcing the return of warmer weather. As we were slowly drifting with the motor off, some passengers released a marigold bowl with a lit candle in the midst…

## India snapshot (#20)

Posted in pictures, Travel with tags , , , , on January 12, 2013 by xi'an

Last step of the Indian trip, in Varanasi, when walking the ghats, with a local. And the Ganges as a background.

## ISBA regional meeting in Varanasi (day 3)

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on January 11, 2013 by xi'an

On the last/my day of the ISBA meeting in Varanasi, I attended a few talks before being kindly driven to the airport (early, too early, but with the unpredictable traffic there, it was better to err on the cautionary side!). In the dynamical model session, Simon Wilson presented a way to approximate posteriors for HMMs based on Chib’s (or Bayes’!) formula, while Jonathan Stroud exposed another approach to state-space model approximation involving a move of the state parameter based on a normal approximation of its conditional given the observable, approximation which seemed acceptable for the cloud analysis model he was processing. Nicolas Chopin then gave a quick introduction to particle MCMC, all the way to SMC². (As a stern chairmain of the session, I know Nicolas felt he did not have enough time but he did a really good job of motivating those different methods, in particular in explaining why the auxiliary variable approach makes the unbiased estimator of the likelihood a valid MCMC method.) Peter Green’s plenary talk was about a emission tomography image analysis whose statistical processing turned into a complex (Bernstein-von Mises) convergence theorem (whose preliminary version I saw in Bristol during Natalia Bochkina’s talk).

Overall, as forewarned by and expected from the program, this ISBA meeting was of the highest scientific quality. (I only wish I had had hindi god abilities to duplicate and attend several parallel sessions at the same time!) Besides, much besides!, the wamr attention paid to everyone by the organisers was just simply un-be-lie-vable! The cultural program went in par with the scientific program. The numerous graduate students and faculty involved in the workshop organisation had a minute knowledge of our schedules and locations, and were constantly anticipating our needs and moves. Almost to a fault, i.e. to a point that was close to embarassing for our cultural habits. I am therefore immensely grateful [personally and as former ISBA president] to all those people that contributed to the success of this ISBA meeting and first and foremost to Professor Satyanshu Upadhyay who worked relentlessly towards this goal during many months! (As a conference organiser, I realise I was and am simply unable to provide this level of welcome to the participants, even for much smaller meetings… The contrast with my previous conference in Berlin could not be more extreme as, for a much higher registration fee, the return was very, very limited.) I will forever (at least until my next reincarnation!) keep the memory of this meeting as a very special one, quite besides giving me the opportunity of my first visit to India